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Chapter 3 Controlled Elliptic Variational Inequalities
Chapter 3
Controlled Elliptic Variational Inequalities
This chapter is concerned with optimal control problems governed by variational inequalities of elliptic type and semilinear elliptic equations. The main emphasis is put on first order necessary conditions of optimality obtained by an approximating regularizing process. Since the optimal control problems governed by nonlinear elliptic equations, and in particular by variational inequalities, are nonconvex and nonsmooth the standard methods to derive first order necessary conditions of optimality are usually inapplicable in this situation. The method we shall use here is to approximate the given problem by a family of smooth optimization problems containing an adapted penalty term and to pass to limit in the corresponding optimality conditions. We shall discuss in detail several controlled free boundary problems to which the general theory is applied, such as the obstacle problem and the Signorini problem. 3.1. Elliptic Variational Inequalities. Existence Theory 3.1.1. Abstract Elliptic Variational Inequalities
Let X be a reflexive Banach space with the dual X * and let A: X + X * be a monotone operator (linear or nonlinear). Let cp: X R be a lower semicontinuous convex function on X, cp f +m. If f is a given element of X, consider the following problem. Find y E X such that --f
(AY,Y
-2)
+4 Y )
- d z ) 5
125
(Y - z , f )
vz E X .
(1.1)
126
3. Controlled Elliptic Variational Inequalities
This is an abstract elliptic variational inequality associated with the operator A and convex function cp, and can be equivalently expressed as AY + dcpo(Y)
3
f
( 1*2)
9
where dp c X x X * is the subdifferential of cp. In the special case where cp = I , is the indicator function of a closed convex subset K of X , i.e., zK(x)
=
{0
+oo
ifxEK, otherwise,
problem (1.1) becomes: Find y
E
K such that
It is useful to notice that if the operator A is itself a subdifferential d+ of X + R, then the variational inequality a continuous convex function (1.1) is equivalent to the minimization problem (the Dirichlet principle)
+:
min{$(z) + cp(z) - ( z , f ) ; z E X }
(1.4)
or, in the case of problem (1.31, min{+(z) - ( z , f);z E K}.
(1.5)
As far as concerns existence in problem (l.l), we note first the following result.
Theorem 1.1. Let A : X + X * be a monotone, demicontinuous operator and let cp: X + R be a lower semicontinuous, proper, convex function. Assume that there exists y o E D(cp) such that
lim ((AY,Y - y o ) + ~ ( Y ) ) / l l Y l l=
llyll-+ 30
+w.
( 1.6)
Then problem (1.1) has at least one solution. Moreover, the set of solutions is bounded, convex, and closed in X . If the operatorA is strictly monotone, i.e., ( A M- Av, u - v) = 0 u = v, then the solution is unique.
+
Proof By Theorem 1.5 in Chapter 2, the operator A dp is maximal monotone in X X X * . Since by condition (1.6) it is also coercive, we conclude (see Corollary 1.2 in Chapter 2) that is surjective. Hence, Eq. (1.2) (equivalently, (1.1)) has at least one solution.
3.1. Elliptic Variational Inequalities. Existence Theory
127
Since the set of all solutions y to (1.1) is ( A + d c p ) - ’ ( f ) , we infer that this set is closed and convex (Proposition 1.1 in Chapter 2). By the coercivity condition (1.61, it is also bounded. Finally, if A (or more generally, if A + d q ) is strictly monotone, then ( A + dcp)-’f consists of a single element. In the special case cp
=
ZK, we have:
Corollary 1.1. Let A: X -+ X * be a monotone demicontinuous operator and let K be a closed convex subset of X . Assume either that there is y o E K such that
lim ( A y ,y - yo)/llyll =
II yll-
(1.7)
+w,
m
or that K is bounded. Then problem (1.3) has at least one solution. The set of all solutions is bounded, convex, and closed. If A is strictb monotone, then the solution to (1.3) is unique.
To be more specific we shall assume in the following that X Hilbert space, X * = V ’ , and VcHcV’
=
V is a (1.8)
algebraically and topologically, where H is a real Hilbert space identified with its own dual. The norms of V and H will be denoted by II * II and I . I, respectively. For u E V and u’ E V’ we denote by (u, u ’ ) the value of u‘ in u; if u, u’ E H , this is the scalar product in H of u and u ‘ . The norm in V‘ will be denoted by II II*. Let A E L ( V , V ’ ) be a linear continuous operator from V to V’ such that, for some w > 0,
-
( A u ,u )
2 wllu112
vu E V .
Very often the operator A is defined by the equation ( u , Au) = a ( u , u )
V U , U EV ,
(1-9)
where a: V x V -+ R is a bilinear continuous functional on V X V such that a( u, u ) 2 wlluIl2
vu E V .
( 1 .lo)
128
3. Controlled Elliptic Variational Inequalities
In terms of a, the variational inequality (1.1) on V becomes
a(y,y
-2)
y
K,
+ V(Y) - cp0)
(Y - z , f )
Vz
E
E
K.
v,
(1.11)
and, E
a(y,y
-
z ) I ( y - z ,f )
Vz
(1.12)
As we shall see later, in applications V is usually a Sobolev space on an open subset R of R"', H = L 2 ( R ) , and A is an elliptic differential operator on R with appropriate homogeneous boundary value conditions. The set K incorporates various unilateral conditions on the domain R or on its boundary dR. By Theorem 2.1 of Chapter 2 we have the following existence result for problem (1.11). Corollary 1.2. Let a: V X V + R be a bilinear continuousfunctional satisfying condition (1.10) and let cp: V + R be a l.s.c., convex proper function. Then, for every f E V' , problem (1.11) has a unique solution y E V. The map f + y is Lipschitzfrom V' to V.
Similarly for problem (1.12): Corollary 1.3. Let a: V X V + R be a bilinear continuousfunctional satisfying condition (1.10) and let K be a closed convex subset of V. Then, for every f E V ' , problem (1.12) has a unique solution y . The map f + y is Lipschitzfrom V ' to V.
A problem of great interest when studying Eq. (1.11) is whether Ay To answer this problem, we define the operator A,: H + H ,
A,y=Ay
fory€D(A,)={uEV;AuEH).
E
H.
(1.13)
The operator A, is positive definite on H and R(Z + A,) = H ( I is the unit operator in H ). (Indeed, by Theorem 1.3 in Chapter 2 the operator Z + A is surjective from V to V'.) Hence, A, is maximal monotone in H X H.
Theorem 1.2. Under the assumptions of Corollary 1.2, suppose in addition that there exists h E H and C E R such that
3.1. Elliptic Variational Inequalities. Existence Theory
Then, i f f
E
129
H , the solution y to (1.11) belongs to D ( A H )and lAyl IC(I +
Proof: Let A ,
E
(1.15)
Ifl).
L ( H , H 1 be the Yosida approximation of A,, i.e.,
A,
=
A-'(Z - (I + AA,)-'),
A
> 0.
Let y E V be the solution to (1.11). If in (1.11) we set z + Ah) and use condition (1.141, we get
=
(I + A A H ) - ' ( y
(Here we have assumed that A is symmetric; the general case follows by Theorem 2.4 in Chap. 2.) We get the estimate IA,+ylIc(1 + I f
I)
VA
> 0,
where C is independent of A and f . This implies that y estimate (1.15) holds. Corollary 1.4.
Zn Corollaty 1.3, assume in addition that f
(I + AA,)-'(y
+ Ah) E K
for some h
E
E
E
D(AH) and
H and
H and all A > 0. (1.16)
Then the solution y to variational inequality (1.3) belongs to D(A,), and the following estimate holds: lAyl IC(I + I f l )
Vf
E
H.
(1.17)
3.1.2. The Obstacle Problem Throughout this section, R is an open and bounded subset of the Euclidean space RN with a smooth boundary dR. In fact, we shall assume that dR is of class C 2 .However, if R is convex this regularity condition on dR is no longer necessary.
130
3. Controlled Elliptic Variational Inequalities
Let V = H'(R), H
=
L2(R), and A : V + V' be defined by N
a2 > 0. If
by
Here, ao,a,, E LYR) for all i, j
c
=
1,. .., N , a,,
N
a , ( x ) 2 0,
2
t,1=1
a,,(x)6,6, 2 w11611~
=
a,,, and
V t E R N ,X
E
a,
(1.20)
where w is some positive constant and 11. IIN is the Euclidean norm in R N . If a 1 = 0, we shall assume that a o ( x ) 2 p > 0 a.e. x E R. The reader will recognize of course in the operator defined by (1.18) the second order elliptic operator
c N
AOY
=
-
( a l l y x , ) , , + aoy
(1.21)
with the boundary value conditions aIy
dY + a2 =0 dU
in d R ,
(1.22)
where d / d u is the outward normal derivative, d
-y
dv
c N
=
i,j=1
a i j y x cos( , u, X i ) .
(1.23)
Similarly, the operator A defined by (1.19) is the differential operator (1.21) with Dirichlet homogeneous conditions: y = 0 in dR.
3.1. Elliptic Variational Inequalities. Existence Theory
131
Let I) E H2(R) be a given function and let K be the closed convex subset of V = H’(R) defined by K
=
{ y E V ;y ( x ) 2 $ ( x ) a.e. x
E
a}.
(1.24)
Note that K # 0 because $+= max($,O) E K . If V = Hd(fl), we shall assume that $(x) 4 0 a.e. x E dR, which will imply as before that K # 0. Let f E V’. Then, by Corollary 1.3, the variational inequality ( 1.25) VZ E K a(y,y -z ) I ( Y - z7f) has a unique solution y E K . Formally, y is the solution to the following boundary value problem known in the literature as the obsfacfeprobfem,
in R + = { x E R ; y ( x ) > $ ( x ) } , A,y2f, y > $ inR,
A,y = f
y = $
inR\R+
dY
- = -
’ du
a* du
in d R + \ dR,
(1.26) (1.27)
Indeed, if $ E C ( a ) and y is a sufficiently smooth solution, then R + is an open subset of R and so for every a E C;(R+) there is p > 0 such that y f p a 2 $ on R, i.e., y f p a E K . Then if take z = y f p a in (1.251, we see that
Hence, A,y = f in 9 ’ ( f l + ) . Now, if we take z = y a , where a
+
E
H’(R) and a
2
0 in R, we get
and therefore A,y 2 f in S’(Sz). The boundary conditions (1.27) are obviously incorporated into the definition of the operator A if a2 = 0. If a2 > 0, then the boundary 0 conditions (1.27) follow from the inequality (1.25) if a,$ + a2 d $ / d u I a.e. in dR (see Theorem 1.3 following). As for the equation d y / d u = d $ / d u in dR+, this is a transmission property that is implied by the conditions y 2 I) in R and y = $ in d R +, if y is smooth enough.
132
3. Controlled Elliptic Variational Inequalities
In problem (1.261, (1.27), the surface d o + \ dQ = S, which separates the domains R + and R \ fi' is not known a priori and is called the free boundary. In classical terms, this problem can be reformulated as follows: Find the free boundary S and the function y that satisfy the system A,y =f y = $ a,y
dY + a2 = 0 dV
inR+, dY a* _ --
inR\R+,
dv
dv
in S,
in dR.
In the variational formulation (1.29, the free boundary S does not appear explicitly but the unknown function y satisfies a nonlinear equation. Once y is known, the free boundary S can be found as the boundary of the coincidence set { x E a; y ( x ) = + ( x ) } . There exists an extensive literature on regularity properties of the solution to the obstacle problem and of the free boundary. We mention in this context the earlier work of Br6zis and Stampacchia [ll and the books of Kinderlehrer and Stampacchia [l] and A. Friedman [l], which contain complete references on the subject. Here, we shall present only a partial result. Proposition 1.1. Assume that aij E C'(fi), a , E Lm(R), and that conditions (1.20) hold. Further, assume that I!,I E H2(R) and
a*
a,@ + a2 - I0 dV
a.e. in dR.
(1.28)
Then for every f E L2(SZ) the solution y to variational inequality (1.25) belongs to H ' ( a )and satisfies the complementarity system
ax. x E R,Y(X) 2 (A,Y(X) -f(X))(Y(X) - *(XI) = 0 A,y(x) 2f(x) a.e. x E SZ,
*(XI,
(1.29)
along with boundary value conditions
(1.30) Moreover, there exists a positive constant C independent off such that IlyIlfP(n) IC(llfIlL2(n)
+ 1).
(1.31)
133
3.1. Elliptic Variational Inequalities. Existence Theory
Proof. We shall apply Corollary 1.4, where H = L 2 ( 0 ) ,V = H ' ( 0 ) (respectively, V = H d ( 0 ) if a2 = O), A is defined by (1.18) (respectively, (1.19)), and K is given by (1.24). Clearly, the operator A,: L 2 ( 0 )-+ L 2 ( 0 )is defined in this case by (AHY)(X)
D(A , ) A
=
=
(A,y)(x)
i y
E H'(
ax.
X E
0 ) ;a l y
We shall verify condition (1.16) with h
> 0 the boundary value problem
0, Y
ED(A,),
dY + a2 = 0 a.e. in d 0 . dV } = A,$.
+ AA,w = y + AA,$ dw alw + a2 - = 0 in d 0 , dV w
To this end, consider for in R ,
which has a unique solution w E D ( A , ) . Multiplying this equation by (w - $)-E H'(R) and integrating on 0, we get, via Green's formula,
( y - $)(w - $)- du 5 0 . Hence, (w - $I-= 0 a.e. in 0 and so w E K as claimed. Then, by Corollary 1.4, we infer that y E D ( A , ) and II&yIlL2(n) 5 C(llfllLqn)
+ l),
and since dR is sufficiently smooth (or R convex) this implies (1.31). Now, if y E D ( A , ) , we have
and so by (1.25) we see that jR
( A , y ( X ) - f( x ) ) ( y ( X ) - Z( x ) ) du
I0
VZ E K .
(1.32)
134
3. Controlled Elliptic Variational Inequalities
The last inequality clearly can be extended by density to all z where
K O= { u If in (1.32) we take z get
E
=
L 2 ( R ) ; u ( x ) 2 + ( x ) a.e. x
=
0).
KO,
(1.33)
+ + a , where a is any positive L 2 ( 0 )function, we
(A,y)(x) -f(x) 2 0
Then, for z
E
E
a.e. x
E
R.
+, (1.32) yields
which completes the proof. We note that under assumptions of Theorem 1.3 the obstacle problem can be equivalently written as ( 1.34)
dZ,o(y)
=
i
u E L2(
I
a ) ;1u( x ) ( y ( x ) - z( x ) ) dr 2 0 v z E KO R
or, equivalently,
where p : R
+
2R is the maximal monotone graph, if r > 0, if r = 0, if r < 0.
(1.35)
Hence, under the conditions of Theorem 1.3, we may equivalently write the variational inequality (1.25) as
a,y
dY + ff2 =0 dV
a.e. in d R ,
(1.36)
3.1. Elliptic Variational Inequalities. Existence Theory
135
arid as seen in Section 2.2, Chapter 2, it is equivalent to the minimization problem
(1.36)' where j : R
is defined by
+
j(r)
=
ifr20 otherwise.
(0
+m
( 1.37)
As seen elsewhere, Eq. (1.36) can be approximated by the smooth boundary value problem A o y + p,(y - $) = f
a.e. in fl, ( 1.38)
where P,(r) = -(l/&)r- ( p, is the Yosida approximation of p). In this context, we have a more general result. Let p be a maximal monotone graph in R X R, and let $ E H'(fl). Let p, = ~ - ' (-l (1 + ~ p ) - 'be ) the Yosida approximation of p. Then, for each f E L'(fl), the boundary value problem AOY + PAY a,y
+ a'
$1 = f
in
JY
in d f l
-= 0 dV
f l 7
( 1.39)
has a unique solution y,f E H'(fl). (Problem (1.39) can be written as A,y p,(y - #) = f , where y -+ p,(y - t,h) is monotone and continuous in ~ ' ( f l ) . )
+
Proposition 1.2.
Assume thaf
Then y,f
+
yf
strongly in ~ ' ( f l )weak& , in ~ ' ( f l ) ,
(1.41)
136
3. Controlled Elliptic Variational Inequalities
where y,f is the solution to boundary value problem (1.39).Moreover, i f f , + f weakly in L’(R), then y,f. + y f
w e a k l y i n H 2 ( R ) ,strong&inH’(R).
ProoJ Let y, = y,f.. Multiplying Eq. (1.39) by p,(y, on R,we get, by Green’s formula,
(1.42)
9 ) and integrating
Hence
Inasmuch as ( d / d u X y , -
9 ) - (a$/&
9)P,(Y,
-
9 ) = -(a1/a2XY,
+ ( a 1 / a 2 ) 9Be() 9 ) s 0 in
-
d f l , we infer that
9 ) P,(y,
-
is bounded in L2(R)
( P,( y, - @)}
and {A,y,) is bounded in L2(R>.We may conclude, therefore, that {y,) is bounded in H2(R) and on a subsequence, again denoted (ye},we have Y,
+
weakly in H 2 ( R ) ,strongly in H ’ ( R ) , (1.43)
Y
P,(Y, - 9 ) 5 A,y,=A,y, -A,y +
weakly in L2(R),
( 1.44)
weaklyin L2(R).
(1.45)
Clearly, we have Aoy+ dY
5=f
a’ - + a , y dU
=
0
a.e. in R a.e. in J R .
On the other hand, if denote by B c L2(R) X L’(fl) the operator By = ( w E L2(fl); w ( x ) E P ( y ( x ) - $ ( x ) ) a.e. in R},then B is maximal monotone and its Yosida approximation B, is given by B,(y)
=
P,(y -
9)
a.e. in R.
3.1. Elliptic Variational Inequalities. Existence Theory
137
Then, by (1.44) and Proposition 1.3 in Chapter 2, we deduce that 6 E By, i.e., ( ( x ) E p ( y ( x ) - @(XI). Hence y = y f , thereby completing the proof of Proposition 1.2. In particular, Condition (1.40) is satisfied if p is given by (1.35) and so Proposition 1.1 can be viewed as a particular case of Proposition 1.2. A simple physical model for the obstacle problem is that of an elastic membrane that occupies a plane domain R and is limited from below by a rigid obstacle J/ whilst it is under the pressure of a vertical force field of density f. We assume that the membrane is clamped along the boundary dR (see Fig. 1.1). It is well-known from linear elasticity that when there is no obstacle the vertical displacement y = y ( x ) , x E R, of the membrane satisfies the Laplace-Poisson equation. In the presence of the obstacle y = @ ( x ) , the deflection y = y ( x ) of the membrane satisfies the system (1.26). More precisely, we have -Ay =f
in { x
-Ayzf,
y z @
y=O
E
R; y ( x ) > @ ( x ) ) , inn,
in dR.
( 1.46)
The contact region { x E R; y ( x ) = @(XI) is not known a priori and its boundary is the free boundary of the problem. Let us consider now the case of two parallel elastic membranes loaded by forces f,, i = 1,2, that act from opposite directions (see Fig. 1.21.The variational inequality characterizing the equilibrium solution y is (see, e.g.,
Figure 1.1.
138
3. Controlled Elliptic Variational Inequalities
Figure 1.2.
Kikuchi and Oden [ll)
VY, . V ( Y , - 21) dU + Pz 5
/-fl(YI
+
- 21)
/ VY,
/n
R
fz(Y2
*
V Y ,
- 22) d X
dU
V(2,
- 22)
7
22)
K7 (1.47)
where f, 2 0, f, 5 0, and
K = { ( y , , y , ) €H,'(R) x H , ' ( R ) ; y , - y , s f a . e . i n R } .
(1.48)
Here, pl, p2 are positive constants, f is the distance between the initial positions of the unloaded membranes, and y , ( x , , x , ) 2 0, y,(x,, x , ) 5 0 are the deflection of the membranes 1 and 2 in ( x , ,x , ) = x . This problem is of the form (1.251, where H = L 2 ( f l )X L2(R), I/ = H,'(R) x H,'(R), K is defined by (1.48), f = ( f l , f,), and
4 Y , 2)
= PI
/ VY, R
v z , dU + Pz for Y
Formally, the solution y free boundary problem
AYI =f,, y, -y, 5 1 - P , A Y , Sf17 y, = 0 , -PI
=
=
/ VY, R
( Y , ,y2),
*
v z , dU
=
(zl, 2,) E I/ x I/.
( y , ,y,) to problem (1.47) is the solution to the
in { x ; Y d X ) -Pz AY, = f 2 in R , -P2 AY, 2f2 in R , y,=O in dR.
-Yz(X)
< f } 7
139
3.1. Elliptic Variational Inequalities. Existence Theory
The free boundary of this problem is the boundary of the contact set { x ; y , ( x ) - y,(x> = I ) . An important success of the theory of elliptic variational inequalities has been the discovery made by C. Baiocchi [l]that the mathematical model of
the water flow through an isotropic homogeneous rectangular dam can be described as an obstacle problem of the type just presented. Let us now briefly described this problem. Denote by D = (0, a ) X (0, b ) the dam and by Do the wetted region (see Fig. 1.3).The boundary S that separates the wetted region Do from the dry region D, = D \ Do is unknown and it is a free boundary. Let z be the piezometric head and let p(xI,x2)be the unknown pressure at the point (x1,x2)E D . We have z = p + x , in D and, by the D'Arcy law (we normalize the coefficients), Az=O
inD,.
(1.49)
Note also that z satisfies the obvious boundary conditions z dz
-=0 dx2
=
h , in AF,
inAB,
z =x2 dZ
- -dU
in S
U
0 in S ,
where d / d u is the normal derivative to S.
GC,
z
=
h , in BC,
(1 S O )
140
3. Controlled Elliptic Variational Inequalities
Introduce the function
and consider the Baiocchi transformation
Lemma 1.1. The function y satisfies the equation Ay
(1.51)
in g'( D)
= XD,
and the conditions
y > 0 in Do
y=O
in B \ D o ,
y =g
in d D , (1.52)
where
in FH
U HL U L C ,
g
=
0
g
=
1 2 -(x2 - h2) 2
We have denoted by
inCB,
g
=
1 2 -(x2 - h,) 2
xD, the characteristic function of
inAF.
Do.
Prooj We shall assume that y E H ' ( D ) and that the free boundary S is sufficiently smooth. Then, if x 2 = a ( x , ) is the equation of S, we have, for every test function cp E C,"(D),
141
3.1. Elliptic Variational Inequalities. Existence Theory
J-YA+=J
Dn
c p ( ~ I ~ ~ 2 ) ~ = x D , ( 4 0 )VcpECXD),
and Eq. (1.51) follows. Conditions (1.52) follow by the definition of y and by (1.50).
W
By Lemma 1.1, we may view y as the solution to the obstacle problem
- A y 2 -1, y 2 0 in D, Ay=1 in(xED;y(x)>O), y=g in d D ,
( 1.53)
or, in the variational form,
Vy * V(y
- u ) clx
+ JD(y
- u ) clx I0,
Vu E K,
(1.54)
where K = { u E H ’ ( D ) ; u = g in dD, u 2 0 in D). By Corollary 1.3, we conclude that problem (1.54) (and consequently the dam problem (1.53)) has a unique solution y E K. The free boundary S can be found solving the equation y(x,, xz) = 0. For sharp regularity properties of y and S, we refer to the book of A. Friedman [l].
142
3. Controlled Elliptic Variational Inequalities
3.1.3. An Elasto-Plastic Problem
Let R be an open domain of RN and let a: H,'(R) defined by
X
Hd(R)
-+
R be
( 1.55)
Introduce the set K
=
( y E Hd(R); IIVy(x)llN I1 a.e. x
E
R},
(1.56)
where II . I I N is the Euclidean norm in RN, and consider the variational inequality y
E
a ( y , y - 2)
K,
I(
y -z , f )
Vz
E
K,
(1.57)
where f E H-'(R). By Corollary 1.3, this problem has a unique solution y . If y is sufficiently smooth, then it follows as for the obstacle problem that -Ay
in { x E R ; I l V y ( x ) l l ~< I} in R, = R \ R e ,
=f
llVyllN = 1
in dR.
y=O
=
Re, (1.58)
The interpretation of problem (1.58) is as follows: The domain can be decomposed in two parts Re (the elastic zone) and R, (the plastic zone). In R e , y satisfies the classical equation of elasticity whilst in R,, I l V y ( x ) l l ~ = 1; the surface S that separates the elastic and plastic zones is a free boundary, which is not known a priori and is one of the unknowns of the problem. This models the elasto-plastic torsion of a cylindrical bar of cross-section R that is subject to an increasing torque. The state y represents in this case the stress potential in R. As noted earlier, (1.57) is equivalent to the minimization problem min{ f a ( z , z ) - ( z ,f ) ; z
E
K).
(1.59)
If f E L2(R) and dR is sufficiently smooth, then the solution y to (1.57) belongs to H2(R)(BrCzis and Stampacchia [l]). It is useful to point out that y
=
lim ye
E'O
in L'(R),
3.1. Elliptic Variational Inequalities. Existence Theory
where y ,
E
143
Hd(fl) is the solution to the boundary value problem -
Ay - div dh,( Vy)
=
in 0,
f
indfl,
y=O where h , : R N .+ R N is defined by
he(u)
=
I"
if l l u l l ~< 1,
(IlullN -
if l l u l l ~2 1,
2E
and ah,: R N -+ R N is its differential, i.e., dh,(u)
=
i"
if IIuIIN < 1,
( u l l u l l ~- 1)
if IlullN 2 1.
EIIUIIN
Let us calculate starting from (1.59) the solution to problem (1.57) in the case where fl = (0,l). If make the substitution
w( x )
=
l X Z ( s) ds,
x E (0, l ) ,
0
then problem (1.59) becomes i n f ( ~ / o l W ' ( x ) d X - / '0w ( x ) l I fX ( s ) d s d X ; w where U = { u E L2(0,1); lu(x)l to (1.60) satisfies the equation
I1
a.e. x
E
where Nu is the normal cone to U. Hence,
y'(x)
=
w(x) =
1
1
E
(0, l)}.Hence, the solution w
144
3. Controlled Elliptic Variational Inequalities
3.1.4. Elliptic Problems with Unilateral Conditions at the Boundary
Consider in R c R N the boundary value problem cy
-
Ay
dY
=f
+ P(Y) dV
3
g
y
=
0
in R,
(1.61)
r, , in r,, in
rl and r2 are two open, smooth, and disjoint parts of dR, rl u r, = dR, f E L2(R), g E L2(rl), c is a positive constant and p is a
where -
maximal monotone graph in R X R. Let j : R -+ R be a lower semicontinuous, convex function such that dj = p. We set V={y~H~(fi);y=oinr,}
and define the operator A: V + V‘ by
p(z)
=
1j (
z ) dz
Vz
E
V,
rl
and let fo
E
V’ be defined by
(fo 7 z > =
1nf ( x ) z ( x ) dx + 1g ( x ) z ( x W
Vz
E
V.
Vz
E
‘v,
rl
By Corollary 1.2, the variational inequality a ( y , y -2)
+ V(Y) - V(Z) 5
( f 0 , Y -2)
(1.63)
has a unique solution y E V. Problem (1.63) can be equivalently written as min{+a(z,z)
+ p(z)
- ( f o , z ) ;z E
v}.
(1.64)
The solution y to (1.63) (equivalently, (1.64)) can be viewed as generalized solution to problem (1.61). Indeed, if in (1.63) we take z = y - a, where a E C t ( R ) , we get (cya
In
+ v y . V a ) dx = 1nfa dx.
3.1. Elliptic Variational Inequalities. Existence Theory
Hence, cy - b y
=
f in g ' ( R ) . Now, multiplying this by y R,we get
z E V , and integrating on
- z,
145
where
More precisely, we have
where ( ;) is the pairing between H112(rl) and H-'I2(r1)(if y E H'(R) and A y E L2(R), then y E H112(rl)and d y / d v E W1I2(rl) (see Lions and Magenes [l]). Then, by (1.631, we see that
( 2 )
4
( j ( y ) -j(z))dx I g - -,y
-z
vz
E
v.
Hence, if g - d y / d v E L2(rl), we may conclude that g - d y / d v a.e. in r l .Otherwise, this simply means that
E
p(y)
where a+: H1'2(rl)-, H-'Iz(Fl) is the subdifferential of the function 6: H112(rl)+ R defined by
In the special case where rl = dR and g = 0, then as seen in Section 2.2, Chapter 2 (Proposition 2.10, y E H2(R) and llylIH2(n) IC(I
+ IlfllL2(n))
Vf E L2(R),
(1.65)
where C is independent of f. Moreover, we have y
=
lim0 y ,
&+
weakly in H 2 ( R ) ,strongly in H ' ( R ) ,
(1.66)
146
3. Controlled Elliptic Variational Inequalities
where y , E H2(R) is the solution to the approximating problem cy, - Ay, dY,
+ p,(y,) dU
=
f
in R,
=
0
in d R ,
=
1 -(r
(1.67)
and p,(r)
E
-
(1
+ E@)-'r)
Vr E
R, E > 0.
In this case, we have a more precise result. Namely: Proposition 1.3. Let f, + f weakly in L2(R). Then the solutiony, E H2(R) to problem (1.67) is weakly convergent in H2(R) and strongly convergent in H'(R) to the solution y f to problem (1.61).
Proof: As seen in the proof of Proposition 2.11, Chapter 2, we have for y , an estimate of the form (1.63, i.e.,
where C is independent of p, (i.e., of denoted E , we have Y, 4 Y
E).
Hence on a subsequence, again
weakly in H ~ ( R )strongly , in H ' ( Q ) ,
dY
weakly in H1I2(an),strongly in L ~ ( R ) ,
P&(Y&)
5
strongly in L~( d R ) ,
Y&
Y
strongly in L ~an). (
dY,
-4 du du +
+
(1.68)
Arguing as in the proof of Proposition 1.2, we see that 6 E p ( y > a.e. in dR. Then, letting E tend to zero in (1.681, we see that y is the solution to W (1.61). We shall consider now some particular cases. If j ( r ) = golrI, r E R, where go is some positive constant, then
3.1. Elliptic Variational Inequalities. Existence Theory
147
and so problem (1.61) becomes
cy dY
- +go
dv
Ay
-
sign y
=f
0
3
in R , in dR.
( 1.69)
Equivalently, cy-Ay=f
inR, y
JY
+ golyl = 0
a.e. in ail.
(1.70)
The boundary conditions can be rewritten as
=o y's o LO
if
1-1
dY dV
< go ,
dY dv
if - = g o ,
a.e. in dR.
dY
if - = - g o , dV
Hence, there are apriori two regions r' and r2on dR where Idy/dvl < go and I d y / d v l = g o , respectively. However, r' and r2 are not known, so problem (1.69) is in fact a free boundary problem and as seen before it has a unique solution y E H2(R). Problem (1.70) models the equilibrium configuration of an elastic body C! that is in unilateral contact with friction on dR (see Duvaut and Lions [l], Chapter IV).
The Signon'ni Problem. Consider now problem (1.61) in the special case g = 0, and where rl = dR, r2= 0, r > 0, r=0, r < 0,
V r ER.
(1.71)
i.e. cy-Ay=f
dY
a.e.inR,yLO, - 2 0 , dV
dY
y-=O dv
a.e.indR. (1.72)
148
3. Controlled Elliptic Variational Inequalities
This is the famous problem of Signorini, which describes the conceptual model of an elastic body R that is in contact with a rigid support body and is subject to volume forces f. These forces produce a deformation of R and a displacement on dR, with the normal component negative or zero. Other unilateral problems of the form (1.61) arise in fluid mechanics with semipermeable boundaries, climatization problems or in thermostat control of heat flow, and we refer the reader to the previously cited book of Duvaut and Lions [l]. In mechanics, one often meets problems in which constitutive laws are given by nonmonotone multivalued mappings that lead to problems of the following type (see P. D. Panagiotopoulos [l]):
where A: V .+ V' is defined by (1.18) and y is the generalized gradient (in the sense of Clarke) of a locally Lipschitz function 4: V + R. In particular, we may take
where j E L";b,(R) is such that lj(01IC(1 + l&Ip-'), 5 E R, and SO (see . s))), R(j(y(x>))I a.e. x E d+(y) c {w E L'(a); W ( X > E [fi(J(Y( (2.75) in Chapter 2). This is a hemivariational inequality, and the existence theory developed in the preceding partially extends to this class of nonlinear problems (see the book [l] by Panagiotopoulos, and the references given there).
3.2. Optimal Control of Elliptic Variational Inequalities
In this section, we shall discuss several optimal control problems governed by semilinear elliptic equations and in particular that governed by elliptic variational inequalities and problems with free boundary. The most important objective of such a treatment is to derive a set of first order conditions for optimality (maximum principle) that is able to give complete information on the optimal control. Since the optimal control problems governed by nonlinear equations are nonsmooth and nonconvex, the standard methods of deriving necessary conditions of
3.2. Optimal Control of Elliptic Variational Inequalities
149
optimality are inapplicable here. The method is in brief the following: One approximates the given problem by a family of smooth optimization problems and afterwards tends to the limit in the corresponding optimality equations. An attractive feature of this approach, which we shall illustrate on some model problems, is that it allows the treatment of optimal control problems governed by a large class of nonlinear equations, even of nonmonotone type and with general cost criteria.
3.2.1. General Formulation of Optimal Control Problems
Let V and H be a pair of real Hilbert spaces such that V is a dense subset of H and
VcHcV’ algebraically and topologically. We have denoted by V’ the dual of V and the notation is that of Section 1.1. Thus (.; ) is the pairing between V and V’ (and the scalar product of H ) and II 1 , I * I are the norms in V and H, respectively. Consider the equation
where A: V + V’ is a linear continuous operator satisfying the coercivity condition ( A u , u ) 2 wllull cp: V
+
2
Vu E V
for some w > 0,
R is a lower semicontinuous, convex function,
(2.2)
dcp: V + V’ is the
subdifferential of cp, B E L(U, V’),and f is a given element of V’. Here, U is another real Hilbert space with the scalar product denoted ( ,. ) and norm I * 10 (the controller space). As seen in Section 3.1 a large class of nonlinear elliptic problems, including problems with free boundary and unilateral conditions at the boundary, can be written in this form. The parameter u is called the control and the corresponding solution y is the state of the system. Equation (2.1) itself will be referred to as the state system or control system.
3. Controlled Elliptic Variational Inequalities
150
The optimal control problem we shall study in this chapter can be put in the following general form:
(PI Minimize the function
on ally
E
V and u
E
U satisfving the state system (2.1).
Here, g: H + R and h: U following conditions: (i)
+
R
are given functions that satisfy the
g is Lipschitz on bounded subsets of H (i.e., g is locally Lipschitz) and
bounded from below by an afine function, i.e.,
where (Y E H and C is a real constraint; (ii) h is convex, lower semicontinuous, and
(iii) B is completely continuousfrom U to V ' ,
The last hypothesis is in particular satisfied if the injection of V into H is compact. Roughly speaking, the object of control theory for system (2.1) is the adjustment of the control parameter u, subject to certain restrictions, such that the corresponding state y has some specified properties or to achieve some goals, which very often are expressed as minimization problems of the form (P). For instance, we might pick one known state y o and seek to find u in a certain closed convex subset Uo c U so that y = y o .Then the least square approach leads us to a problem of the type (PI, where
g(y)
=
31y
-
y0l2
and
h(u) =
if u E U,, elsewhere.
Other control problems such as that of finding the control u such that the free boundary of Eq. (2.1) (if this equation is a problem with free boundary) is as close as possible to a given surface S, though more
3.2. Optimal Control of Elliptic Variational Inequalities
151
complicated, also admit an adequate formulation in terms of (PI. This will be discussed further in a later section. Now let us briefly discuss existence in problem (PI. A pair (y*, u * ) E V x U for which the infimum in (P) is attained is called optimalpair and the control u* is referred as optimal control. Proposition 2.1. optimal pair.
Under assumptions (i)-(iii) problem (PI has at least one
Pro08 For every u E U we shall denote by y" E U the solution to Eq. (2.1). Note first that the map u -+ y" is weakly-strongly continuous from U to V. Indeed, if {u,} c U is weakly convergent to u in U, by assumption (iii) we see that
Bu,
+ Bu
strongly in V'
whilst by (2.1) and (2.2) we have wIIyUn- yumI12I IIBu, because d q is monotone in V lim y'n
n+m
=y
Now letting n tend to
X
-
B~,II*IIyUn- yUmII
V ' . Hence,
exists in the strong topology of V .
+m
in
Ay"n
+ d q ( y".)
E Bu,
+f ,
we conclude that y = y", as claimed. Now let d = inf{g(y") + h(u); u E U ) . By assumptions (i), (ii), it follows that d > - w . Now, let (u,} c U be such that d I g(y,)
+ h ( u , ) I d + n-'
Vn,
where y, = y".. By (i) and (ii) we see that {u,) is bounded in U, and so on a subsequence (for simplicity again denoted (u,}) we have u, + u*
y,
-+
y*
weakly in U , strongly in V ,
because B is completely continuous. Since as seen in the preceding y* = y"' and g(y*) h(u*) = d (because g is continuous and h is weakly lower semicontinuous), we infer that (y*, u * ) is an optimal pair for problem (PI.
+
152
3. Controlled Elliptic Variational Inequalities
From the previous proof it is clear that as far as existence is concerned, hypotheses (i)-(iii) are too strong. For instance, it suffices to assume that g is merely continuous from V to R.Also, if g is convex, assumption (iii) is no longer necessary because g is, in this case, weakly lower semicontinuous on V. On the other hand, it is clear that the convexity assumption on h cannot be dispensed with since it assures the weak lower semicontinuity of the function u + g ( y " ) + h(u), a property that for infinite dimensional controllers space U is absolutely necessary to attain the infimum in problem (PI.
3.2.2. A General Approach to the Maximum Principle Here we shall discuss a general approach to obtain first order necessary conditions of optimality for optimal control problems and in particular for problem (PI. Most of the optimal control problems that arise in applications can be represented in the following abstract form:
(6) Minimize the functional on all ( y , u ) E X
X
U, subject to state equation F ( y ,U)
=
0.
(2.7)
Here, L : X X U + R is a continuous function, h: U + R = I - m, +..I is convex and lower semicontinuous, and F : X X U + Y is a given operator, where X, Y,U are Banach spaces. We shall assume of course that for every u E U, Eq. (2.7) has a unique solution y " E X. Let us first assume that the map u + y" is Giteaux differentiable on U and that its differential in u E U , z = D,y"(u), is the solution to the equation
where Fy and F, are the differentials of y -+ F ( y , u ) and u + F ( y , u ) , respectively. (This happens always if the operator F is differentiable.)
3.2. Optimal Control of Elliptic Variational Inequalities
153
Let ( y * ,u * ) be an optimal pair for problem (PI. We have A - l ( L ( y U * + A " ,+~ *A u ) - L ( Y * , u * ) ) + A-'(h(u* Vu E U, A
+ Au)
-
h ( u * ) )2 0.
> 0, and if L is (GPteaux) differentiable on X X U this yields
(L,(Y*,U*),z)
+ (L,(y*,u*),u) + h'(u*,u)2 0
VUE U , (2.9)
where h ' is the directional derivative of h and (.,. 1, ( * ,. ) are the dualities between X, X * and U, U*, respectively. Denote by F; the adjoint of F y , and further assume that the linear equation (2.10)
q y * , u * ) p = L,(Y*, u * ) (the dual state equation) has a solution p we see that
E
(Ly(y*,u*),z)= (p,F,(y*,u*)z)
Y*. Then, by (2.8) and (2.10)
=
-(F;L(y*,u*)p,v),
and substituting in (2.9) yields VUE
(F;L(Y*>U*)P - L,(y*,u*),u) Ih'(u*,u)
u.
By Proposition 2.8 in Chapter 2, this implies
F,*( y * , u * ) p - L,( y * , u*) E dh( u * ) .
(2.11)
Equations (2.71, (2.101, and (2.11) taken together represent the first order optimality system for problem (P), and if F is linear and L is convex these are also sufficient for optimality. If L and F are nonsmooth, one might hope to obtain a maximum principle type result in terms of generalized derivatives of L and F by using the following method: Assume that the control space U is reflexive and that there are a family (F,),,, of smooth operators from X X U to Y and a set of differentiable functions L,: X X U + R such that
(i)'
For every E > 0 and u E U the equation F,(y, u ) = 0 has a unique solution y =y," and there exists p 2 1 such that lly:llx
I
C(1
+ lull;)
Vu E
u.
(2.12)
'
154
3. Controlled Elliptic Variational Inequalities
(ii)' If {u,} is weakly convergent to u in U then { y , " ~ is } strongly convergent to y" in X and liminf L,(y,Ue,u,) &+
0
2
(2.13)
L(y",u).
Moreover, lim, L,(y,", u> = L ( y " , u ) V u E U. (iiiY L,(y, u> 2 -C(lyl + lul) V ( y ,u> E X X U , where C is independent of E . ~
Let ( y * , u * ) be any optimal pair in problem penalized problem
i
1
(P,) inf ~ , ( y , u )+ h ( u ) +
-IU
2P
- u*IZP;
(el. We consider the
F,(Y,u)
= O,U E
1
u
,
and assume that this problem has a solution (y,,u,) (this happens, for instance, if the function u -+ L,(y,U, u ) is weakly lower semicontinuous). Then, we have Lemma 2.1. For
E
-+
0, we have u, + u*
y, Pro05
For all
E
-+
y*
strongly in U , strongly in X .
> 0 we have
and by condition (2.12) and assumption (iiiY, {u,) is bounded in U and so, on a subsequence E, 0. -+
weakly in U ,
u&n-+ ii y,,
-+
y
=y z
strongly in X .
Since h is weakly lower semicontinuous, we have lim inf h( u,,) 2 h( U ) E" +
and so, by assumption (ii)',
0
3.2. Optimal Control of Elliptic Variational Inequalities
155
Hence, J = y * , U = u * , and u,, -+ u* strongly in U on some subsequence {qJ.Since the limit u* is independent of the subsequence, we conclude that u, + u* strongly in U, as claimed. Now, since problem (6,) is a smooth optimization problem of the form (61, (y,,u,) satisfy an optimality system of the type (2.10), (2.11). More precisely, there is p , E Y * that satisfies along with (y,, u,) the system
F&(Y&, U & > = 07 ( F & ) ; ( Y &U, & ) P &= ( L & ) J Y CU"), , (F,),*(y,,u,)p,
-
( L & ) u ( Y & , uE &W ) u , ) + J ( u , - u*)Iu,
- u*I:p-27
(2.14)
where J : U + U* is the duality mapping of the space U. By virtue of Lemma 2.1, we may view (2.14) as an approximating optimality system for problem (P). If one could obtain from (2.14) sufficiently sharp a priori estimates on p , , one might pass to limit in (2.14) to obtain a system of first order optimality conditions for problem (6). We shall see that this is possible in most of the important cases, but let us see first how this scheme looks for problem (P) considered in the previous section. We shall assume that g and h satisfy assumptions (9, (ii) (with the eventual exception of condition (2.511, B E L ( U , V r ) , f~ V r , and the injection of V into H is compact. Let ( y * ,u * ) be an optimal pair in problem (PI. Then, we associate with this pair the penalized optimal control problem (adapted penalty):
Here, g" is defined by (see formula (2.79) in Chapter 2)
where P,u = Cy,, u,e,, u = Cy=, u,e,, A,,T = Cy=,u,e,, {e;} an orthonorma1 basis in H , and cp": V + R is a family of convex functions of class C2 on V such that cp"(u) 2 - C ( llull
lim cp"(u) = cp(u)
E-0
+ 1) Vu E V
Vu E V , E
> 0,
156
3. Controlled Elliptic Variational Inequalities
whilst, for any weakly convergent sequence u,
+
u
in V ,
lim inf cp"( u,) 2 cp( u ) . E'O
By Proposition 2.1, we know that problem (P,) has at least one solution ( y , , u,). Since our conditions on cpE clearly imply assumptions (i)', (ii)', (iii)' of Lemma 2.1, where X = H, Y = V ' , and F,(Y,U) =AY L & ( Y ,u ) we conclude that, for
E +
u,
+
y,
+ y*
+ Vcp"(Y> - B u -f,
=g"(y)
+ h(u),
0,
strongly in U , strongly in H, weakly in V .
u*
On the other hand, the optimality system for problem (P,) is (see (2.14)) AY, + Vcp"(Y , ) -
=
Bu, + f,
A*P&- V*cp"(Y&)P& = Vg"(y,), B*p, E d h ( u , ) + U, - u*.
(2.16) (2.17) (2.18)
Note that since V*cp"(y,) E L(V,V ' ) is a positive operator, Eq. (2.17) has a unique solution p , E V. Moreover, since {Vg"(y,)}is bounded on H (because g is locally Lipschitz), we have the estimate wllp,ll IIVg"(y,)l Hence, on a subsequence
E, +
P&+P A*p, + A*p Vg"(y,)
-3
5
c
v.5 > 0.
0, we have weakly in V , strongly in H, weakly in V', weakly in H.
By Proposition 2.15 in Chapter 2 we know that 6 E dg(y*),where dg is the generalized gradient of g whilst by (2.18) it follows that B*p E dh(u*) or, equivalently, u* E dh*(B*p). Summarizing, we have proved: Proposition 2.2. Let ( y * ,u * ) be an optimalpair orproblem (PI. Then there is p E V such that
-A*p - 77 E d g ( y * ) , u* E dh*( B * p ) ,
(2.19)
3.2. Optimal Control of Elliptic Variational Inequalities
where in V.
=
w - lim,+ oV2cp"(y,)p, in V ' , p ,
+
p , and y,
+
157
y * weakly
We may view (2.19) as the optimality system of problem (P). There is a large variety of possibilities in choosing the approximating family {cp"). One of these is to take cpE of the form (2.151, i.e.,
where n
=
[ E - ' I , P,, : V -,X,,An: R"
+
V and
(2.21)
or
if cp is lower semicontinuous on H (this happens, for instance, if cp is coercive on V ) . One of the interesting features of the adapted penalty problem (P,) is that every sequence u, of corresponding optimal controllers is strongly convergent to u*. If instead of (P,) we consider the problem
(P")
min{g"(y)
+ h( u ) ; Ay + Vcp"(y )
= Bu
+f
u E U},
then under assumptions (i)-(iii), (P") admits at least one optimal pair ( y E ,u E )and, on a sequence E, + 0, uE* + u: yen
+yf
weakly in H, strongly in V ,
where (yf ,u f ) is an optimal pair of (PI. The result remains true if replace h by a smooth approximation, for instance, h, as in (2.22), and one might try to calculate u s by a gradient type algorithm taking in account that the differential &$ of the function 4: u + R,
4(u) =g"((A
+ Vcp")-'(Bu +f)) + h , ( u ) ,
u E
U,
158
3. Controlled Elliptic Variational Inequalities
is given by d + ( u ) = dh,(u) - B*p,
where -A*p - V'cps( y ) p Ay
=
Vg"( u),
+ V p E (y ) = Bu + f .
Now, we shall present another approach to problem (PI, apparently different from the scheme developed in the preceding, but keeping similar features. To this end, we shall assume that, beside the assumptions (i) and (ii), the injection of V into H is compact, cp is lower semicontinuous on H, B E L(U, H I , f E H, and hypothesis (1.14) holds. Then, as seen earlier, the solution y to variational inequality (2.1) belongs to D ( A , ) , i.e., Ay E H. Let (y*, u * ) E D ( A , ) X U be an optimal pair for problem (P). We associate with (y*, u * ) the family of optimization problems
(Q,)
inf(g(y) + h ( u ) + e-l(cp(y) + cp*(u)
+ +lu - U * l 2 + 31.
-
(Y,.))
- u*Ih},
where the infimum is taken on the set of all (y, u , u ) E V subject to
X
U
X
H,
Ay=Bu-u+f. Here, u* = Bu* gate of cp.
+ f - Ay* E dcp(y*) c H
and cp*: H
+
R
is the conju-
Lemma 2.2. Problem Q, has at least one solution (y,, u,, u,) H.
E
V
X
U
X
(2.23) Since {u,} is bounded in H and {y,) is bounded in V,we may pass to limit in the previous inequality to get the existence of a minimum point. W
3.2. Optimal Control of Elliptic Variational Inequalities
Lemma 2.3. For
E +
159
0, u,
-,u*
strongly in U ,
u,
-+
u*
strongly in H ,
y,
+
y*
strongly in I/.
(2.24)
Proof: We have
+ E-'(cp(Y,> + cp*(V,)
g(Y€) + N u , )
+~(IU,
-
u * Iu2
-
(Y&lU&))
+ Iu, - u*I2) _< g ( y * ) + ' h ( u * )
VE > 0, (2.25)
because (see Proposition 2.5 in Chapter 2) cp(Y*) + 'p*(v*)= ( Y * , . * ) . Note also that
cpo(Y,)
+ cp*(.,)
Hence, on a subsequence
E, +
V
weakly in H,
-+
v.2 > 0.
2 0
(2.26)
0, we have
weakly in U ,
yen + j j E,,
(Y,,.,)
uEn+ ii u,.
NOW,letting
-
strongly in H, weakly in I/.
(2.27)
tend to zero in (2.25) and using (2.26) and (2.27) we get
Ig(y*)
+ h ( u * ) = inf P,
(2.28)
respectively, cp(Y)
+ cp*(V)
- (J,V) =
0,
because cp and cp* are weakly lower semicontinuous and
2 ( f , A J ) - (jj,Bii
+f)
=
-(J,V).
Then, by Proposition 2.5, Chapter 2, we infer that V E dcp(jj), and so by (2.28) we get (2.24).
160
3. Controlled Elliptic Variational Inequalities
Lemma 2.4. Let ( y e ,u,, u,) be optimal for problem p,. Then there is p , E V that satisfies the system
Here, dcp: V
+
-A*P, v,
E
B*p,
E
M Y & )+ &-‘(dcp(Y€) M Y , - EP, + E(V* -
(2.29)
dh(u,)
(2.31)
+ U, - u*.
(2.30)
7I),.
V’ is the subdifferential of cp: V + R.
Proof: Let us denote by L,(y, u , u ) the function L A ( y ? u , u )= g ^ ( y ) + h ( u ) + & - ‘ ( P A ( y )
+ cp*(v)
+ +(Iu - U * l 2 + Iu - U * I h ) + +(Iu - u,IU2 + IU - L J , ~ ~ ) ,
where g ” is defined by (2.15) and problem
(ph
(y,.))
A > 0,
by (2.22). Clearly, for every A > 0 the
inf(L,(y,u,u);u~U,u~H,Ay=Bu-u+f}
(2.32)
has a solution ( y A u, ” , u ” ) .We have Lh(YA7UA,Uh)
< g ” ( Y c ) + h ( u ~ +) YE) + q * ( u . )-
+ +(I.,
-
U*l2
+ Iu,
- U*lh)
(Y~7’e))
VA > 0.
Since {u”}),{u”}are bounded in U and H , respectively, we may assume that U” +
ii
weakly in U
u”
+
V
weakly in H
yA
+
7
weakly in V , strongly in H.
Then we have g ” ( y ” )+ g ( y ) , liminf cpA(yA)2 c p ( j j ) , A-0
liminf cp*(u”) 2 c p * ( ~ ) , A-rO
liminf h ( u ” ) 2 h ( E ) A+O
161
3.2. Optimal Control of Elliptic Variational Inequalities
This yields
+ &-I( <
Hence,
+ cp*(cv)
cp( J )
-
( 1 , V ) ) + +(I.
- U*l2
+ l i i - .*I;)
inf Q,.
1=y,, ii = u,, V = uE and for A
+
0
U”
-+
u,
strongly in U ,
u”
-+
u,
strongly in H,
Y ” -+YE
strongly in I/.
(2.33)
Now, since problem (2.32)is smooth one can easily find the corresponding optimality conditions. Indeed, we have
(vgA(yA),z)+ h r ( U h , W )
+ &-‘((vVA(YA),z) + ( c p * ) r ( u Ae, ) - ( 2 , ~ ” ) ( y A ,e ) ) + (d- u * , e )
+ ( u ” - u * , w ) + ( u A- U & , W ) 2 0
for all w
E
(2.34)
U, 8 E H and z satisfying the equation A Z = B W - ~ .
Now, let p ”
E I/
be the solution to
-A*p”
=
VgA(yA)+ &-‘(VpA(y”)
- u”).
(2.35)
Substituting (2.35)in (2.34), we get
+
+
- ( B * ~ ~ , w ) @ , P A ) h r ( u A , w )+ d ( ( c p * ) ’ ( d , e ) - (y^,e))
e)
+(dfor all w
E
U, 0
E
+ (u^
-
u*, e )
+ (u” - u * , w ) + (u” - u , , w ) 2 0
H. This yields
B*p” E dh(u”) + 2 ~ -” U, - u*,
(2.36)
-PA
(2.37)
E
& - * ( d c p * ( u ” )- y A )
+ 2uA - u* - u,.
We may equivalently write (2.37)as lJA E
acp(yA - &PA
+ & ( U * + U& - 2u”)).
(2.38)
162
3. Controlled Elliptic Variational inequalities
Then, substituting in (2.35) and multiplying the resulting equation by ( I + A dp)F'yA - yA+ & p A- E(U* + u, - 2q), we get the estimate VA
IlpAll IC
> 0,
because d p is monotone and (Vg*} is uniformly bounded on bounded subsets. Hence on a subsequence, A + 0, we have weakly in V , weakly in V ' ,
P A+P& A* p A+ A*p,
whilst by Propositions 2.15 and 1.3, in Chapter 2, VgA(yA)+ 6 E dg(y,)
weakly in H ,
7 E dp( y,)
weakly in V'
and VpA(y ^)
+
because {VpA(yA)) is bounded in V' and y A (2.36) it follows that B*p, E d h ( u , )
+ y,
strongly in V. Finally, by
+ U, - u*,
and V& E dcp(Y, - E P ,
+
E(U*
-
.&>>
thereby completing the proof. By virtue of Lemma 2.4 we may view (2.29H2.31) as an approximating optimality system for problem (P). Let 6, = u* - us. Then, we may rewrite (2.29) as -A*P,
E
MY,)
+ E-'(dv(Y&)
-
MY,
- E P , + &O&)>* (2.39)
If multiply this equation by p , - 0, and use the monotonicity of dcp along with the coercivity condition (2.21, we get 11p,11 5
because (Vg(yJ is bounded. Hence, there is a sequence
E, +
+
U," + u* +
u*
E
VE
> 0,
0 such that
weakly in V , strongly in H , weakly in V ' , strongly in U ,
P&, + P A*P&" A*P
%
c
dcp(y*)
strongly in H .
3.2. Optimal Control of Elliptic Variational Inequalities
Then, letting
E = E,
163
tend to zero in (2.29)-(2.31), we get -A*p B*p
E E
d g ( y * ) + w*, dh(u*),
(2.40)
where
in the weak topology of V ' . These conditions can be made more explicit in the specific problems that will be studied in what follows. 3.2.3. Optimal Control of Semilinear Elliptic Equations In this section, we shall study the following particular case of problem (P): Minimizeg(y) + h ( u ) o n a l l y subject to A,y
E H,'(R)
n H 2 ( R )andu
+ p ( y ) = f + Bu
E
U , (2.42)
in R , in dR.
y = o
(2.43)
Here, A , is the elliptic differential operator (1.211, i.e., N
AOY
=
-
C
i,j = 1
where a i j ,a,, E L"(R), a i j = aji for all i , j N
C
i,j=l
(2.44)
(aij(x)yx,)x,+ a , ( x ) ~ ,
a i j ( x ) t i t j2 W I I ~ I I ~
=
1,. . ., N , a , 2 0 a.e. in R,
~tE R
~ , a.e. x
E
a,
for some w > 0, p is a monotonically increasing continuous function on R, f E L2(R), and B E L(U, L2(R)). We shall assume that the functions g and h satisfy assumptions (i) and (ii) of problem (P). More precisely, we shall assume: (a) The function g : L2(R) + R is Lipschitz on bounded subsets and there are C E R and f,, E L2(R) such that
d Y )
2 (f0,Y) +
c
VY E L 2 ( R ) *
(2.45)
164
3. Controlled Elliptic Variational Inequalities
(b) The function h: U -+ R is convex, lower semicontinuous, and satisfies condition (2.5). Problem (2.421, (2.43) is a particular case of problem (P), where H L2(R), V = H,'(R), A : V -+ V' is defined by (1.191, i.e.,
=
( 2.46)
and ~ ( y =)
1j ( y ( x ) ) du, n
y
E
H,'(R);
dj
=
p, j : R
-+
R . (2.47)
The function g : L2(fl) -+ R may arise as an integral functional of the form (2.48) where g o : R x R -+ R is measurable in x and Lipschitz in y . We shall assume that R is a bounded and open subset of R N , either with smooth boundary (of class C2 for instance) or convex. As seen earlier, this implies that for every u E U, problem (2.43) has a unique solution y E H,'(R) n H2(R). Moreover, if h satisfies condition (2.9, then problem (2.42) has at least one optimal pair ( y , u). Theorem 2.1 following is a maximum principle type result for this problem.
Theorem 2.1. Let ( y * ,u * ) be any optimal pair in problem (2.42), (2.431, where p is a monotonically increasing locally Lipschitz function on R, and g , h satisjj assumptions (a), ( b ) . Then there exist functions p E H,'(R), q E L'(R), and 6 E L 2 ( R )such that A p E (L"(R))* and
-(Ap), T(X) EP(X)
If either 1 I N
-
q
=
6
a.e. in R ,
W y * ( x > > , 6(x) E d g ( y ( x ) ) B*p E dh(u*). I 3 or
(2.49) a*e*x
E
a,
(2.50) (2.51)
p satisfies the condition
P ' ( r ) IC ( l p ( r ) l + Irl
+ 1)
a.e. r
E
R,
(2.52)
3.2. Optimal Control of Elliptic Variational Inequalities
then Ap
E L’(R)
165
and Eq. (2.49) becomes -Ap - 77
=
6
a.e. in 0.
(2.53)
Here, d p and dg are the Clarke generalized gradients of p and g, respectively (see Section 2.3 in Chapter 21, (L”(R))* is the dual space of Lm(R)and (Ap), is the absolutely continuous part of Ap E (L”(R))*. We can apply the Lebesgue decomposition theorem to the elements p of the space (,!,“(a))*: p = pa+ ps,where pa E L’(R) is the absolutely continuous part of p and p, the singular part (see, e.g., Ioffe and Levin [l]). This means that there exists an increasing sequence of measurable sets R, C R such that m(R \ R , ) k 2 m 0 and p,(cp) = 0 for all cp E LYR) having support in R,. Thus, (2.49) should be understood in the following sense: There exists a singular measure vs E (L”(R))* such that Ap = v, - 77 - 6 , where 77 E L’(R) and 6 E L2(R) satisfy Eq. (2.50).
Proof of Theorem 2.1. We shall use the approach described in Section 2.2 by approximating problem (2.41) by a family of problems of the form (Qe). Namely, we shall consider the approximating problem: Minimize
Ay
=
BU
-
v
+f.
(2.55)
Here, u* = Bu* + f - Ay* E dcp(y*) = p(y*) a.e. in R and dcp c L2(R) X L2(R) is the subdifferential of cp given by (2.471, as a 1.s.c. convex function from L ~ ( R to ) R. Similarly, cp*(u) =sup{(u,p) - cp(u); UEL2(R))
=
jR j * ( u ( x ) ) dx,
UEL*(R).
(Throughout this section [ - I 2 denotes L2 norm and the L2 scalar product.) Without any loss of generality, we may assume that p(0) = 0. ( - , a )
166
3. Controlled Elliptic Variational Inequalities
As seen in Lemmas 2.2 and 2.3, problem (2.54) has at least one solution ( y , , u,, u,) E (H,'(R) n H2(R)) x U x L2(R) and, for E -+ 0, u,
+
u*
strongly in U ,
u,
+
u*
strongly in L ~a), ( stronglyin H,'(R) n H 2 ( a ) .
y, + y *
(2.56)
Now, by Lemma 2.4 it follows that there are p, E H,'(Rn) 17 H2(R) and
& E L2(R) such that -AP,
=
u,
=
(&(XI
B*p,
E
5,
+
&-I(
P ( Y & ) - v,)
+
P ( y , - ~ p , E ( U * - u,)) dg(y,(x)) dh(u,) + U, - u*.
a.e. in R, a.e.in R,
(2.57) (2.58)
a.e. in R ,
(2.59) (2.60)
To pass to limit in system (2.57)-(2.60), we need some a priori estimates on p,. These come down to multiplying Eq. (2.57) by p, and integrating on Q. We have
2 -115,1121IP,112
-
(I1 P(y,)ll2 + Ilu,ll2)llu*
-
v,l12
V€ > 0.
Since { &} is bounded in L2(R>(because g is locally Lipschitz), we have Ilp,llH:(n, Ic
VE
> 0,
(2.61)
and so we may suppose that p,
Ap,
+p
weakly in H,' ( R),strongly in L2(0),
+ Ap
weakly in H-'
Extracting a further subsequence P,,(X)
.,Ax)
(a).
E,, +
P(X) - u * ( x ) -+ 0 +
y,,(x) + y * ( x )
(2.62)
0, we can assume by (2.56) that ax. x
E
a,
a.e. x
E
R,
a.e. x E R.
(2.63)
3.2. Optimal Control of Elliptic Variational Inequalities
167
Now, by the Egorov theorem, for every S > 0 there is a measurable subset E, c R such that rn(E,) I 6 and the sequences p,,, u,. - u*, y,, are bounded in Lm(R\ E,) and uniformly convergent on R \ E,. Now multiply Eq. (2.57) by LA(p , ) where 5, is a smooth approximation of signum function, i.e., for A > 0, for
ldr)
r 2 A,
for - A < r < A,
=
-1
for
r
I -A.
We have
and, therefore,
:1 P(Y.s) < :/
~5-l
-
&-'
-
u ~ ) l , ( -p ~E ( U *
P(Y,)
+115,11L~(n,
-
- .&>)
u . ) ( lA(p&- &(.*
- u . ) ) - lA(pe))
V& > 0.
Now, let S > 0 be arbitrary and let E8 c R be such that rn(E,) I 6 and the convergences in (2.63) are uniform on R \ E,. Then, letting E = E, tend to zero in the previous inequality, we see that
<
c + 2 limsup E ; ~ E" +
0
IP(y,,)
- v,,l
h,
and this yields
IP(y,,) - v,,lh,
(2.64)
3. Controlled Elliptic Variational Inequalities
168
where C is independent of 6 (for 6 sufficiently small). On the other hand, since ( / 3 ( y E n )- uEn}is bounded in L2(R), we have
L
I P ( Y & , ) - U&,I du
Ic6"2
where C is independent of n and 6. Then, for 6 =
E,)?,
(2.64) yields (2.65)
from which (2.66) Since
is bounded in L 2 ( f l ) ,we may assume that weakly in L2(R)
ten+ 6
and, by Proposition 2.13 in Chapter 2, 6 E d g ( y * ) . Similarly, letting E, tend to zero in (2.60) we get (2.51). By the estimates (2.65) and (2.66), it follows that there exists a generalized subsequence of &, say &, such that weak star in ( Lw(a))*,
ApEA+ p,
1
-(
P ( y e A )- u,)
-+
7
weakstar in (L"(R))*,
(2.67)
&A
where p
= Ap
on LYR) n H,'(R) and in particular on C,"(R). We have -p, = 7
+ 4.
(2.68)
On the other hand, by the mean formula (Corollary 2.1, Chapter 21, we have 1 -(P(Y&,)
-
&A
=
e,(P~A - ( .*
- ueA))
R,
E @ ( z , ) , ycAI zA 5 yEA- &,(pEA - u* + uJ). Now since 0, and P(yEA)- uE,>are uniformly bounded on R \ E,, we may assume that
where 0, &;I(
OEA+ 0
weak star in Lw(R \ E,)
(2.69)
and 1
-( &A
P(yJ
- u,)
+
Op
weak star in Lm(R\ E , ) .
(2.70)
169
3.2. Optimal Control of Elliptic Variational Inequalities
By (2.69) we have
This yields ( P o is the directional derivative of p )
Hence, V w E R, a.e. x E
O(x)w I po(y*(x),w)
R
E,,
and therefore O ( x ) E d p ( y * ( x ) ) a.e. x E R \ E,. We have therefore proved that 77 E Lm(R\ E,) and ~ ( x E) d p ( y * ( x ) ) p ( x ) a.e. x E R \ E,. Then, by (2.68), we see that the restriction of p to R \ E, belongs to LYR \ E , ) and a.e. x
pa(x ) = p( x ) E - d p ( y * ( x ) ) p ( x ) - (( x )
E
R
\
E, .
Since 6 is arbitrary, we conclude that -&Ax)
E
dP(Y*(X))P(X)
a*e*xE
+ ((x)
a,
as claimed. Suppose now that 1 I N I 3. Then, by the Sobolev imbedding theorem, H2(R) c C ( a ) a n d so y, are uniformly bounded on On the other hand, we have, by (2.57),
a.
4EP,)
=
-&,
-
P(Y&) +
LL
and, since { p ( y , ) - u,) is bounded in L2(R), we infer that { ~ p , ), , is bounded in H2(R)and therefore in C(a). Finally, we have y, -
EP&
+ E(U*
-
U&)
E p-l(v,),
y* E p - ' ( u * ) .
Subtracting and multiplying by u, - u*, we get EIU*
- U&l I ly,l
+ elp,l
a.e. in R
170
3. Controlled Elliptic Variational Inequalities
and, since p is locally Lipschitz, this implies that IAp,l
I15,l
+ L(lp,l + Iv,
-
.*I)
a.e. x
E
a.
Hence, {Ap,} is bounded in L2(R) and we conclude, therefore, that p E H,'(R) n H2(R), 7 E L2(R), and Ap = (Ap*),, as claimed. Assume now that condition (2.52) holds. We shall prove that {q, = ( 1 / ~ )p(y,) ( - u,)} is weakly compact in L'(R). We have
3.2. Optimal Control of Elliptic Variational Inequalities
171
One might obtain the same result if one uses the approach described in Proposition 2.2. Namely, we approximate problem (2.42), (2.43) by the following family of optimal control problems: Minirnizeg"(y)
+ h ( u ) + $ I U - U*I;
o n d ( y , u ) E (H,'(R) nH2(a))x U ,
(2.72)
subject to Ay where A y fined by
= A,y
+ p " ( y ) = Bu + f
in R ,
(2.73)
with D ( A ) = H,'(R) n H2(R), and p"
E
CYR) is de-
Here, p,(r) = ~ - ' ( r (1 + & p ) - ' r ) , r E R,and p is a C: mollifier in R, i.e., p E C"(R), p ( r ) = 0 for Irl > 1, p ( r ) = p ( - r ) V r E R, / p ( t ) dt = 1. Note that p is monotonically increasing, Lipschitz, and
I P E ( r ) - p,(r)l
I2 8
Vr
E
R.
(2.75)
We are in the situation described in Section 2.2, where
Throughout the sequel, we set B E = ( BE)'. Let ( y &u,) , be optimal for problem (2.72). Then we have u,
-+
u*
strongly in H i ( R ) , weakly in H 2 ( R ) ,
Y, +Y* p"(y,)
+
strongly in U,
p(y*)
weaklyin L 2 ( R ) .
(2.76)
Indeed, from the inequality g"( y,)
+ h( u,) + +IU" - u*l;
Ig"(
y"')
+ h( u * )
172
3. Controlled Elliptic Variational Inequalities
( y " is the solution to Eq. (2.7311, we see that {u,) is bounded in U. Hence, on a subsequence E,, + 0, we have
u,"
+
weakly in U .
U
On the other hand, we may write Eq. (2.73) as
AY, + P & ( Y & )= Bu, + f + P & ( Y & ) - P E ( Y & ) and so, by Proposition 1.2, y,, + j j
strongly in H ' ( 0 ) and weakly in
H2(a), where j j is the solution to (2.43), where u = U. This yields g ( j j ) + h ( ~+) limsup +1uEn- u*Ic I g ( y * ) + h ( u * ) = inf(P). En -+
0
Hence, us" -+ u* strongly in U and j = y * , U = u*. The sequence being arbitrary, we have (2.76). Moreover, there is p , E H,'(R) n H2(a) such that
-4, - B"(Y,>P,
=
Vg"(y,)
B*p,
E
dh(u,)
in
a,
E,,
(2.77)
+ U , - u*.
(2.78)
Now, multiplying Eq. (2.77) by p , and sign p,, we get the estimate
Ilp,ll2Hd(n, +
/-I B"(Y,)P,l dx I c
Hence on a subsequence, again denoted
p, Vg"(y,)
+p +
5
we have
weakly in H; ( a),strongly in
( a),
weakly in L 2 ( a ) ,
and, on a generalized subsequence
be^( y,,)p,,
E,
v.5 > 0.
+
v
{EJ,
weak star in ( L"( a))*.
Hence, -Ap - q = 5 E d g ( y * ) in a. Now, by the Egorov theorem, for each S > 0 there exists a measurable such that m ( E , ) I 6 and y * , p E Lm(a \ E,) and subset E, of
y,( x )
+ y*( x ) ,
p,( x )
Since {( BE(y,)) is bounded in subsequence
b " ( y , ) +f,
+p ( x
)
uniformly on
Lm(a \ E,), weak star in
\
E,.
we may assume that, on a
Lm(a \ E6).
3.2. Optimal Control of Elliptic Variational Inequalities
173
Then, by Lemma 2.5 following, we infer that f s ( x ) E d p ( y * ( x ) ) a.e. x E Cl \ E, and so 77,(x) = f , ( x ) p ( x > E
ap(Y*(x)>P(x>
a*e*x
E
\E,*
The last part of Theorem 2.1 follows as in the previous proof since { b E ( y E ) }is bounded in Lm(CL)if 1 I N < 3 (because I b”(y,)I I 1 p’(y,)I < C in a ) and is weakly compact in L’(R) if p satisfies condition (2.52). The main ingredient to pass to limit in the approximating optimality system is Lemma 2.5 following, which has intrinsic interest.
Lemma 2.5. Let X be a locally compact space and let u be a positive measure on X such that u( X ) < 03. Let yE E L’( X ; u ) be such that y, + y strong& in L ’ ( X ; u ) and @“(y,) + f o
weaklyin L’(x; u ) .
Then fo(x)E dp(y(x))
Prooj
a.e. x E X .
On a subsequence, again denoted ye, we have y,(x) + y ( x )
v x E X \A,
u ( A ) = 0.
On the other hand, by Mazur’s theorem there is a sequence o, of convex combinations of { b E ( y E )such ) that on+ fo
strongly in L’( X ; u ) ,
where w,(x) = X i a,!, pEc(yi(x)).Here, I, is a finite set of positive n = 1. Hence, there is a integers in [n,+ M[, yi =y,, and a: 2 0, C i E I a,!, subsequence, again denoted o,,such that o,(x) -+ f o ( x ) V x E Cl \ B , u ( B ) = 0. In formula (2.74) we make the substitution t = r - c28 to get ps(r)= - & - z / p , ( t ) p ( y ) d t - /pE(-Eze)p(e)d0
and this yields b E ( r )= . ~ - ‘ / ~ + “ ~ P r, -( tt ) p ~ ( ~ ) d t r- c2 = --E-~
/ b E ( r - & ) p f ( 0 ) do.
174
3. Controlled Elliptic Variational Inequalities
where 8; 2 0, C ~ 8;L= ~1. Hence,
B
Y;( x 1)
(2.79) where
On the other hand, we have
Hence, l(e;e;)-'((i
+ Eip)-'Yi
- (1
+ Eip)-l(yi
-
&;e;)> - I ( I
cEi
Because be, uniformly bounded on every bounded subset ( p is locally Lipschitz). Then, by (2.791, it follows that
bet(y i ( x ) )
c 6;q; m,
=
k= 1
+'-yj,
where
'y; +
0 as i
+ 03.
(2.80)
3.2. Optimal Control of Elliptic Variational Inequalities
175
On the other hand, 8; - yi + 0 uniformly with respect to k, and by (2.80) we have bEi(yi(x))h _<
m,
k= 1
8 ; P 0 ( ~ ; , h+) -yih
Vh
E
R.
Hence, limsupbEi(yi(x))h _< P o ( y i ( x ) , h )
V h E R.
i-rm
Finally,
It should be said that the methods of Section 2.2 are applicable to more general problems of the form (2.42), for instance, for the optimal control problem: Minirnizeg(y)
+ h ( u ) on all ( y , u ) E ( w ~ , R) P (n W2vP(R)) x U , 1 IP < 03,
(2.81)
subject to -Ay
+ P ( y ) = f + Bu y=o
a.e. in 0, in a R ,
(2.82)
where f E LP(R), B E L(U, LP(R)), g locally Lipschitz on LP(R), h is convex and 1.s.c. on U, and P is monotone and locally Lipschitz. One gets a result of the type of Theorem 2.1 by using the approximating control process (2.721, i.e.:
-Ay
+ p"(y)
= f + Bu
y=o
in R, in dR
(2.83)
176
3. Controlled Elliptic Variational Inequalities
Then one writes the optimality system and passes to limit as in the proof of Theorem 2.1. 3.2.4. Optimal Control of the Obstacle Problem We shall study here the following problem: Minimizeg(y) + h ( u ) o n a l l ( y , u )E ( H ~ ( R n ) H2(R))X U , y E K , ( 2.84) subject to a ( y , y - z ) I( f where a: H,'(R) set (1.24).
X
H,'(R)
+
+ Bu,y
-z)
Vz
E
(2.85)
K,
R is defined by (1.19) and K is the convex
As seen in the preceding, (2.85) is equivalent to the obstacle problem (A,y -f
-
B u ) ( y - @)
=
a.e. in R ,
0
A,y - f - Bu 2 0 , y 2 @
a.e. in
a,
in dR.
y=O
(2.86)
Here, B E L(U, L2(fl)), f E L 2 ( f l ) ,and g : L 2 ( R )+ R, h: U assumptions (a) and (b) in Section 2.3.
+
R satisfy
Theorem 2.2. Let ( y * ,u * ) be an optimal pair for problem (2.84). Then there exist p E H,'(fl) with A p E (L"(R))*and 6 E L2(R) such that $, E dg(y*) and
+
a.e. in [ x; y*( x ) > @( x ) ] , (Ap), 6 = 0 p ( A y * - Bu* - f ) = 0 a.e. in R , a ( p , X(Y* -
@)I + ( 6 7 ( Y * - @ ) x )= 0 B*p
E
dh(u*),
a ( P , P ) + ( 6 , P ) I 0. Zf 1 I N
I3,
vx E C ' ( h ) ,
(2.87) (2.88) (2.89) (2.90) (2.91)
then Eq. (2.87) reduces to ( A p + 6 ) ( y * - @)
=
Oin R .
(2.92)
177
3.2. Optimal Control of Elliptic Variational Inequalities
We have denoted by A the operator A, with the domain D ( A ) = H,'(R) n H 2 ( C l ) and by ( A p ) , the absolutely continuous part of A*p. If N I3, then y* E H2(R) c C(n) and so A p ( y * - #) is well-defined as an element of (L"(R))*. In particular, (2.92) implies that Ap* = in [ x ; y * ( x ) > +(x)l. The system (2.871, (2.89) represents a quasivariational inequality of elliptic type.
Proof of Theorem 2.2. Consider the penalized problem (2.93) subject to Ay
+ P E ( Y - S)
=
Bu
(2.94)
+f,
where
In other'words, p" is defined by formula (2.741, where p is the graph (1.35). By Proposition 2.1, problem (2.93) has at least one optimal pair (u,,y , ) E ux ( ~ ~ n ( H~(R)). 0 ) Arguing as in the proof of Theorem 2.1 (see problem (2.72)), it follows by Proposition 1.2 that
u,
+
u*
stronglyin H,'(R),weaklyin H 2 ( R ) ,
y, + y *
p " ( y , - 9)
strongly in U ,
+ Bu* - Ay*
weakly in L 2 ( n ) .
+f
Moreover, we have for (2.93) the optimality system (see (2.771, (2.78)) -AP, - P ( Y & - S I P , B*p,
E
=
W(Y,)
+
d h ( ~ , ) U, - u*.
in
(2.96) (2.97)
3. Controlled Elliptic Variational Inequalities
178
Then multiplying Eq. (2.96) first by p, and then by sign p , , we get the estimate
c.
IcIIvg"(y,)lL2(n) I
Hence, there is a subsequence, again denoted P, Vg"(y,) Letting
E
+
-+
such that
P
weakly in H i ( R),strongly in L2(R ) ,
6 E dg(y*)
weakly in L 2 ( R ) .
tend to zero in (2.97), we get B*p
Now, let bY
E,
(2.98)
&: R
.+
R and 7,:R
On the other hand, we have
--j
E
dh(u*).
R be the measurable functions defined
179
3.2. Optimal Control of Elliptic Variational Inequalities
This yields IP&P"(Y&-
$11 I EIP&PE(Y&- $ W - ' l Y & - $16, +E-'ly& -
+ 2clp,l
$177,)
a.e. in R . (2.101)
We note that p " ( y , - $)q& = &-'(ye - $177, + C q , remain in a bounded subset of L2(R), whilst by the definition of 6, wee see that ~-'ly~(x )$(x)l&.(x)
a.e. x
IE
E
R.
Since ( p , P " ( y , - #)) is bounded in L'(R), it follows by (2.99) and (2.101) that, for some subsequence E,, + 0, p & , ( x ) p E f l ( Y & , ( x-) $ ( x ) )
+
0
ax. x
E
a,
(2.102)
whilst p,, p " n ( y E n- $)
+p ( f
+ Bu* - A y * )
weakly in L'( R).
Together with (2.102) and the Egorov theorem, the latter yields a.e.inR,
p(f+Bu* - A y * ) = O and therefore
stronglyin L'(S1). (2.103)
p E a p e n ( y e n- $) -+p(f+ Bu* - A y * )
Then, by (2.1001, we see that ( y E n- $ ) P " n ( y , , - $ ) p , ,
Inasmuch as I p " ( y , - $) (2.103), and (2.104) that
+
0
+ &-'(y8 - $)-I
(Y&n- $)+ P Y Y , , - $ ) P & n
+
Since (yen +&JY&"
-
$)+E
0
stronglyin L'(fl). (2.104) I C E , it follows by (2.99),
strongly in
Jw).
Hi(R), applying Green's formula in (2.96) yields
$I+ x ) + ( v g Y Y , , ) , ( Y & " - @ I +x )
-+
0
vx E c l m
Since p,, + p weakly in Hd(R) and (yen - $ ) + + y * - $ strongly in H'(R), we get (2.89). Regarding inequality (2.91), it is an immediate consequence of Eq. (2.96) because we have
(4+, V g & ( Y & ) , P &I) 0
V& > 0-
180
3. Controlled Elliptic Variational Inequalities
Now, selecting a further subsequence, if necessary, we may assume that y,,(x)
+
a.e. x
y*(x)
E
R.
On the other hand, by estimate (2.98) it follows that there is p of E, such that and a generalized subsequence p"~(y,,- $ ) p ,
weakstar in
-+ p
E
(L"(R))*
(,!,"(a))*.
This implies that Ap admits an extension as element of (L"(R))*, and we have -A*p - p
=
5 E dg(y*).
Now, by Egorov's theorem, for every 6 > 0 there is a measurable subset E, of R such that rn(E,) I 6, y* - $ is bounded on R \ E, = R, and
y,"
- $ -+
y*
- $
uniformly on R,.
Then, by (2.104), it follows that p(y* - $)
=
0 in R,, i.e.,
On the other hand, there is an increasing sequence and ps = 0 on L"(Rk). Hence,
rn(n \ Rk)I k-'
(ak)such
that
Thus, ( y * - $ ) p a = 0 a.e. in R,, and letting 6 tend to zero we infer that ( y * - $ ) p a = 0. Hence,
-(Ap), If 1 s N
=
5E
a.e. in [ y * >
dg(y*)
$1.
s 3, then H2(R) c C ( n ) and so y,(x)
-+ y
*(x)
uniformly on
a.
Since $ E H2(R) c C(n), it follows by (2.104) that ( y * - $ ) p
(Y*
-
$)(AP + 5 )
This completes the proof of Theorem 2.2.
=
0.
=
0, i.e.,
181
3.2. Optimal Control of Elliptic Variational Inequalities
Remark 2.1. Theorems 2.1 and 2.2 remain valid if one assumes that ( y * , u * ) is merely local optimal in problem (2.42) (respectively, (2.841, i.e., g(y*) + h(u*)I g ( Y ) + h(u)
for all ( y , u ) satisfying (2.43)) (respectively, (2.85)) and such that lu - u*Iu < r. Indeed, in problem (2.72) (respectively, (2.94)) replace the cost functional by g"(y)
+ h ( u ) + aIu - u*1;,
where a is sufficiently large that Iim sup g"( y,"') &+
0
+ h( u * ) I ar'.
Then Iu, - u*Iu I r for all E > 0 and this implies as before that u, strongly in U.The rest of the proof remains unchanged.
+
u*
Problems of the form (2.84) arise in a large variety of situations, and we now pause briefly to present on such an example. Consider the model, already described in Section 2.2, of an elastic plane membrane clamped along the boundary an, inflated from above by a vertical field of forces with density u and limited from below by a rigid obstacle y = $ ( x ) < 0, Vx E n (see Fig. 1.1). We have a desired shape of the membrane, given by the distribution y = yo(x> of the deflection, and we look for a control parameter u subject to constraints. lu(x)l I p
a.e. x
E
n,
(2.105)
such that the system response y" has a minimum deviation of y o . For instance, we may consider the problem of minimizing the functional / , ( y ( x ) - y0(x))' dw on all ( y , u ) E (H,'(R) n H 2 ( n ) )X L'(fl), subject to control constraint (2.105) and to state equation (2.861, where f = u. This is a problem of the form (2.84), where A, = - A , B = I , U = L'(n), f = 0, and
h(u) =
if lu(x)l I p a.e. x otherwise.
E
n,
182
3. Controlled Elliptic Variational Inequalities
By Proposition 2.1, this problem has at least one solution ( y * , u * ) whilst by Theorem 2.2 such a solution must satisfy the optimality system Ay*
+ u*
=
y*
=
0
+,
y* = O
in R + = ( x
R; y*(x) > + ( x ) } , a.e. in R, = R \ R', A + + u* I0 indR,
Ap =y* -yo
p(u*
+ Ay*) = 0
E
a.e. in R + ,
a.e. in R ,
u* = p sign p
(2.106) p
(2.107)
=
in d R ,
0
(2.108)
a.e. in R .
(2.109)
Assume that IA+I # p a.e. in 0. Then, by Eq. (2.108), we see that p R,. Hence, p is the solution to boundary value problem in R + , A p = y* - y o p=O inR,, p=O
indR.
=
0 in
(2.110)
This system could be solved numerically using an algorithm of the following type: yi 2
+,
( A y , + u i ) ( y i - +) = 0 Ay, + u , I0 a.e. in R
Ap, = y i - y o
ui+
a.e. in 0, y,
=
0
in d R ,
in Ri = ( x E R ; y i ( x ) > + ( x ) } , pi = 0 in d R i , =
p sign p i
a.e. in R i .
Let us assume now that y o E HdCCl) n H2(R> and A y o ( x ) 2 max(A+(x), p ) a.e. x E 0. Then, by Eqs. (2.106), we see that A(y* - y o ) I0 a.e. in R and, therefore, by the maximum principle y* - y o 2 0 in a. Then, assuming that R + is smooth enough, we deduce by (2.110) and by virtue of the maximum principle that p < 0 a.e. in a+,and therefore, by (2.1091, u* = - p
inR+
Hence, y* E H,'(R) n H2(R) satisfies the variational inequality Ay* = p
inat,
Ay* < p
+
y*
=J!,I
in d R + ,
y* >
y*
=
in R,
y* = O
+
inR, inR+, indR,
(2.111)
183
3.2. Optimal Control of Elliptic Variational Inequalities
from which we may determine R+. For instance, if R = ( 0 , l ) and = - 1, then clearly the solution to (2.111) is convex and so it is of the following form: y * ( x ) = -1
forO
y*(O) = y * ( l ) = 0
-1
<0
forx
E
(0,a) u ( b , l )
=
R+.
Hence for x
E
[a, b),
for x
E
[a,b],
and one determines constants a, b, c, and C , from the continuity conditions y*(a) =y*,(b)
=
-1
(y*)’(a)
=
(y*)\(b)
=
0.
A problem of great interest in the study of a physical system modeled by the obstacle problem is that of controlling the incidence set R, = ( x E R, y ( x ) = $ ( X I } or its boundary dR,. For instance, in the case of the contact problem (1.46) recalled in the preceding a problem of interest would be to find the force field f (viewed as a control parameter) such that the set of all points where the membrane is in contact with the obstacle be as close as possible (in a certain acceptable sense) to a given measurable subset D of 0. The least squares approach to this problem leads us to consider the optimal control problem: Minimize (2.112)
on all ( y , u ) E H,’(R) X U subject to (2.85). Here, ,yo is the characteristic function of D and xy is that of the set R; y ( x ) = $ ( X I } . However, since the function y + xy is not locally Lipschitz on L2(R) we shall replace it by g,(y) = A / ( ( y - $I++ A), A > 0, and so we shall consider the problem:
{x E
184
3. Controlled Elliptic Variational Inequalities
To be more specific, we shall assume that A, is given by h: L2(a> + h(u)
=
{L
=
-A, B
if lu(x)l I p a.e. in otherwise.
= I,
a,
This problem has at least one solution (y, , u,) and since, for A g,(y)
+
x,
f = 0, and
+
0,
strongly in L2(a )
for every y E L 2 ( a ) ,it is readily seen that for A + 0, u, u* weak star in Lm(a), y, + y* weakly in H2(a), where (y*, u * ) is an optimal pair for problem (2.112). On the other hand, the optimality system of (2.87H2.90) is in this case Ay,
+ u, = 0
y, 2 $, Ay,
pIp,I +p, A $
+ u,
I0
=
at = [ y , = $1, a.e. in a.
a.e. in
0
=
$1
a:, a.e. in a ,
a.e. in [y, >
uA = p sign p,
This problem can be treated as in the previous examples. Now we shall study problem (2.84) in the case where g is a continuous for instance, convex function on
~(a),
g ( y ) = IlY -Yollccn,, The subdifferential d g : C ( a ) Chapter 1, Section 1)
a;
+
Y
E
cm.
(2.113)
M ( a ) of g is given by (see Example 4 in
where M , = (xo E Iz(x,)l = Ilzll~~n,}. is a bounded, open, and smooth For simplicity, we shall assume that domain of R3, and A, = -A.
185
3.2. Optimal Control of Elliptic Variational Inequalities
Theorem 2.3. Let ( y * ,u * ) be optimal in problem (2.84) where g is given by and (2.113). Then there existsp E Wdvq(R),1 < q < 3/2, with A p E p E dg(y*) such that
Ap
=
p ( x)( Ay*( x)
p in { x E
R;y * ( x ) > @ ( x ) ) ,
+ Bu*( x ) + f( x)) B*p
E
=
dh(u*).
0
a.e. x
E
R,
Proog Since the proof is essentially the same as that of Theorem 2.2, it will be sketched only. We consider the approximating control problem:
Minimizeg,(y) + h ( u ) + n~ ~ ( x 0 Usatisfy ) ) -Ay
31.
- u*It,
+ p " ( y - @)
subject to having all ( y , u ) E (H,'(R) = Bu
+f
in R .
Here, g , is the usual convex regularization of g ( z ) = llz - y o l l ~ - ( ni.e., ),
Let (y,, u,) be optimal in the preceding problem. Then we have
p"(y,
-
u,
+
Y,
+
Y* @) + Bu*
and there are p,
Ap,
-
E
strongly in U ,
u*
+ by* +f
strongly in H,'(R) n ~
'(n),
weakly in L2(R),
H,'(R) n H'(R) such that
B E ( y , - @ ) p ,= Vg,(y,)
in ~ , B * P E , dh(u,) + u,
-
u*.
Since sup{ll,$ IIL~(n);5 E d g ( y ) } I1, we have
l l v g & ( y & ) l l LI ~ (c~ )
V& > 0.
Then, multiplying the latter equation by sign p , and integrating on R,we get
186
3. Controlled Elliptic Variational Inequalities
Now, let hi E La(R),i = 0,1,2 (see Section 3.21, the problem
has a unique solution 8
E
CY
> 3. Then, as mentioned in Chapter 2
H,'(R) n LYR) and
and, therefore,
Ilp&llw;4(n)Ic where l/q quence,
+ l/a
=
V,F
> 0,
1, i.e., 1 < q < 3/2. Hence, on a generalized subsep,
+
Q&(Y&) j 6 ( y &- $ ) p ,
p
weakly in W , . q (R ) ,
CL
vaguely in M ( Q ,
v
vaguely in
+
-+
~(fi).
We have, therefore, in g'(fl),
Ap
=
v+ p
B*p
E
dh(u*).
Since y, + y * uniformly on R (because H2(R) c C ( 1 ) compactly), we have v
=
0
in { x
E
R; y*(x) > $ ( x ) } .
Then, arguing as in the proof of Theorem 2.2, we get
p(x)(Bu*(x) Finally, letting
E
+ Ay*(x) + f ( x ) ) = 0
a.e. x
tend to zero in the obvious inequality
E
R.
187
3.2. Optimal Control of Elliptic Variational Inequalities
we infer that
Remark 2.2. In applications, the function g : L2(R) + R that occurs in the payoff of problem (P) and subsequent optimal control problems considered here is usually an integral functional of the form
where g o :R x R -+ R is measurable in x and locally Lipschitz in y , whilst condition (i) or (a) requires that go be global Lipschitz in y. However, it turns out that most of the optimality results established here remain valid if instead of (i), one merely assumes that (i)' go 2 0 on R X R, go(.,O) E L'(R), and for each r > 0 there is h, E L ' ( 0 ) such that l g o ( x , y ) - g o ( x , z)l I h,(x)ly - zI
a.e. x
E
a,
for all y , z E R such that lyl, IzI 5 r. (ii)' There exists some positive constants a, C , , C , and p such that
a.e. x
E
E
L'(R)
R,y
E
R.
For the form and the proof of the optimality conditions under these general assumptions on g we refer to author's work [3].
3.2.5. Elliptic Control Problems with Nonlinear Boundary Conditions We will study here the following problem:
(2.114)
188
3. Controlled Elliptic Variational Inequalities
on all ( y , u ) E H2(R) X U, subject to
y - Ay
=
f + Bu
dY
+p(y)3 0 dU
a.e. in R, (2.1 15)
a.e.in d R ,
where B E L(U, L2(R)), f E L2(R), and p c R X R is a maximal monotone graph. The functions g: L2(R) + R and h: U + R satisfy assumptions (a), (b) of Section 2.3. We know by Proposition 2.1 that if h satisfies the coercivity condition (2.5) then this problem admits at least one solution. Let ( y * , u * ) be any optimal pair for problem (2.114). Then, using the standard approach, we associate with (2.114) the adapted penalized problem: Minimize
g"( y )
+ h( u ) + $1.
(2.116)
- u*I21,
on all ( y , u ) E H 2 ( n ) x U, subject to
y - Ay
=
f + Bu
dY
+ P"(Y> 3 0 dU
a.e. in R, (2.117)
a.e. in d R ,
where p" is defined by (2.74). Let (y,*,u,*)be an optimal pair for problem (2.116). Then, by the inequality
g"( y , )
+ h( U & ) + ;Iu& - u*I2 Ig"( y,"') + h( u * )
(y," is the solution to (2.117)), we deduce as in the previous cases that u,
+
u*
y, + y *
strongly in U weaklyin H2(R),stronglyin H ' ( R ) .
Indeed, by Proposition 1.3 if u, + ii weakly in U then y , H2(R), where y" is the solution to (2.115). This yields limsup &+
0
IU, -
u*12 = o
-+
(2.118)
y" weakly in
3.2. Optimal Control of Elliptic Variational Inequalities
189
because lim inf, h(u,) 2 h(E) and g " ( y , ) + g ( y " ) . Now the optimal pair ( y , , u,) satisfies, along with some p , E H2(R),the first order optimality system ~
+ Ap, = Vg"(y,)
-p,
dP&
-+ dV
PYYJP, B*p,
E
=
a.e. in R , a.e. in d R ,
0
+
(2.1 19)
d h ( ~ , ) U, - u*.
Now, multiplying Eq. (2.119) first by p , and then by sign p , , we get the estimate (2.120) Hence, on a subsequence, again denoted
-+
we have
weakly in H'(
P, + P Vg"(y,)
E,
a),
weaklyin L ' ( S Z ) ,
6 E dg(y*)
strongly in L ~ ( R )weakly , in H ' ( R ) , (2.121)
P,+P and, by (2.118), P"(Y,)
+
weakly in L'(
770
where q o ( x ) E p ( y * ( x ) ) a.e. x Note also that, by (2.120),
E
an),
dSZ.
weak star in (L"( all))*,
P e ~ ( y S , ) p s+ , p
(2.122)
where ( E J is a generalized subsequence (directed subset) of ( E } . Now, letting E tend to zero in Eq. (2.1191, we see that p satisfies the system -p A p E dg(y*) a.e. in R , (2.123) dP - + p = o in d R ,
+
dV
B*p
E
(2.124)
dh(u*).
We note that since p E H'(SZ) and A p well-defined by the formula
E
L2(R), d p / d v
E
H-'/'(dR) is
190
3. Controlled Elliptic Variational Inequalities
where yocp is the trace of cp to dR. The boundary condition in (2.123) means of course that dP
-(
dV
cp)
+ p( cp)
4
vcp E L"( an) n ~ l / ~ ( d R ) ,
o
and in particular it makes sense in 9 ' ( d R ) . These equations can be made more explicit in some specific situations.
Theorem 2.4. Let ( y * , u * ) be optimal for problem (2.114), where p is monotonically increasing and locally Lipschitz. Then there are p E H'(R), q E L'(dR), and 6 E L 2 ( R ) such that p - A p E L2(R>, dp/dv E (L"(dR))*and inR, -p+Ap=( ((x) E dg( y*( x)) a.e. x E 0, (2.125)
(
+q =0
T(X) E
a.e. in d a ,
P(Y*(X>)P(X) ax. x B*p E dh(u*).
E
(2.126) (2.127)
dR,
If either 1 I N I 3 or p satisfies condition (2.52) then d p / d v and Eq. (2.126) becomes dP
dV
+ d p ( y * )p
3
0
E L'(dR)
a.e. in d R .
(2.128)
Here, ( d p / d v ) , is the absolutely continuous part of dp/dv. &oo$ By the Egorov theorem, for every 6 > 0 there is E, c dR such that m(E,) I 6, p,, y , are uniformly bounded on dR \ E,, and
y, p,
y* +p +
and, on a subsequence
{E,)
PYY,") +fo
uniformly on dR uniformly on dR
\
\
E6, E,
c {E}, weak star in L'( dR
\
Eo).
Then, by Lemma 2.5, we infer that f o ( x ) E dp(y*(x>>a.e. x and so, by (2.122) (see the proof of Theorem 2.1), pa(x)
E
d p ( y * ( x)) p ( x)
a.e. x
E
dR
\
E, .
E
dR
\
E,
3.2. Optimal Control of Elliptic Variational Inequalities
191
Since 6 is arbitrary, Eq. (2.126) follows. Now, if 1 I N I 3 then H2(R) c C(n), and so ( y J is bounded in C(fi). We may conclude, therefore, that
l p " ( y & ) lI c
vx
E
dR.
This implies that { b"(y,)) is weak star compact in L Y R ) and so 77 = = - d p / d u E LYdR). If condition (2.52) holds then we derive as in the ] proof of Theorem 2.1, via the Dunford-Pettis criterion, that ( b E ( y E ) is weakly compact in L ' ( R ) . Hence p E L'(dR), and the proof of Theorem 2.4 is complete. Now we shall consider the case where p is defined by (1.71). Then, Eq. (2.13) reduces to the Signorini problem
y
-
Ay
=
Bu
+f
a.e. in R , y dY
dY -20,
0,
a.e. in d R , (2.129)
y-=o
dU
2
dU
which models the equilibrium of an elastic body in contact with a rigid supporting body. The control of displacement y is achieved through a distributed field of forces with density Bu. Theorem 2.5. Let ( y * , u * ) E H2(R) X U be an optimal pair for problem (2.114) governed by Signorini system (2.129). Then there exist functions p E H'(R) and 6 E L 2 ( R ) such that d p / d u E (L"(dR))*, 6 E dg(y*), and
-p
+ Ap = 6 B*p
E
a.e. in R,
(2.130)
(2.132)
h( u * ) .
If 1 I N I 3, then y * -JP dU
=o
a.e. in d R .
(2.133)
Prooj The proof is almost identical with that of Theorem 2.2. However, we sketch it for reader's convenience.
192
3. Controlled Elliptic Variational Inequalities
Note that in this case P" is given by formula (2.95). Let A,: dR + R be the measurable functions
A&(X) =
We have
i
0
if y , ( x ) >
- E ~ ,
1
if y , ( x ) 4
--E
2
6,: dR
and
.
I P&( x 1P "( Y , ( x 11I I E I P&( x 1B ( Y , ( x 11I( E - ' I Y&( x ) I 6, ( x 1
+ 2~lp,(x)l
+~-'Iy,(x)A,(x)l)
+R
a.e. x
E
dR.
Since { P"(y,)) is bounded in L2(dR) and P " ( Y & ) A & ( x )= E - ' Y & ( x ) A & ( x ) +
EA&(X) /l
0
we infer that { ~ - l y , A , ) is bounded in L2(dR).Note also that ~-'Iy,l 6, I E a.e. in dR and, since ( b " ( y , ) p , ) is bounded in L'(dR), there is a sequence E,, + 0 such that p,,(x)pEn(yE,(x))
+
0
a.e. x
E
dR.
Then, by (2.120, we conclude that
P&,P Y Y & , )
+
-P
dY *
-= O dV
strongly in L'( an).
Finally, by (2.134) we see that yEnBCn(yEn)pEn +o
strongly in L ' ( ~ R ) .
(2.135)
Now, by the Egorov theorem, for every 6 > 0 there exists E, c dR such that rn(E,) I 8, ye" are uniformly bounded on dR \ E, and yCn+ y*
uniformly on dR
\
E,.
3.2. Optimal Control of Elliptic Variational Inequalities
193
By virtue of (2.fZZl this impfies that y*p
=
in dR
0
\
Es
and, arguing as in the proof of Theorem 2.2, we see that y * p , dR, which along with (2.123) yields (2.131). If 1 I N I 3, then H2(R) c C(n) and y,
+ y*
uniformly on
=
0 a.e. in
a.
Then, by (2.122) and (2.13.9, we deduce that Y*P
=
0,
where y * p is the product of the measure p y * E C(dR). H
E
(L”(dR))* with the function
3.2.6. Control and Observation on the Boundary We consider here the following problem: Minimize
(2.136)
in R ,
Y-AY=f
-+p(y) dY
dv
3
B,u
in
rl,
y
=
0
in
r,,
(2.137)
where p c R X R is maximal monotone, B, E L(U, L2(rl)), f E L2(R), and aR = rl u r,, where rl and I’, are smooth disjoint parts of dR. The functions g, : L2(R) + R and h : U + R satisfy conditions (a), (b) of Section 2.3, whilst g o : rl x R + R is measurable in x , differentiable in y , and
194
3. Controlled Elliptic Variational Inequalities
As seen in Section 1.4, for each u E U, Eq. (2.137) has a unique solution y E H'(R) (if n F2 = 0, then y E H2(fl)).As a matter of fact (2.136) is a problem of the form (PI considered in Section 2.1, where
r,
g(Y)
= gdx)
so(x,Y(x))
+
VY
E
I
and V ,A are defined as in Section 1.4 (see (1.62)). Let ( y * , u * ) be an arbitrary optimal pair for problem (2.136). Then, following the general approach developed in Section 2.3, consider the penalized problem:
subject to in 0
Y-AY=f dY
dU
+ p & ( y )= Bou
in
rl,
y
=
in
0
r2.
Here, g ; is defined by (2.15) and p" by (2.74).
By a standard argument, it follows that, for
E
-+
u,
+
u*
strongly in U ,
y,
-+
y*
weakly in
y*
strongly in L ~d(a ) ,
y, p"(y,)
--f
-+
fo
HI(
0,
a),strongly in L'( a),
weakly in L2(an),
(2.140)
where f o ( x ) E P ( y * ( x ) ) a.e. x E dR. On the other hand, the optimality principle for problem (2.138) has the form dP,
-+
dv
P, - AP€ = v g ; ( Y & )
in
a,
B"(Y,>P&= - v g o ( x Y Y € )
in
rl,
0
in
r2,
P, B*p,
E
=
dh(u,)
+ U, - u*.
( 2.141a)
(2.14lb)
3.2. Optimal Control of Elliptic Variational Inequalities
195
Now, multiplying Eq. (2.141a) by p , and sign p , , (more precisely, with [ ( p e l , where 5 is a smooth approximation of sign), we find the estimates
and so on a subsequence, again denoted
Letting
E
dP
we have
tend to zero in Eqs. (2.141a), we get
dU
E,
+ p = -Vgo(x,y*) B*p
E
in
rl,
p
=
0
in
r,,
(2.142) (2.143)
dh(u*),
where p E (L"(T,))*. One can give explicit forms for these equations if p is locally Lipschitz or if p is the graph of the form (1.71). Since the proofs are identical with that of Theorem 2.2 and 2.4 respectively, we only mention the results. Theorem 2.6. Let ( y * ,u * ) be optimal in problem (2.136), (2.137), where /3 is monotonically increasing and satisfies condition (2.52). Then there exists p E H'(R) such that A p E L 2 ( R ) ,d p / d u E L2(rl), -p
+ Ap E dgl(y*) B,*p E d h ( u * ) .
a.e. in R ,
(2.144)
196
3. Controlled Elliptic Variational Inequalities
Consider now the case where /3 is given by (1.71). In this case, the state equation (2.137) reduces to the unilateral problem inR,
y-Ay=f
dY
LO, - - B o u > O , dV
y
=
0
in I‘,.
(2.145)
Theorem 2.7. Let ( y * ,u * ) be optimal forproblem (2.136) governed by state equations (2.145). Then there is p E H’(R) such that A p E L2(R), d p / d u E (L”(I‘,))*and -p
+ Ap E
-)
p ( B o u * - dY*
dgl(y*)
a.e. in R , a.e. in ( x E
=
0
a.e. in
rl ; y * ( x ) > 0},
rl,
B,*p E d h ( u * ) . Similar results can be obtained if one considers problems of the form (2.135) governed by variational inequalities on R with Dirichlet boundary control (see V. Barbu [71, p. 107). Bibliographical Notes and Remarks Section 1. The results of this section are classical and can be found in standard texts and monographs devoted to variational inequalities (see, for instance, J. L. Lions [l], Kinderlehrer and G. Stampacchia [l], A. Friedman [l], and C. M. Elliott and J. R. Ockendon [l]). For other recent results on analysis and shape of free boundary in elliptic variational inequalities and nonlinear elliptic boundary value problems, we refer to J. Diaz [l]. Section 2. Most of the results presented in this section rely on author’s work [3,7]. For other related results, we refer the reader to the works of A. Friedman [2], V. Barbu and D. Tiba [l],V. Barbu and Ph. Korman [l],D. Tiba [l], G. MoroSanu and Zheng-Xu He [l], and L. Nicolaescu [l]. An attractive feature of the approach used here is that it allows the treatment of more general problems such as optimal control problems governed by
Bibliographical Notes and Remarks
197
not well-posed systems (J. L. Lions [2]) or by hemivariational inequalities (Haslinger and Panagiotopoulos [ 11, Panagiotopoulos [2]). A different approach to first order necessary conditions for optimal control problem governed by elliptic variational inequalities is due to F. Mignot [l] (see also Mignot and Puel [l]), and relies on the concept of conical derivative of the map u + y". Let us briefly describe such a result, for the optimal control problem (2.84), where J!,I = 0, U = L2(R), B = I , and
= {rp E H,'(R), rp 2 0 in 2,; uf) = O}, where y" is the solution to (2.85), then ( y * , u * ) is optimal if and only if there is p E -Sue such that u* = p and
If Z,
=
( x E R, y " ( x ) = 0} and S,
(rp, Aoy" -
(Mignot and Puel [l]). A different approach related to the method developed in Section 2.2 (see problem Q,) has been used by Bermudez and Saguez ([1-4]). In a few words, the idea is to transform the original problem in a linear optimal control problem with nonconvex state constraints and to apply to this problem the abstract Lagrange multiplier rule in infinite dimensional spaces. A different approach involving Eckeland variational principle was used by J. Yong [4]. Optimal controllers for the dam problem were studied by A. Friedman et' a f . [ 11. Optimality conditions for problems governed by general variational inequalities in infinite dimensional spaces were obtained in the work of Shuzong Shi [l] and Barbu and Tiba [l]. For some earlier results on optimal control problems governed by variational inequalities and nonlinear partial differential equations, we refer to J. L. Lions [4] (see also [2]). There is an extensive literature on optimal control of free boundary in elliptic variational inequalities containing control parameters on variable domains (shape optimization). We mention in this direction the works of Ch. Saguez [l], J. Haslinger and P. Neittaanmaki [l], P. Neittaanmaki, et a f . [l], J. P. Zolesio [l], V. Barbu and A. Friedman [l], W. B. Lui and J. E. Rubio [l], Hlavacek, et al. [l], and Hoffman and Haslinger [l]. A standard shape optimization problem involving free boundaries is the following: Let R, be a domain in R N that depends upon a control variable u E U and let its boundary dR, = ro u r,, To n T, = 0, where r0 is prescribed inde-
198
3. Controlled Elliptic Variational Inequalities
pendently of u. The problem is to find u E U such that robecomes the free boundary and R, the noncoincidence set of a given obstacle problem in a domain R 3 R,, for instance: A y = f in [ y > 01, A y s f in R,, y > 0 in R, , y = 1 in r, . Such a problem is studied by the methods of this chapter in the work of Barbu and Friedman [l] and an explicit form of the solution is found in a particular case. A different approach has been developed in Barbu and Tiba [2] and Barbu and Stojanovic [l]. The idea is to reduce the problem to a linear control problem of the following type: Find u E U such that d y / d v = 0 in r0and y > 0 in R,, where y is the solution to Dirichlet problem A z = f in r, , z = 0 in r0,z = 1 in r, .
First order necessary conditions for state constraint optimal control problems governed by semilinear elliptic problems have been obtained by Bonnans and Casas [l] using methods of convex analysis (see also Bonnans and Tiba [l]).
Controlled Elliptic Variational Inequalities
This chapter is concerned with optimal control problems governed by variational inequalities of elliptic type and semilinear elliptic equations. The main emphasis is put on first order necessary conditions of optimality obtained by an approximating regularizing process. Since the optimal control problems governed by nonlinear elliptic equations, and in particular by variational inequalities, are nonconvex and nonsmooth the standard methods to derive first order necessary conditions of optimality are usually inapplicable in this situation. The method we shall use here is to approximate the given problem by a family of smooth optimization problems containing an adapted penalty term and to pass to limit in the corresponding optimality conditions. We shall discuss in detail several controlled free boundary problems to which the general theory is applied, such as the obstacle problem and the Signorini problem. 3.1. Elliptic Variational Inequalities. Existence Theory 3.1.1. Abstract Elliptic Variational Inequalities
Let X be a reflexive Banach space with the dual X * and let A: X + X * be a monotone operator (linear or nonlinear). Let cp: X R be a lower semicontinuous convex function on X, cp f +m. If f is a given element of X, consider the following problem. Find y E X such that --f
(AY,Y
-2)
+4 Y )
- d z ) 5
125
(Y - z , f )
vz E X .
(1.1)
126
3. Controlled Elliptic Variational Inequalities
This is an abstract elliptic variational inequality associated with the operator A and convex function cp, and can be equivalently expressed as AY + dcpo(Y)
3
f
( 1*2)
9
where dp c X x X * is the subdifferential of cp. In the special case where cp = I , is the indicator function of a closed convex subset K of X , i.e., zK(x)
=
{0
+oo
ifxEK, otherwise,
problem (1.1) becomes: Find y
E
K such that
It is useful to notice that if the operator A is itself a subdifferential d+ of X + R, then the variational inequality a continuous convex function (1.1) is equivalent to the minimization problem (the Dirichlet principle)
+:
min{$(z) + cp(z) - ( z , f ) ; z E X }
(1.4)
or, in the case of problem (1.31, min{+(z) - ( z , f);z E K}.
(1.5)
As far as concerns existence in problem (l.l), we note first the following result.
Theorem 1.1. Let A : X + X * be a monotone, demicontinuous operator and let cp: X + R be a lower semicontinuous, proper, convex function. Assume that there exists y o E D(cp) such that
lim ((AY,Y - y o ) + ~ ( Y ) ) / l l Y l l=
llyll-+ 30
+w.
( 1.6)
Then problem (1.1) has at least one solution. Moreover, the set of solutions is bounded, convex, and closed in X . If the operatorA is strictly monotone, i.e., ( A M- Av, u - v) = 0 u = v, then the solution is unique.
+
Proof By Theorem 1.5 in Chapter 2, the operator A dp is maximal monotone in X X X * . Since by condition (1.6) it is also coercive, we conclude (see Corollary 1.2 in Chapter 2) that is surjective. Hence, Eq. (1.2) (equivalently, (1.1)) has at least one solution.
3.1. Elliptic Variational Inequalities. Existence Theory
127
Since the set of all solutions y to (1.1) is ( A + d c p ) - ’ ( f ) , we infer that this set is closed and convex (Proposition 1.1 in Chapter 2). By the coercivity condition (1.61, it is also bounded. Finally, if A (or more generally, if A + d q ) is strictly monotone, then ( A + dcp)-’f consists of a single element. In the special case cp
=
ZK, we have:
Corollary 1.1. Let A: X -+ X * be a monotone demicontinuous operator and let K be a closed convex subset of X . Assume either that there is y o E K such that
lim ( A y ,y - yo)/llyll =
II yll-
(1.7)
+w,
m
or that K is bounded. Then problem (1.3) has at least one solution. The set of all solutions is bounded, convex, and closed. If A is strictb monotone, then the solution to (1.3) is unique.
To be more specific we shall assume in the following that X Hilbert space, X * = V ’ , and VcHcV’
=
V is a (1.8)
algebraically and topologically, where H is a real Hilbert space identified with its own dual. The norms of V and H will be denoted by II * II and I . I, respectively. For u E V and u’ E V’ we denote by (u, u ’ ) the value of u‘ in u; if u, u’ E H , this is the scalar product in H of u and u ‘ . The norm in V‘ will be denoted by II II*. Let A E L ( V , V ’ ) be a linear continuous operator from V to V’ such that, for some w > 0,
-
( A u ,u )
2 wllu112
vu E V .
Very often the operator A is defined by the equation ( u , Au) = a ( u , u )
V U , U EV ,
(1-9)
where a: V x V -+ R is a bilinear continuous functional on V X V such that a( u, u ) 2 wlluIl2
vu E V .
( 1 .lo)
128
3. Controlled Elliptic Variational Inequalities
In terms of a, the variational inequality (1.1) on V becomes
a(y,y
-2)
y
K,
+ V(Y) - cp0)
(Y - z , f )
Vz
E
E
K.
v,
(1.11)
and, E
a(y,y
-
z ) I ( y - z ,f )
Vz
(1.12)
As we shall see later, in applications V is usually a Sobolev space on an open subset R of R"', H = L 2 ( R ) , and A is an elliptic differential operator on R with appropriate homogeneous boundary value conditions. The set K incorporates various unilateral conditions on the domain R or on its boundary dR. By Theorem 2.1 of Chapter 2 we have the following existence result for problem (1.11). Corollary 1.2. Let a: V X V + R be a bilinear continuousfunctional satisfying condition (1.10) and let cp: V + R be a l.s.c., convex proper function. Then, for every f E V' , problem (1.11) has a unique solution y E V. The map f + y is Lipschitzfrom V' to V.
Similarly for problem (1.12): Corollary 1.3. Let a: V X V + R be a bilinear continuousfunctional satisfying condition (1.10) and let K be a closed convex subset of V. Then, for every f E V ' , problem (1.12) has a unique solution y . The map f + y is Lipschitzfrom V ' to V.
A problem of great interest when studying Eq. (1.11) is whether Ay To answer this problem, we define the operator A,: H + H ,
A,y=Ay
fory€D(A,)={uEV;AuEH).
E
H.
(1.13)
The operator A, is positive definite on H and R(Z + A,) = H ( I is the unit operator in H ). (Indeed, by Theorem 1.3 in Chapter 2 the operator Z + A is surjective from V to V'.) Hence, A, is maximal monotone in H X H.
Theorem 1.2. Under the assumptions of Corollary 1.2, suppose in addition that there exists h E H and C E R such that
3.1. Elliptic Variational Inequalities. Existence Theory
Then, i f f
E
129
H , the solution y to (1.11) belongs to D ( A H )and lAyl IC(I +
Proof: Let A ,
E
(1.15)
Ifl).
L ( H , H 1 be the Yosida approximation of A,, i.e.,
A,
=
A-'(Z - (I + AA,)-'),
A
> 0.
Let y E V be the solution to (1.11). If in (1.11) we set z + Ah) and use condition (1.141, we get
=
(I + A A H ) - ' ( y
(Here we have assumed that A is symmetric; the general case follows by Theorem 2.4 in Chap. 2.) We get the estimate IA,+ylIc(1 + I f
I)
VA
> 0,
where C is independent of A and f . This implies that y estimate (1.15) holds. Corollary 1.4.
Zn Corollaty 1.3, assume in addition that f
(I + AA,)-'(y
+ Ah) E K
for some h
E
E
E
D(AH) and
H and
H and all A > 0. (1.16)
Then the solution y to variational inequality (1.3) belongs to D(A,), and the following estimate holds: lAyl IC(I + I f l )
Vf
E
H.
(1.17)
3.1.2. The Obstacle Problem Throughout this section, R is an open and bounded subset of the Euclidean space RN with a smooth boundary dR. In fact, we shall assume that dR is of class C 2 .However, if R is convex this regularity condition on dR is no longer necessary.
130
3. Controlled Elliptic Variational Inequalities
Let V = H'(R), H
=
L2(R), and A : V + V' be defined by N
a2 > 0. If
by
Here, ao,a,, E LYR) for all i, j
c
=
1,. .., N , a,,
N
a , ( x ) 2 0,
2
t,1=1
a,,(x)6,6, 2 w11611~
=
a,,, and
V t E R N ,X
E
a,
(1.20)
where w is some positive constant and 11. IIN is the Euclidean norm in R N . If a 1 = 0, we shall assume that a o ( x ) 2 p > 0 a.e. x E R. The reader will recognize of course in the operator defined by (1.18) the second order elliptic operator
c N
AOY
=
-
( a l l y x , ) , , + aoy
(1.21)
with the boundary value conditions aIy
dY + a2 =0 dU
in d R ,
(1.22)
where d / d u is the outward normal derivative, d
-y
dv
c N
=
i,j=1
a i j y x cos( , u, X i ) .
(1.23)
Similarly, the operator A defined by (1.19) is the differential operator (1.21) with Dirichlet homogeneous conditions: y = 0 in dR.
3.1. Elliptic Variational Inequalities. Existence Theory
131
Let I) E H2(R) be a given function and let K be the closed convex subset of V = H’(R) defined by K
=
{ y E V ;y ( x ) 2 $ ( x ) a.e. x
E
a}.
(1.24)
Note that K # 0 because $+= max($,O) E K . If V = Hd(fl), we shall assume that $(x) 4 0 a.e. x E dR, which will imply as before that K # 0. Let f E V’. Then, by Corollary 1.3, the variational inequality ( 1.25) VZ E K a(y,y -z ) I ( Y - z7f) has a unique solution y E K . Formally, y is the solution to the following boundary value problem known in the literature as the obsfacfeprobfem,
in R + = { x E R ; y ( x ) > $ ( x ) } , A,y2f, y > $ inR,
A,y = f
y = $
inR\R+
dY
- = -
’ du
a* du
in d R + \ dR,
(1.26) (1.27)
Indeed, if $ E C ( a ) and y is a sufficiently smooth solution, then R + is an open subset of R and so for every a E C;(R+) there is p > 0 such that y f p a 2 $ on R, i.e., y f p a E K . Then if take z = y f p a in (1.251, we see that
Hence, A,y = f in 9 ’ ( f l + ) . Now, if we take z = y a , where a
+
E
H’(R) and a
2
0 in R, we get
and therefore A,y 2 f in S’(Sz). The boundary conditions (1.27) are obviously incorporated into the definition of the operator A if a2 = 0. If a2 > 0, then the boundary 0 conditions (1.27) follow from the inequality (1.25) if a,$ + a2 d $ / d u I a.e. in dR (see Theorem 1.3 following). As for the equation d y / d u = d $ / d u in dR+, this is a transmission property that is implied by the conditions y 2 I) in R and y = $ in d R +, if y is smooth enough.
132
3. Controlled Elliptic Variational Inequalities
In problem (1.261, (1.27), the surface d o + \ dQ = S, which separates the domains R + and R \ fi' is not known a priori and is called the free boundary. In classical terms, this problem can be reformulated as follows: Find the free boundary S and the function y that satisfy the system A,y =f y = $ a,y
dY + a2 = 0 dV
inR+, dY a* _ --
inR\R+,
dv
dv
in S,
in dR.
In the variational formulation (1.29, the free boundary S does not appear explicitly but the unknown function y satisfies a nonlinear equation. Once y is known, the free boundary S can be found as the boundary of the coincidence set { x E a; y ( x ) = + ( x ) } . There exists an extensive literature on regularity properties of the solution to the obstacle problem and of the free boundary. We mention in this context the earlier work of Br6zis and Stampacchia [ll and the books of Kinderlehrer and Stampacchia [l] and A. Friedman [l], which contain complete references on the subject. Here, we shall present only a partial result. Proposition 1.1. Assume that aij E C'(fi), a , E Lm(R), and that conditions (1.20) hold. Further, assume that I!,I E H2(R) and
a*
a,@ + a2 - I0 dV
a.e. in dR.
(1.28)
Then for every f E L2(SZ) the solution y to variational inequality (1.25) belongs to H ' ( a )and satisfies the complementarity system
ax. x E R,Y(X) 2 (A,Y(X) -f(X))(Y(X) - *(XI) = 0 A,y(x) 2f(x) a.e. x E SZ,
*(XI,
(1.29)
along with boundary value conditions
(1.30) Moreover, there exists a positive constant C independent off such that IlyIlfP(n) IC(llfIlL2(n)
+ 1).
(1.31)
133
3.1. Elliptic Variational Inequalities. Existence Theory
Proof. We shall apply Corollary 1.4, where H = L 2 ( 0 ) ,V = H ' ( 0 ) (respectively, V = H d ( 0 ) if a2 = O), A is defined by (1.18) (respectively, (1.19)), and K is given by (1.24). Clearly, the operator A,: L 2 ( 0 )-+ L 2 ( 0 )is defined in this case by (AHY)(X)
D(A , ) A
=
=
(A,y)(x)
i y
E H'(
ax.
X E
0 ) ;a l y
We shall verify condition (1.16) with h
> 0 the boundary value problem
0, Y
ED(A,),
dY + a2 = 0 a.e. in d 0 . dV } = A,$.
+ AA,w = y + AA,$ dw alw + a2 - = 0 in d 0 , dV w
To this end, consider for in R ,
which has a unique solution w E D ( A , ) . Multiplying this equation by (w - $)-E H'(R) and integrating on 0, we get, via Green's formula,
( y - $)(w - $)- du 5 0 . Hence, (w - $I-= 0 a.e. in 0 and so w E K as claimed. Then, by Corollary 1.4, we infer that y E D ( A , ) and II&yIlL2(n) 5 C(llfllLqn)
+ l),
and since dR is sufficiently smooth (or R convex) this implies (1.31). Now, if y E D ( A , ) , we have
and so by (1.25) we see that jR
( A , y ( X ) - f( x ) ) ( y ( X ) - Z( x ) ) du
I0
VZ E K .
(1.32)
134
3. Controlled Elliptic Variational Inequalities
The last inequality clearly can be extended by density to all z where
K O= { u If in (1.32) we take z get
E
=
L 2 ( R ) ; u ( x ) 2 + ( x ) a.e. x
=
0).
KO,
(1.33)
+ + a , where a is any positive L 2 ( 0 )function, we
(A,y)(x) -f(x) 2 0
Then, for z
E
E
a.e. x
E
R.
+, (1.32) yields
which completes the proof. We note that under assumptions of Theorem 1.3 the obstacle problem can be equivalently written as ( 1.34)
dZ,o(y)
=
i
u E L2(
I
a ) ;1u( x ) ( y ( x ) - z( x ) ) dr 2 0 v z E KO R
or, equivalently,
where p : R
+
2R is the maximal monotone graph, if r > 0, if r = 0, if r < 0.
(1.35)
Hence, under the conditions of Theorem 1.3, we may equivalently write the variational inequality (1.25) as
a,y
dY + ff2 =0 dV
a.e. in d R ,
(1.36)
3.1. Elliptic Variational Inequalities. Existence Theory
135
arid as seen in Section 2.2, Chapter 2, it is equivalent to the minimization problem
(1.36)' where j : R
is defined by
+
j(r)
=
ifr20 otherwise.
(0
+m
( 1.37)
As seen elsewhere, Eq. (1.36) can be approximated by the smooth boundary value problem A o y + p,(y - $) = f
a.e. in fl, ( 1.38)
where P,(r) = -(l/&)r- ( p, is the Yosida approximation of p). In this context, we have a more general result. Let p be a maximal monotone graph in R X R, and let $ E H'(fl). Let p, = ~ - ' (-l (1 + ~ p ) - 'be ) the Yosida approximation of p. Then, for each f E L'(fl), the boundary value problem AOY + PAY a,y
+ a'
$1 = f
in
JY
in d f l
-= 0 dV
f l 7
( 1.39)
has a unique solution y,f E H'(fl). (Problem (1.39) can be written as A,y p,(y - #) = f , where y -+ p,(y - t,h) is monotone and continuous in ~ ' ( f l ) . )
+
Proposition 1.2.
Assume thaf
Then y,f
+
yf
strongly in ~ ' ( f l )weak& , in ~ ' ( f l ) ,
(1.41)
136
3. Controlled Elliptic Variational Inequalities
where y,f is the solution to boundary value problem (1.39).Moreover, i f f , + f weakly in L’(R), then y,f. + y f
w e a k l y i n H 2 ( R ) ,strong&inH’(R).
ProoJ Let y, = y,f.. Multiplying Eq. (1.39) by p,(y, on R,we get, by Green’s formula,
(1.42)
9 ) and integrating
Hence
Inasmuch as ( d / d u X y , -
9 ) - (a$/&
9)P,(Y,
-
9 ) = -(a1/a2XY,
+ ( a 1 / a 2 ) 9Be() 9 ) s 0 in
-
d f l , we infer that
9 ) P,(y,
-
is bounded in L2(R)
( P,( y, - @)}
and {A,y,) is bounded in L2(R>.We may conclude, therefore, that {y,) is bounded in H2(R) and on a subsequence, again denoted (ye},we have Y,
+
weakly in H 2 ( R ) ,strongly in H ’ ( R ) , (1.43)
Y
P,(Y, - 9 ) 5 A,y,=A,y, -A,y +
weakly in L2(R),
( 1.44)
weaklyin L2(R).
(1.45)
Clearly, we have Aoy+ dY
5=f
a’ - + a , y dU
=
0
a.e. in R a.e. in J R .
On the other hand, if denote by B c L2(R) X L’(fl) the operator By = ( w E L2(fl); w ( x ) E P ( y ( x ) - $ ( x ) ) a.e. in R},then B is maximal monotone and its Yosida approximation B, is given by B,(y)
=
P,(y -
9)
a.e. in R.
3.1. Elliptic Variational Inequalities. Existence Theory
137
Then, by (1.44) and Proposition 1.3 in Chapter 2, we deduce that 6 E By, i.e., ( ( x ) E p ( y ( x ) - @(XI). Hence y = y f , thereby completing the proof of Proposition 1.2. In particular, Condition (1.40) is satisfied if p is given by (1.35) and so Proposition 1.1 can be viewed as a particular case of Proposition 1.2. A simple physical model for the obstacle problem is that of an elastic membrane that occupies a plane domain R and is limited from below by a rigid obstacle J/ whilst it is under the pressure of a vertical force field of density f. We assume that the membrane is clamped along the boundary dR (see Fig. 1.1). It is well-known from linear elasticity that when there is no obstacle the vertical displacement y = y ( x ) , x E R, of the membrane satisfies the Laplace-Poisson equation. In the presence of the obstacle y = @ ( x ) , the deflection y = y ( x ) of the membrane satisfies the system (1.26). More precisely, we have -Ay =f
in { x
-Ayzf,
y z @
y=O
E
R; y ( x ) > @ ( x ) ) , inn,
in dR.
( 1.46)
The contact region { x E R; y ( x ) = @(XI) is not known a priori and its boundary is the free boundary of the problem. Let us consider now the case of two parallel elastic membranes loaded by forces f,, i = 1,2, that act from opposite directions (see Fig. 1.21.The variational inequality characterizing the equilibrium solution y is (see, e.g.,
Figure 1.1.
138
3. Controlled Elliptic Variational Inequalities
Figure 1.2.
Kikuchi and Oden [ll)
VY, . V ( Y , - 21) dU + Pz 5
/-fl(YI
+
- 21)
/ VY,
/n
R
fz(Y2
*
V Y ,
- 22) d X
dU
V(2,
- 22)
7
22)
K7 (1.47)
where f, 2 0, f, 5 0, and
K = { ( y , , y , ) €H,'(R) x H , ' ( R ) ; y , - y , s f a . e . i n R } .
(1.48)
Here, pl, p2 are positive constants, f is the distance between the initial positions of the unloaded membranes, and y , ( x , , x , ) 2 0, y,(x,, x , ) 5 0 are the deflection of the membranes 1 and 2 in ( x , ,x , ) = x . This problem is of the form (1.251, where H = L 2 ( f l )X L2(R), I/ = H,'(R) x H,'(R), K is defined by (1.48), f = ( f l , f,), and
4 Y , 2)
= PI
/ VY, R
v z , dU + Pz for Y
Formally, the solution y free boundary problem
AYI =f,, y, -y, 5 1 - P , A Y , Sf17 y, = 0 , -PI
=
=
/ VY, R
( Y , ,y2),
*
v z , dU
=
(zl, 2,) E I/ x I/.
( y , ,y,) to problem (1.47) is the solution to the
in { x ; Y d X ) -Pz AY, = f 2 in R , -P2 AY, 2f2 in R , y,=O in dR.
-Yz(X)
< f } 7
139
3.1. Elliptic Variational Inequalities. Existence Theory
The free boundary of this problem is the boundary of the contact set { x ; y , ( x ) - y,(x> = I ) . An important success of the theory of elliptic variational inequalities has been the discovery made by C. Baiocchi [l]that the mathematical model of
the water flow through an isotropic homogeneous rectangular dam can be described as an obstacle problem of the type just presented. Let us now briefly described this problem. Denote by D = (0, a ) X (0, b ) the dam and by Do the wetted region (see Fig. 1.3).The boundary S that separates the wetted region Do from the dry region D, = D \ Do is unknown and it is a free boundary. Let z be the piezometric head and let p(xI,x2)be the unknown pressure at the point (x1,x2)E D . We have z = p + x , in D and, by the D'Arcy law (we normalize the coefficients), Az=O
inD,.
(1.49)
Note also that z satisfies the obvious boundary conditions z dz
-=0 dx2
=
h , in AF,
inAB,
z =x2 dZ
- -dU
in S
U
0 in S ,
where d / d u is the normal derivative to S.
GC,
z
=
h , in BC,
(1 S O )
140
3. Controlled Elliptic Variational Inequalities
Introduce the function
and consider the Baiocchi transformation
Lemma 1.1. The function y satisfies the equation Ay
(1.51)
in g'( D)
= XD,
and the conditions
y > 0 in Do
y=O
in B \ D o ,
y =g
in d D , (1.52)
where
in FH
U HL U L C ,
g
=
0
g
=
1 2 -(x2 - h2) 2
We have denoted by
inCB,
g
=
1 2 -(x2 - h,) 2
xD, the characteristic function of
inAF.
Do.
Prooj We shall assume that y E H ' ( D ) and that the free boundary S is sufficiently smooth. Then, if x 2 = a ( x , ) is the equation of S, we have, for every test function cp E C,"(D),
141
3.1. Elliptic Variational Inequalities. Existence Theory
J-YA+=J
Dn
c p ( ~ I ~ ~ 2 ) ~ = x D , ( 4 0 )VcpECXD),
and Eq. (1.51) follows. Conditions (1.52) follow by the definition of y and by (1.50).
W
By Lemma 1.1, we may view y as the solution to the obstacle problem
- A y 2 -1, y 2 0 in D, Ay=1 in(xED;y(x)>O), y=g in d D ,
( 1.53)
or, in the variational form,
Vy * V(y
- u ) clx
+ JD(y
- u ) clx I0,
Vu E K,
(1.54)
where K = { u E H ’ ( D ) ; u = g in dD, u 2 0 in D). By Corollary 1.3, we conclude that problem (1.54) (and consequently the dam problem (1.53)) has a unique solution y E K. The free boundary S can be found solving the equation y(x,, xz) = 0. For sharp regularity properties of y and S, we refer to the book of A. Friedman [l].
142
3. Controlled Elliptic Variational Inequalities
3.1.3. An Elasto-Plastic Problem
Let R be an open domain of RN and let a: H,'(R) defined by
X
Hd(R)
-+
R be
( 1.55)
Introduce the set K
=
( y E Hd(R); IIVy(x)llN I1 a.e. x
E
R},
(1.56)
where II . I I N is the Euclidean norm in RN, and consider the variational inequality y
E
a ( y , y - 2)
K,
I(
y -z , f )
Vz
E
K,
(1.57)
where f E H-'(R). By Corollary 1.3, this problem has a unique solution y . If y is sufficiently smooth, then it follows as for the obstacle problem that -Ay
in { x E R ; I l V y ( x ) l l ~< I} in R, = R \ R e ,
=f
llVyllN = 1
in dR.
y=O
=
Re, (1.58)
The interpretation of problem (1.58) is as follows: The domain can be decomposed in two parts Re (the elastic zone) and R, (the plastic zone). In R e , y satisfies the classical equation of elasticity whilst in R,, I l V y ( x ) l l ~ = 1; the surface S that separates the elastic and plastic zones is a free boundary, which is not known a priori and is one of the unknowns of the problem. This models the elasto-plastic torsion of a cylindrical bar of cross-section R that is subject to an increasing torque. The state y represents in this case the stress potential in R. As noted earlier, (1.57) is equivalent to the minimization problem min{ f a ( z , z ) - ( z ,f ) ; z
E
K).
(1.59)
If f E L2(R) and dR is sufficiently smooth, then the solution y to (1.57) belongs to H2(R)(BrCzis and Stampacchia [l]). It is useful to point out that y
=
lim ye
E'O
in L'(R),
3.1. Elliptic Variational Inequalities. Existence Theory
where y ,
E
143
Hd(fl) is the solution to the boundary value problem -
Ay - div dh,( Vy)
=
in 0,
f
indfl,
y=O where h , : R N .+ R N is defined by
he(u)
=
I"
if l l u l l ~< 1,
(IlullN -
if l l u l l ~2 1,
2E
and ah,: R N -+ R N is its differential, i.e., dh,(u)
=
i"
if IIuIIN < 1,
( u l l u l l ~- 1)
if IlullN 2 1.
EIIUIIN
Let us calculate starting from (1.59) the solution to problem (1.57) in the case where fl = (0,l). If make the substitution
w( x )
=
l X Z ( s) ds,
x E (0, l ) ,
0
then problem (1.59) becomes i n f ( ~ / o l W ' ( x ) d X - / '0w ( x ) l I fX ( s ) d s d X ; w where U = { u E L2(0,1); lu(x)l to (1.60) satisfies the equation
I1
a.e. x
E
where Nu is the normal cone to U. Hence,
y'(x)
=
w(x) =
1
1
E
(0, l)}.Hence, the solution w
144
3. Controlled Elliptic Variational Inequalities
3.1.4. Elliptic Problems with Unilateral Conditions at the Boundary
Consider in R c R N the boundary value problem cy
-
Ay
dY
=f
+ P(Y) dV
3
g
y
=
0
in R,
(1.61)
r, , in r,, in
rl and r2 are two open, smooth, and disjoint parts of dR, rl u r, = dR, f E L2(R), g E L2(rl), c is a positive constant and p is a
where -
maximal monotone graph in R X R. Let j : R -+ R be a lower semicontinuous, convex function such that dj = p. We set V={y~H~(fi);y=oinr,}
and define the operator A: V + V‘ by
p(z)
=
1j (
z ) dz
Vz
E
V,
rl
and let fo
E
V’ be defined by
(fo 7 z > =
1nf ( x ) z ( x ) dx + 1g ( x ) z ( x W
Vz
E
V.
Vz
E
‘v,
rl
By Corollary 1.2, the variational inequality a ( y , y -2)
+ V(Y) - V(Z) 5
( f 0 , Y -2)
(1.63)
has a unique solution y E V. Problem (1.63) can be equivalently written as min{+a(z,z)
+ p(z)
- ( f o , z ) ;z E
v}.
(1.64)
The solution y to (1.63) (equivalently, (1.64)) can be viewed as generalized solution to problem (1.61). Indeed, if in (1.63) we take z = y - a, where a E C t ( R ) , we get (cya
In
+ v y . V a ) dx = 1nfa dx.
3.1. Elliptic Variational Inequalities. Existence Theory
Hence, cy - b y
=
f in g ' ( R ) . Now, multiplying this by y R,we get
z E V , and integrating on
- z,
145
where
More precisely, we have
where ( ;) is the pairing between H112(rl) and H-'I2(r1)(if y E H'(R) and A y E L2(R), then y E H112(rl)and d y / d v E W1I2(rl) (see Lions and Magenes [l]). Then, by (1.631, we see that
( 2 )
4
( j ( y ) -j(z))dx I g - -,y
-z
vz
E
v.
Hence, if g - d y / d v E L2(rl), we may conclude that g - d y / d v a.e. in r l .Otherwise, this simply means that
E
p(y)
where a+: H1'2(rl)-, H-'Iz(Fl) is the subdifferential of the function 6: H112(rl)+ R defined by
In the special case where rl = dR and g = 0, then as seen in Section 2.2, Chapter 2 (Proposition 2.10, y E H2(R) and llylIH2(n) IC(I
+ IlfllL2(n))
Vf E L2(R),
(1.65)
where C is independent of f. Moreover, we have y
=
lim0 y ,
&+
weakly in H 2 ( R ) ,strongly in H ' ( R ) ,
(1.66)
146
3. Controlled Elliptic Variational Inequalities
where y , E H2(R) is the solution to the approximating problem cy, - Ay, dY,
+ p,(y,) dU
=
f
in R,
=
0
in d R ,
=
1 -(r
(1.67)
and p,(r)
E
-
(1
+ E@)-'r)
Vr E
R, E > 0.
In this case, we have a more precise result. Namely: Proposition 1.3. Let f, + f weakly in L2(R). Then the solutiony, E H2(R) to problem (1.67) is weakly convergent in H2(R) and strongly convergent in H'(R) to the solution y f to problem (1.61).
Proof: As seen in the proof of Proposition 2.11, Chapter 2, we have for y , an estimate of the form (1.63, i.e.,
where C is independent of p, (i.e., of denoted E , we have Y, 4 Y
E).
Hence on a subsequence, again
weakly in H ~ ( R )strongly , in H ' ( Q ) ,
dY
weakly in H1I2(an),strongly in L ~ ( R ) ,
P&(Y&)
5
strongly in L~( d R ) ,
Y&
Y
strongly in L ~an). (
dY,
-4 du du +
+
(1.68)
Arguing as in the proof of Proposition 1.2, we see that 6 E p ( y > a.e. in dR. Then, letting E tend to zero in (1.681, we see that y is the solution to W (1.61). We shall consider now some particular cases. If j ( r ) = golrI, r E R, where go is some positive constant, then
3.1. Elliptic Variational Inequalities. Existence Theory
147
and so problem (1.61) becomes
cy dY
- +go
dv
Ay
-
sign y
=f
0
3
in R , in dR.
( 1.69)
Equivalently, cy-Ay=f
inR, y
JY
+ golyl = 0
a.e. in ail.
(1.70)
The boundary conditions can be rewritten as
=o y's o LO
if
1-1
dY dV
< go ,
dY dv
if - = g o ,
a.e. in dR.
dY
if - = - g o , dV
Hence, there are apriori two regions r' and r2on dR where Idy/dvl < go and I d y / d v l = g o , respectively. However, r' and r2 are not known, so problem (1.69) is in fact a free boundary problem and as seen before it has a unique solution y E H2(R). Problem (1.70) models the equilibrium configuration of an elastic body C! that is in unilateral contact with friction on dR (see Duvaut and Lions [l], Chapter IV).
The Signon'ni Problem. Consider now problem (1.61) in the special case g = 0, and where rl = dR, r2= 0, r > 0, r=0, r < 0,
V r ER.
(1.71)
i.e. cy-Ay=f
dY
a.e.inR,yLO, - 2 0 , dV
dY
y-=O dv
a.e.indR. (1.72)
148
3. Controlled Elliptic Variational Inequalities
This is the famous problem of Signorini, which describes the conceptual model of an elastic body R that is in contact with a rigid support body and is subject to volume forces f. These forces produce a deformation of R and a displacement on dR, with the normal component negative or zero. Other unilateral problems of the form (1.61) arise in fluid mechanics with semipermeable boundaries, climatization problems or in thermostat control of heat flow, and we refer the reader to the previously cited book of Duvaut and Lions [l]. In mechanics, one often meets problems in which constitutive laws are given by nonmonotone multivalued mappings that lead to problems of the following type (see P. D. Panagiotopoulos [l]):
where A: V .+ V' is defined by (1.18) and y is the generalized gradient (in the sense of Clarke) of a locally Lipschitz function 4: V + R. In particular, we may take
where j E L";b,(R) is such that lj(01IC(1 + l&Ip-'), 5 E R, and SO (see . s))), R(j(y(x>))I a.e. x E d+(y) c {w E L'(a); W ( X > E [fi(J(Y( (2.75) in Chapter 2). This is a hemivariational inequality, and the existence theory developed in the preceding partially extends to this class of nonlinear problems (see the book [l] by Panagiotopoulos, and the references given there).
3.2. Optimal Control of Elliptic Variational Inequalities
In this section, we shall discuss several optimal control problems governed by semilinear elliptic equations and in particular that governed by elliptic variational inequalities and problems with free boundary. The most important objective of such a treatment is to derive a set of first order conditions for optimality (maximum principle) that is able to give complete information on the optimal control. Since the optimal control problems governed by nonlinear equations are nonsmooth and nonconvex, the standard methods of deriving necessary conditions of
3.2. Optimal Control of Elliptic Variational Inequalities
149
optimality are inapplicable here. The method is in brief the following: One approximates the given problem by a family of smooth optimization problems and afterwards tends to the limit in the corresponding optimality equations. An attractive feature of this approach, which we shall illustrate on some model problems, is that it allows the treatment of optimal control problems governed by a large class of nonlinear equations, even of nonmonotone type and with general cost criteria.
3.2.1. General Formulation of Optimal Control Problems
Let V and H be a pair of real Hilbert spaces such that V is a dense subset of H and
VcHcV’ algebraically and topologically. We have denoted by V’ the dual of V and the notation is that of Section 1.1. Thus (.; ) is the pairing between V and V’ (and the scalar product of H ) and II 1 , I * I are the norms in V and H, respectively. Consider the equation
where A: V + V’ is a linear continuous operator satisfying the coercivity condition ( A u , u ) 2 wllull cp: V
+
2
Vu E V
for some w > 0,
R is a lower semicontinuous, convex function,
(2.2)
dcp: V + V’ is the
subdifferential of cp, B E L(U, V’),and f is a given element of V’. Here, U is another real Hilbert space with the scalar product denoted ( ,. ) and norm I * 10 (the controller space). As seen in Section 3.1 a large class of nonlinear elliptic problems, including problems with free boundary and unilateral conditions at the boundary, can be written in this form. The parameter u is called the control and the corresponding solution y is the state of the system. Equation (2.1) itself will be referred to as the state system or control system.
3. Controlled Elliptic Variational Inequalities
150
The optimal control problem we shall study in this chapter can be put in the following general form:
(PI Minimize the function
on ally
E
V and u
E
U satisfving the state system (2.1).
Here, g: H + R and h: U following conditions: (i)
+
R
are given functions that satisfy the
g is Lipschitz on bounded subsets of H (i.e., g is locally Lipschitz) and
bounded from below by an afine function, i.e.,
where (Y E H and C is a real constraint; (ii) h is convex, lower semicontinuous, and
(iii) B is completely continuousfrom U to V ' ,
The last hypothesis is in particular satisfied if the injection of V into H is compact. Roughly speaking, the object of control theory for system (2.1) is the adjustment of the control parameter u, subject to certain restrictions, such that the corresponding state y has some specified properties or to achieve some goals, which very often are expressed as minimization problems of the form (P). For instance, we might pick one known state y o and seek to find u in a certain closed convex subset Uo c U so that y = y o .Then the least square approach leads us to a problem of the type (PI, where
g(y)
=
31y
-
y0l2
and
h(u) =
if u E U,, elsewhere.
Other control problems such as that of finding the control u such that the free boundary of Eq. (2.1) (if this equation is a problem with free boundary) is as close as possible to a given surface S, though more
3.2. Optimal Control of Elliptic Variational Inequalities
151
complicated, also admit an adequate formulation in terms of (PI. This will be discussed further in a later section. Now let us briefly discuss existence in problem (PI. A pair (y*, u * ) E V x U for which the infimum in (P) is attained is called optimalpair and the control u* is referred as optimal control. Proposition 2.1. optimal pair.
Under assumptions (i)-(iii) problem (PI has at least one
Pro08 For every u E U we shall denote by y" E U the solution to Eq. (2.1). Note first that the map u -+ y" is weakly-strongly continuous from U to V. Indeed, if {u,} c U is weakly convergent to u in U, by assumption (iii) we see that
Bu,
+ Bu
strongly in V'
whilst by (2.1) and (2.2) we have wIIyUn- yumI12I IIBu, because d q is monotone in V lim y'n
n+m
=y
Now letting n tend to
X
-
B~,II*IIyUn- yUmII
V ' . Hence,
exists in the strong topology of V .
+m
in
Ay"n
+ d q ( y".)
E Bu,
+f ,
we conclude that y = y", as claimed. Now let d = inf{g(y") + h(u); u E U ) . By assumptions (i), (ii), it follows that d > - w . Now, let (u,} c U be such that d I g(y,)
+ h ( u , ) I d + n-'
Vn,
where y, = y".. By (i) and (ii) we see that {u,) is bounded in U, and so on a subsequence (for simplicity again denoted (u,}) we have u, + u*
y,
-+
y*
weakly in U , strongly in V ,
because B is completely continuous. Since as seen in the preceding y* = y"' and g(y*) h(u*) = d (because g is continuous and h is weakly lower semicontinuous), we infer that (y*, u * ) is an optimal pair for problem (PI.
+
152
3. Controlled Elliptic Variational Inequalities
From the previous proof it is clear that as far as existence is concerned, hypotheses (i)-(iii) are too strong. For instance, it suffices to assume that g is merely continuous from V to R.Also, if g is convex, assumption (iii) is no longer necessary because g is, in this case, weakly lower semicontinuous on V. On the other hand, it is clear that the convexity assumption on h cannot be dispensed with since it assures the weak lower semicontinuity of the function u + g ( y " ) + h(u), a property that for infinite dimensional controllers space U is absolutely necessary to attain the infimum in problem (PI.
3.2.2. A General Approach to the Maximum Principle Here we shall discuss a general approach to obtain first order necessary conditions of optimality for optimal control problems and in particular for problem (PI. Most of the optimal control problems that arise in applications can be represented in the following abstract form:
(6) Minimize the functional on all ( y , u ) E X
X
U, subject to state equation F ( y ,U)
=
0.
(2.7)
Here, L : X X U + R is a continuous function, h: U + R = I - m, +..I is convex and lower semicontinuous, and F : X X U + Y is a given operator, where X, Y,U are Banach spaces. We shall assume of course that for every u E U, Eq. (2.7) has a unique solution y " E X. Let us first assume that the map u + y" is Giteaux differentiable on U and that its differential in u E U , z = D,y"(u), is the solution to the equation
where Fy and F, are the differentials of y -+ F ( y , u ) and u + F ( y , u ) , respectively. (This happens always if the operator F is differentiable.)
3.2. Optimal Control of Elliptic Variational Inequalities
153
Let ( y * ,u * ) be an optimal pair for problem (PI. We have A - l ( L ( y U * + A " ,+~ *A u ) - L ( Y * , u * ) ) + A-'(h(u* Vu E U, A
+ Au)
-
h ( u * ) )2 0.
> 0, and if L is (GPteaux) differentiable on X X U this yields
(L,(Y*,U*),z)
+ (L,(y*,u*),u) + h'(u*,u)2 0
VUE U , (2.9)
where h ' is the directional derivative of h and (.,. 1, ( * ,. ) are the dualities between X, X * and U, U*, respectively. Denote by F; the adjoint of F y , and further assume that the linear equation (2.10)
q y * , u * ) p = L,(Y*, u * ) (the dual state equation) has a solution p we see that
E
(Ly(y*,u*),z)= (p,F,(y*,u*)z)
Y*. Then, by (2.8) and (2.10)
=
-(F;L(y*,u*)p,v),
and substituting in (2.9) yields VUE
(F;L(Y*>U*)P - L,(y*,u*),u) Ih'(u*,u)
u.
By Proposition 2.8 in Chapter 2, this implies
F,*( y * , u * ) p - L,( y * , u*) E dh( u * ) .
(2.11)
Equations (2.71, (2.101, and (2.11) taken together represent the first order optimality system for problem (P), and if F is linear and L is convex these are also sufficient for optimality. If L and F are nonsmooth, one might hope to obtain a maximum principle type result in terms of generalized derivatives of L and F by using the following method: Assume that the control space U is reflexive and that there are a family (F,),,, of smooth operators from X X U to Y and a set of differentiable functions L,: X X U + R such that
(i)'
For every E > 0 and u E U the equation F,(y, u ) = 0 has a unique solution y =y," and there exists p 2 1 such that lly:llx
I
C(1
+ lull;)
Vu E
u.
(2.12)
'
154
3. Controlled Elliptic Variational Inequalities
(ii)' If {u,} is weakly convergent to u in U then { y , " ~ is } strongly convergent to y" in X and liminf L,(y,Ue,u,) &+
0
2
(2.13)
L(y",u).
Moreover, lim, L,(y,", u> = L ( y " , u ) V u E U. (iiiY L,(y, u> 2 -C(lyl + lul) V ( y ,u> E X X U , where C is independent of E . ~
Let ( y * , u * ) be any optimal pair in problem penalized problem
i
1
(P,) inf ~ , ( y , u )+ h ( u ) +
-IU
2P
- u*IZP;
(el. We consider the
F,(Y,u)
= O,U E
1
u
,
and assume that this problem has a solution (y,,u,) (this happens, for instance, if the function u -+ L,(y,U, u ) is weakly lower semicontinuous). Then, we have Lemma 2.1. For
E
-+
0, we have u, + u*
y, Pro05
For all
E
-+
y*
strongly in U , strongly in X .
> 0 we have
and by condition (2.12) and assumption (iiiY, {u,) is bounded in U and so, on a subsequence E, 0. -+
weakly in U ,
u&n-+ ii y,,
-+
y
=y z
strongly in X .
Since h is weakly lower semicontinuous, we have lim inf h( u,,) 2 h( U ) E" +
and so, by assumption (ii)',
0
3.2. Optimal Control of Elliptic Variational Inequalities
155
Hence, J = y * , U = u * , and u,, -+ u* strongly in U on some subsequence {qJ.Since the limit u* is independent of the subsequence, we conclude that u, + u* strongly in U, as claimed. Now, since problem (6,) is a smooth optimization problem of the form (61, (y,,u,) satisfy an optimality system of the type (2.10), (2.11). More precisely, there is p , E Y * that satisfies along with (y,, u,) the system
F&(Y&, U & > = 07 ( F & ) ; ( Y &U, & ) P &= ( L & ) J Y CU"), , (F,),*(y,,u,)p,
-
( L & ) u ( Y & , uE &W ) u , ) + J ( u , - u*)Iu,
- u*I:p-27
(2.14)
where J : U + U* is the duality mapping of the space U. By virtue of Lemma 2.1, we may view (2.14) as an approximating optimality system for problem (P). If one could obtain from (2.14) sufficiently sharp a priori estimates on p , , one might pass to limit in (2.14) to obtain a system of first order optimality conditions for problem (6). We shall see that this is possible in most of the important cases, but let us see first how this scheme looks for problem (P) considered in the previous section. We shall assume that g and h satisfy assumptions (9, (ii) (with the eventual exception of condition (2.511, B E L ( U , V r ) , f~ V r , and the injection of V into H is compact. Let ( y * ,u * ) be an optimal pair in problem (PI. Then, we associate with this pair the penalized optimal control problem (adapted penalty):
Here, g" is defined by (see formula (2.79) in Chapter 2)
where P,u = Cy,, u,e,, u = Cy=, u,e,, A,,T = Cy=,u,e,, {e;} an orthonorma1 basis in H , and cp": V + R is a family of convex functions of class C2 on V such that cp"(u) 2 - C ( llull
lim cp"(u) = cp(u)
E-0
+ 1) Vu E V
Vu E V , E
> 0,
156
3. Controlled Elliptic Variational Inequalities
whilst, for any weakly convergent sequence u,
+
u
in V ,
lim inf cp"( u,) 2 cp( u ) . E'O
By Proposition 2.1, we know that problem (P,) has at least one solution ( y , , u,). Since our conditions on cpE clearly imply assumptions (i)', (ii)', (iii)' of Lemma 2.1, where X = H, Y = V ' , and F,(Y,U) =AY L & ( Y ,u ) we conclude that, for
E +
u,
+
y,
+ y*
+ Vcp"(Y> - B u -f,
=g"(y)
+ h(u),
0,
strongly in U , strongly in H, weakly in V .
u*
On the other hand, the optimality system for problem (P,) is (see (2.14)) AY, + Vcp"(Y , ) -
=
Bu, + f,
A*P&- V*cp"(Y&)P& = Vg"(y,), B*p, E d h ( u , ) + U, - u*.
(2.16) (2.17) (2.18)
Note that since V*cp"(y,) E L(V,V ' ) is a positive operator, Eq. (2.17) has a unique solution p , E V. Moreover, since {Vg"(y,)}is bounded on H (because g is locally Lipschitz), we have the estimate wllp,ll IIVg"(y,)l Hence, on a subsequence
E, +
P&+P A*p, + A*p Vg"(y,)
-3
5
c
v.5 > 0.
0, we have weakly in V , strongly in H, weakly in V', weakly in H.
By Proposition 2.15 in Chapter 2 we know that 6 E dg(y*),where dg is the generalized gradient of g whilst by (2.18) it follows that B*p E dh(u*) or, equivalently, u* E dh*(B*p). Summarizing, we have proved: Proposition 2.2. Let ( y * ,u * ) be an optimalpair orproblem (PI. Then there is p E V such that
-A*p - 77 E d g ( y * ) , u* E dh*( B * p ) ,
(2.19)
3.2. Optimal Control of Elliptic Variational Inequalities
where in V.
=
w - lim,+ oV2cp"(y,)p, in V ' , p ,
+
p , and y,
+
157
y * weakly
We may view (2.19) as the optimality system of problem (P). There is a large variety of possibilities in choosing the approximating family {cp"). One of these is to take cpE of the form (2.151, i.e.,
where n
=
[ E - ' I , P,, : V -,X,,An: R"
+
V and
(2.21)
or
if cp is lower semicontinuous on H (this happens, for instance, if cp is coercive on V ) . One of the interesting features of the adapted penalty problem (P,) is that every sequence u, of corresponding optimal controllers is strongly convergent to u*. If instead of (P,) we consider the problem
(P")
min{g"(y)
+ h( u ) ; Ay + Vcp"(y )
= Bu
+f
u E U},
then under assumptions (i)-(iii), (P") admits at least one optimal pair ( y E ,u E )and, on a sequence E, + 0, uE* + u: yen
+yf
weakly in H, strongly in V ,
where (yf ,u f ) is an optimal pair of (PI. The result remains true if replace h by a smooth approximation, for instance, h, as in (2.22), and one might try to calculate u s by a gradient type algorithm taking in account that the differential &$ of the function 4: u + R,
4(u) =g"((A
+ Vcp")-'(Bu +f)) + h , ( u ) ,
u E
U,
158
3. Controlled Elliptic Variational Inequalities
is given by d + ( u ) = dh,(u) - B*p,
where -A*p - V'cps( y ) p Ay
=
Vg"( u),
+ V p E (y ) = Bu + f .
Now, we shall present another approach to problem (PI, apparently different from the scheme developed in the preceding, but keeping similar features. To this end, we shall assume that, beside the assumptions (i) and (ii), the injection of V into H is compact, cp is lower semicontinuous on H, B E L(U, H I , f E H, and hypothesis (1.14) holds. Then, as seen earlier, the solution y to variational inequality (2.1) belongs to D ( A , ) , i.e., Ay E H. Let (y*, u * ) E D ( A , ) X U be an optimal pair for problem (P). We associate with (y*, u * ) the family of optimization problems
(Q,)
inf(g(y) + h ( u ) + e-l(cp(y) + cp*(u)
+ +lu - U * l 2 + 31.
-
(Y,.))
- u*Ih},
where the infimum is taken on the set of all (y, u , u ) E V subject to
X
U
X
H,
Ay=Bu-u+f. Here, u* = Bu* gate of cp.
+ f - Ay* E dcp(y*) c H
and cp*: H
+
R
is the conju-
Lemma 2.2. Problem Q, has at least one solution (y,, u,, u,) H.
E
V
X
U
X
(2.23) Since {u,} is bounded in H and {y,) is bounded in V,we may pass to limit in the previous inequality to get the existence of a minimum point. W
3.2. Optimal Control of Elliptic Variational Inequalities
Lemma 2.3. For
E +
159
0, u,
-,u*
strongly in U ,
u,
-+
u*
strongly in H ,
y,
+
y*
strongly in I/.
(2.24)
Proof: We have
+ E-'(cp(Y,> + cp*(V,)
g(Y€) + N u , )
+~(IU,
-
u * Iu2
-
(Y&lU&))
+ Iu, - u*I2) _< g ( y * ) + ' h ( u * )
VE > 0, (2.25)
because (see Proposition 2.5 in Chapter 2) cp(Y*) + 'p*(v*)= ( Y * , . * ) . Note also that
cpo(Y,)
+ cp*(.,)
Hence, on a subsequence
E, +
V
weakly in H,
-+
v.2 > 0.
2 0
(2.26)
0, we have
weakly in U ,
yen + j j E,,
(Y,,.,)
uEn+ ii u,.
NOW,letting
-
strongly in H, weakly in I/.
(2.27)
tend to zero in (2.25) and using (2.26) and (2.27) we get
Ig(y*)
+ h ( u * ) = inf P,
(2.28)
respectively, cp(Y)
+ cp*(V)
- (J,V) =
0,
because cp and cp* are weakly lower semicontinuous and
2 ( f , A J ) - (jj,Bii
+f)
=
-(J,V).
Then, by Proposition 2.5, Chapter 2, we infer that V E dcp(jj), and so by (2.28) we get (2.24).
160
3. Controlled Elliptic Variational Inequalities
Lemma 2.4. Let ( y e ,u,, u,) be optimal for problem p,. Then there is p , E V that satisfies the system
Here, dcp: V
+
-A*P, v,
E
B*p,
E
M Y & )+ &-‘(dcp(Y€) M Y , - EP, + E(V* -
(2.29)
dh(u,)
(2.31)
+ U, - u*.
(2.30)
7I),.
V’ is the subdifferential of cp: V + R.
Proof: Let us denote by L,(y, u , u ) the function L A ( y ? u , u )= g ^ ( y ) + h ( u ) + & - ‘ ( P A ( y )
+ cp*(v)
+ +(Iu - U * l 2 + Iu - U * I h ) + +(Iu - u,IU2 + IU - L J , ~ ~ ) ,
where g ” is defined by (2.15) and problem
(ph
(y,.))
A > 0,
by (2.22). Clearly, for every A > 0 the
inf(L,(y,u,u);u~U,u~H,Ay=Bu-u+f}
(2.32)
has a solution ( y A u, ” , u ” ) .We have Lh(YA7UA,Uh)
< g ” ( Y c ) + h ( u ~ +) YE) + q * ( u . )-
+ +(I.,
-
U*l2
+ Iu,
- U*lh)
(Y~7’e))
VA > 0.
Since {u”}),{u”}are bounded in U and H , respectively, we may assume that U” +
ii
weakly in U
u”
+
V
weakly in H
yA
+
7
weakly in V , strongly in H.
Then we have g ” ( y ” )+ g ( y ) , liminf cpA(yA)2 c p ( j j ) , A-0
liminf cp*(u”) 2 c p * ( ~ ) , A-rO
liminf h ( u ” ) 2 h ( E ) A+O
161
3.2. Optimal Control of Elliptic Variational Inequalities
This yields
+ &-I( <
Hence,
+ cp*(cv)
cp( J )
-
( 1 , V ) ) + +(I.
- U*l2
+ l i i - .*I;)
inf Q,.
1=y,, ii = u,, V = uE and for A
+
0
U”
-+
u,
strongly in U ,
u”
-+
u,
strongly in H,
Y ” -+YE
strongly in I/.
(2.33)
Now, since problem (2.32)is smooth one can easily find the corresponding optimality conditions. Indeed, we have
(vgA(yA),z)+ h r ( U h , W )
+ &-‘((vVA(YA),z) + ( c p * ) r ( u Ae, ) - ( 2 , ~ ” ) ( y A ,e ) ) + (d- u * , e )
+ ( u ” - u * , w ) + ( u A- U & , W ) 2 0
for all w
E
(2.34)
U, 8 E H and z satisfying the equation A Z = B W - ~ .
Now, let p ”
E I/
be the solution to
-A*p”
=
VgA(yA)+ &-‘(VpA(y”)
- u”).
(2.35)
Substituting (2.35)in (2.34), we get
+
+
- ( B * ~ ~ , w ) @ , P A ) h r ( u A , w )+ d ( ( c p * ) ’ ( d , e ) - (y^,e))
e)
+(dfor all w
E
U, 0
E
+ (u^
-
u*, e )
+ (u” - u * , w ) + (u” - u , , w ) 2 0
H. This yields
B*p” E dh(u”) + 2 ~ -” U, - u*,
(2.36)
-PA
(2.37)
E
& - * ( d c p * ( u ” )- y A )
+ 2uA - u* - u,.
We may equivalently write (2.37)as lJA E
acp(yA - &PA
+ & ( U * + U& - 2u”)).
(2.38)
162
3. Controlled Elliptic Variational inequalities
Then, substituting in (2.35) and multiplying the resulting equation by ( I + A dp)F'yA - yA+ & p A- E(U* + u, - 2q), we get the estimate VA
IlpAll IC
> 0,
because d p is monotone and (Vg*} is uniformly bounded on bounded subsets. Hence on a subsequence, A + 0, we have weakly in V , weakly in V ' ,
P A+P& A* p A+ A*p,
whilst by Propositions 2.15 and 1.3, in Chapter 2, VgA(yA)+ 6 E dg(y,)
weakly in H ,
7 E dp( y,)
weakly in V'
and VpA(y ^)
+
because {VpA(yA)) is bounded in V' and y A (2.36) it follows that B*p, E d h ( u , )
+ y,
strongly in V. Finally, by
+ U, - u*,
and V& E dcp(Y, - E P ,
+
E(U*
-
.&>>
thereby completing the proof. By virtue of Lemma 2.4 we may view (2.29H2.31) as an approximating optimality system for problem (P). Let 6, = u* - us. Then, we may rewrite (2.29) as -A*P,
E
MY,)
+ E-'(dv(Y&)
-
MY,
- E P , + &O&)>* (2.39)
If multiply this equation by p , - 0, and use the monotonicity of dcp along with the coercivity condition (2.21, we get 11p,11 5
because (Vg(yJ is bounded. Hence, there is a sequence
E, +
+
U," + u* +
u*
E
VE
> 0,
0 such that
weakly in V , strongly in H , weakly in V ' , strongly in U ,
P&, + P A*P&" A*P
%
c
dcp(y*)
strongly in H .
3.2. Optimal Control of Elliptic Variational Inequalities
Then, letting
E = E,
163
tend to zero in (2.29)-(2.31), we get -A*p B*p
E E
d g ( y * ) + w*, dh(u*),
(2.40)
where
in the weak topology of V ' . These conditions can be made more explicit in the specific problems that will be studied in what follows. 3.2.3. Optimal Control of Semilinear Elliptic Equations In this section, we shall study the following particular case of problem (P): Minimizeg(y) + h ( u ) o n a l l y subject to A,y
E H,'(R)
n H 2 ( R )andu
+ p ( y ) = f + Bu
E
U , (2.42)
in R , in dR.
y = o
(2.43)
Here, A , is the elliptic differential operator (1.211, i.e., N
AOY
=
-
C
i,j = 1
where a i j ,a,, E L"(R), a i j = aji for all i , j N
C
i,j=l
(2.44)
(aij(x)yx,)x,+ a , ( x ) ~ ,
a i j ( x ) t i t j2 W I I ~ I I ~
=
1,. . ., N , a , 2 0 a.e. in R,
~tE R
~ , a.e. x
E
a,
for some w > 0, p is a monotonically increasing continuous function on R, f E L2(R), and B E L(U, L2(R)). We shall assume that the functions g and h satisfy assumptions (i) and (ii) of problem (P). More precisely, we shall assume: (a) The function g : L2(R) + R is Lipschitz on bounded subsets and there are C E R and f,, E L2(R) such that
d Y )
2 (f0,Y) +
c
VY E L 2 ( R ) *
(2.45)
164
3. Controlled Elliptic Variational Inequalities
(b) The function h: U -+ R is convex, lower semicontinuous, and satisfies condition (2.5). Problem (2.421, (2.43) is a particular case of problem (P), where H L2(R), V = H,'(R), A : V -+ V' is defined by (1.191, i.e.,
=
( 2.46)
and ~ ( y =)
1j ( y ( x ) ) du, n
y
E
H,'(R);
dj
=
p, j : R
-+
R . (2.47)
The function g : L2(fl) -+ R may arise as an integral functional of the form (2.48) where g o : R x R -+ R is measurable in x and Lipschitz in y . We shall assume that R is a bounded and open subset of R N , either with smooth boundary (of class C2 for instance) or convex. As seen earlier, this implies that for every u E U, problem (2.43) has a unique solution y E H,'(R) n H2(R). Moreover, if h satisfies condition (2.9, then problem (2.42) has at least one optimal pair ( y , u). Theorem 2.1 following is a maximum principle type result for this problem.
Theorem 2.1. Let ( y * ,u * ) be any optimal pair in problem (2.42), (2.431, where p is a monotonically increasing locally Lipschitz function on R, and g , h satisjj assumptions (a), ( b ) . Then there exist functions p E H,'(R), q E L'(R), and 6 E L 2 ( R )such that A p E (L"(R))* and
-(Ap), T(X) EP(X)
If either 1 I N
-
q
=
6
a.e. in R ,
W y * ( x > > , 6(x) E d g ( y ( x ) ) B*p E dh(u*). I 3 or
(2.49) a*e*x
E
a,
(2.50) (2.51)
p satisfies the condition
P ' ( r ) IC ( l p ( r ) l + Irl
+ 1)
a.e. r
E
R,
(2.52)
3.2. Optimal Control of Elliptic Variational Inequalities
then Ap
E L’(R)
165
and Eq. (2.49) becomes -Ap - 77
=
6
a.e. in 0.
(2.53)
Here, d p and dg are the Clarke generalized gradients of p and g, respectively (see Section 2.3 in Chapter 21, (L”(R))* is the dual space of Lm(R)and (Ap), is the absolutely continuous part of Ap E (L”(R))*. We can apply the Lebesgue decomposition theorem to the elements p of the space (,!,“(a))*: p = pa+ ps,where pa E L’(R) is the absolutely continuous part of p and p, the singular part (see, e.g., Ioffe and Levin [l]). This means that there exists an increasing sequence of measurable sets R, C R such that m(R \ R , ) k 2 m 0 and p,(cp) = 0 for all cp E LYR) having support in R,. Thus, (2.49) should be understood in the following sense: There exists a singular measure vs E (L”(R))* such that Ap = v, - 77 - 6 , where 77 E L’(R) and 6 E L2(R) satisfy Eq. (2.50).
Proof of Theorem 2.1. We shall use the approach described in Section 2.2 by approximating problem (2.41) by a family of problems of the form (Qe). Namely, we shall consider the approximating problem: Minimize
Ay
=
BU
-
v
+f.
(2.55)
Here, u* = Bu* + f - Ay* E dcp(y*) = p(y*) a.e. in R and dcp c L2(R) X L2(R) is the subdifferential of cp given by (2.471, as a 1.s.c. convex function from L ~ ( R to ) R. Similarly, cp*(u) =sup{(u,p) - cp(u); UEL2(R))
=
jR j * ( u ( x ) ) dx,
UEL*(R).
(Throughout this section [ - I 2 denotes L2 norm and the L2 scalar product.) Without any loss of generality, we may assume that p(0) = 0. ( - , a )
166
3. Controlled Elliptic Variational Inequalities
As seen in Lemmas 2.2 and 2.3, problem (2.54) has at least one solution ( y , , u,, u,) E (H,'(R) n H2(R)) x U x L2(R) and, for E -+ 0, u,
+
u*
strongly in U ,
u,
+
u*
strongly in L ~a), ( stronglyin H,'(R) n H 2 ( a ) .
y, + y *
(2.56)
Now, by Lemma 2.4 it follows that there are p, E H,'(Rn) 17 H2(R) and
& E L2(R) such that -AP,
=
u,
=
(&(XI
B*p,
E
5,
+
&-I(
P ( Y & ) - v,)
+
P ( y , - ~ p , E ( U * - u,)) dg(y,(x)) dh(u,) + U, - u*.
a.e. in R, a.e.in R,
(2.57) (2.58)
a.e. in R ,
(2.59) (2.60)
To pass to limit in system (2.57)-(2.60), we need some a priori estimates on p,. These come down to multiplying Eq. (2.57) by p, and integrating on Q. We have
2 -115,1121IP,112
-
(I1 P(y,)ll2 + Ilu,ll2)llu*
-
v,l12
V€ > 0.
Since { &} is bounded in L2(R>(because g is locally Lipschitz), we have Ilp,llH:(n, Ic
VE
> 0,
(2.61)
and so we may suppose that p,
Ap,
+p
weakly in H,' ( R),strongly in L2(0),
+ Ap
weakly in H-'
Extracting a further subsequence P,,(X)
.,Ax)
(a).
E,, +
P(X) - u * ( x ) -+ 0 +
y,,(x) + y * ( x )
(2.62)
0, we can assume by (2.56) that ax. x
E
a,
a.e. x
E
R,
a.e. x E R.
(2.63)
3.2. Optimal Control of Elliptic Variational Inequalities
167
Now, by the Egorov theorem, for every S > 0 there is a measurable subset E, c R such that rn(E,) I 6 and the sequences p,,, u,. - u*, y,, are bounded in Lm(R\ E,) and uniformly convergent on R \ E,. Now multiply Eq. (2.57) by LA(p , ) where 5, is a smooth approximation of signum function, i.e., for A > 0, for
ldr)
r 2 A,
for - A < r < A,
=
-1
for
r
I -A.
We have
and, therefore,
:1 P(Y.s) < :/
~5-l
-
&-'
-
u ~ ) l , ( -p ~E ( U *
P(Y,)
+115,11L~(n,
-
- .&>)
u . ) ( lA(p&- &(.*
- u . ) ) - lA(pe))
V& > 0.
Now, let S > 0 be arbitrary and let E8 c R be such that rn(E,) I 6 and the convergences in (2.63) are uniform on R \ E,. Then, letting E = E, tend to zero in the previous inequality, we see that
<
c + 2 limsup E ; ~ E" +
0
IP(y,,)
- v,,l
h,
and this yields
IP(y,,) - v,,lh,
(2.64)
3. Controlled Elliptic Variational Inequalities
168
where C is independent of 6 (for 6 sufficiently small). On the other hand, since ( / 3 ( y E n )- uEn}is bounded in L2(R), we have
L
I P ( Y & , ) - U&,I du
Ic6"2
where C is independent of n and 6. Then, for 6 =
E,)?,
(2.64) yields (2.65)
from which (2.66) Since
is bounded in L 2 ( f l ) ,we may assume that weakly in L2(R)
ten+ 6
and, by Proposition 2.13 in Chapter 2, 6 E d g ( y * ) . Similarly, letting E, tend to zero in (2.60) we get (2.51). By the estimates (2.65) and (2.66), it follows that there exists a generalized subsequence of &, say &, such that weak star in ( Lw(a))*,
ApEA+ p,
1
-(
P ( y e A )- u,)
-+
7
weakstar in (L"(R))*,
(2.67)
&A
where p
= Ap
on LYR) n H,'(R) and in particular on C,"(R). We have -p, = 7
+ 4.
(2.68)
On the other hand, by the mean formula (Corollary 2.1, Chapter 21, we have 1 -(P(Y&,)
-
&A
=
e,(P~A - ( .*
- ueA))
R,
E @ ( z , ) , ycAI zA 5 yEA- &,(pEA - u* + uJ). Now since 0, and P(yEA)- uE,>are uniformly bounded on R \ E,, we may assume that
where 0, &;I(
OEA+ 0
weak star in Lw(R \ E,)
(2.69)
and 1
-( &A
P(yJ
- u,)
+
Op
weak star in Lm(R\ E , ) .
(2.70)
169
3.2. Optimal Control of Elliptic Variational Inequalities
By (2.69) we have
This yields ( P o is the directional derivative of p )
Hence, V w E R, a.e. x E
O(x)w I po(y*(x),w)
R
E,,
and therefore O ( x ) E d p ( y * ( x ) ) a.e. x E R \ E,. We have therefore proved that 77 E Lm(R\ E,) and ~ ( x E) d p ( y * ( x ) ) p ( x ) a.e. x E R \ E,. Then, by (2.68), we see that the restriction of p to R \ E, belongs to LYR \ E , ) and a.e. x
pa(x ) = p( x ) E - d p ( y * ( x ) ) p ( x ) - (( x )
E
R
\
E, .
Since 6 is arbitrary, we conclude that -&Ax)
E
dP(Y*(X))P(X)
a*e*xE
+ ((x)
a,
as claimed. Suppose now that 1 I N I 3. Then, by the Sobolev imbedding theorem, H2(R) c C ( a ) a n d so y, are uniformly bounded on On the other hand, we have, by (2.57),
a.
4EP,)
=
-&,
-
P(Y&) +
LL
and, since { p ( y , ) - u,) is bounded in L2(R), we infer that { ~ p , ), , is bounded in H2(R)and therefore in C(a). Finally, we have y, -
EP&
+ E(U*
-
U&)
E p-l(v,),
y* E p - ' ( u * ) .
Subtracting and multiplying by u, - u*, we get EIU*
- U&l I ly,l
+ elp,l
a.e. in R
170
3. Controlled Elliptic Variational Inequalities
and, since p is locally Lipschitz, this implies that IAp,l
I15,l
+ L(lp,l + Iv,
-
.*I)
a.e. x
E
a.
Hence, {Ap,} is bounded in L2(R) and we conclude, therefore, that p E H,'(R) n H2(R), 7 E L2(R), and Ap = (Ap*),, as claimed. Assume now that condition (2.52) holds. We shall prove that {q, = ( 1 / ~ )p(y,) ( - u,)} is weakly compact in L'(R). We have
3.2. Optimal Control of Elliptic Variational Inequalities
171
One might obtain the same result if one uses the approach described in Proposition 2.2. Namely, we approximate problem (2.42), (2.43) by the following family of optimal control problems: Minirnizeg"(y)
+ h ( u ) + $ I U - U*I;
o n d ( y , u ) E (H,'(R) nH2(a))x U ,
(2.72)
subject to Ay where A y fined by
= A,y
+ p " ( y ) = Bu + f
in R ,
(2.73)
with D ( A ) = H,'(R) n H2(R), and p"
E
CYR) is de-
Here, p,(r) = ~ - ' ( r (1 + & p ) - ' r ) , r E R,and p is a C: mollifier in R, i.e., p E C"(R), p ( r ) = 0 for Irl > 1, p ( r ) = p ( - r ) V r E R, / p ( t ) dt = 1. Note that p is monotonically increasing, Lipschitz, and
I P E ( r ) - p,(r)l
I2 8
Vr
E
R.
(2.75)
We are in the situation described in Section 2.2, where
Throughout the sequel, we set B E = ( BE)'. Let ( y &u,) , be optimal for problem (2.72). Then we have u,
-+
u*
strongly in H i ( R ) , weakly in H 2 ( R ) ,
Y, +Y* p"(y,)
+
strongly in U,
p(y*)
weaklyin L 2 ( R ) .
(2.76)
Indeed, from the inequality g"( y,)
+ h( u,) + +IU" - u*l;
Ig"(
y"')
+ h( u * )
172
3. Controlled Elliptic Variational Inequalities
( y " is the solution to Eq. (2.7311, we see that {u,) is bounded in U. Hence, on a subsequence E,, + 0, we have
u,"
+
weakly in U .
U
On the other hand, we may write Eq. (2.73) as
AY, + P & ( Y & )= Bu, + f + P & ( Y & ) - P E ( Y & ) and so, by Proposition 1.2, y,, + j j
strongly in H ' ( 0 ) and weakly in
H2(a), where j j is the solution to (2.43), where u = U. This yields g ( j j ) + h ( ~+) limsup +1uEn- u*Ic I g ( y * ) + h ( u * ) = inf(P). En -+
0
Hence, us" -+ u* strongly in U and j = y * , U = u*. The sequence being arbitrary, we have (2.76). Moreover, there is p , E H,'(R) n H2(a) such that
-4, - B"(Y,>P,
=
Vg"(y,)
B*p,
E
dh(u,)
in
a,
E,,
(2.77)
+ U , - u*.
(2.78)
Now, multiplying Eq. (2.77) by p , and sign p,, we get the estimate
Ilp,ll2Hd(n, +
/-I B"(Y,)P,l dx I c
Hence on a subsequence, again denoted
p, Vg"(y,)
+p +
5
we have
weakly in H; ( a),strongly in
( a),
weakly in L 2 ( a ) ,
and, on a generalized subsequence
be^( y,,)p,,
E,
v.5 > 0.
+
v
{EJ,
weak star in ( L"( a))*.
Hence, -Ap - q = 5 E d g ( y * ) in a. Now, by the Egorov theorem, for each S > 0 there exists a measurable such that m ( E , ) I 6 and y * , p E Lm(a \ E,) and subset E, of
y,( x )
+ y*( x ) ,
p,( x )
Since {( BE(y,)) is bounded in subsequence
b " ( y , ) +f,
+p ( x
)
uniformly on
Lm(a \ E,), weak star in
\
E,.
we may assume that, on a
Lm(a \ E6).
3.2. Optimal Control of Elliptic Variational Inequalities
173
Then, by Lemma 2.5 following, we infer that f s ( x ) E d p ( y * ( x ) ) a.e. x E Cl \ E, and so 77,(x) = f , ( x ) p ( x > E
ap(Y*(x)>P(x>
a*e*x
E
\E,*
The last part of Theorem 2.1 follows as in the previous proof since { b E ( y E ) }is bounded in Lm(CL)if 1 I N < 3 (because I b”(y,)I I 1 p’(y,)I < C in a ) and is weakly compact in L’(R) if p satisfies condition (2.52). The main ingredient to pass to limit in the approximating optimality system is Lemma 2.5 following, which has intrinsic interest.
Lemma 2.5. Let X be a locally compact space and let u be a positive measure on X such that u( X ) < 03. Let yE E L’( X ; u ) be such that y, + y strong& in L ’ ( X ; u ) and @“(y,) + f o
weaklyin L’(x; u ) .
Then fo(x)E dp(y(x))
Prooj
a.e. x E X .
On a subsequence, again denoted ye, we have y,(x) + y ( x )
v x E X \A,
u ( A ) = 0.
On the other hand, by Mazur’s theorem there is a sequence o, of convex combinations of { b E ( y E )such ) that on+ fo
strongly in L’( X ; u ) ,
where w,(x) = X i a,!, pEc(yi(x)).Here, I, is a finite set of positive n = 1. Hence, there is a integers in [n,+ M[, yi =y,, and a: 2 0, C i E I a,!, subsequence, again denoted o,,such that o,(x) -+ f o ( x ) V x E Cl \ B , u ( B ) = 0. In formula (2.74) we make the substitution t = r - c28 to get ps(r)= - & - z / p , ( t ) p ( y ) d t - /pE(-Eze)p(e)d0
and this yields b E ( r )= . ~ - ‘ / ~ + “ ~ P r, -( tt ) p ~ ( ~ ) d t r- c2 = --E-~
/ b E ( r - & ) p f ( 0 ) do.
174
3. Controlled Elliptic Variational Inequalities
where 8; 2 0, C ~ 8;L= ~1. Hence,
B
Y;( x 1)
(2.79) where
On the other hand, we have
Hence, l(e;e;)-'((i
+ Eip)-'Yi
- (1
+ Eip)-l(yi
-
&;e;)> - I ( I
cEi
Because be, uniformly bounded on every bounded subset ( p is locally Lipschitz). Then, by (2.791, it follows that
bet(y i ( x ) )
c 6;q; m,
=
k= 1
+'-yj,
where
'y; +
0 as i
+ 03.
(2.80)
3.2. Optimal Control of Elliptic Variational Inequalities
175
On the other hand, 8; - yi + 0 uniformly with respect to k, and by (2.80) we have bEi(yi(x))h _<
m,
k= 1
8 ; P 0 ( ~ ; , h+) -yih
Vh
E
R.
Hence, limsupbEi(yi(x))h _< P o ( y i ( x ) , h )
V h E R.
i-rm
Finally,
It should be said that the methods of Section 2.2 are applicable to more general problems of the form (2.42), for instance, for the optimal control problem: Minirnizeg(y)
+ h ( u ) on all ( y , u ) E ( w ~ , R) P (n W2vP(R)) x U , 1 IP < 03,
(2.81)
subject to -Ay
+ P ( y ) = f + Bu y=o
a.e. in 0, in a R ,
(2.82)
where f E LP(R), B E L(U, LP(R)), g locally Lipschitz on LP(R), h is convex and 1.s.c. on U, and P is monotone and locally Lipschitz. One gets a result of the type of Theorem 2.1 by using the approximating control process (2.721, i.e.:
-Ay
+ p"(y)
= f + Bu
y=o
in R, in dR
(2.83)
176
3. Controlled Elliptic Variational Inequalities
Then one writes the optimality system and passes to limit as in the proof of Theorem 2.1. 3.2.4. Optimal Control of the Obstacle Problem We shall study here the following problem: Minimizeg(y) + h ( u ) o n a l l ( y , u )E ( H ~ ( R n ) H2(R))X U , y E K , ( 2.84) subject to a ( y , y - z ) I( f where a: H,'(R) set (1.24).
X
H,'(R)
+
+ Bu,y
-z)
Vz
E
(2.85)
K,
R is defined by (1.19) and K is the convex
As seen in the preceding, (2.85) is equivalent to the obstacle problem (A,y -f
-
B u ) ( y - @)
=
a.e. in R ,
0
A,y - f - Bu 2 0 , y 2 @
a.e. in
a,
in dR.
y=O
(2.86)
Here, B E L(U, L2(fl)), f E L 2 ( f l ) ,and g : L 2 ( R )+ R, h: U assumptions (a) and (b) in Section 2.3.
+
R satisfy
Theorem 2.2. Let ( y * ,u * ) be an optimal pair for problem (2.84). Then there exist p E H,'(fl) with A p E (L"(R))*and 6 E L2(R) such that $, E dg(y*) and
+
a.e. in [ x; y*( x ) > @( x ) ] , (Ap), 6 = 0 p ( A y * - Bu* - f ) = 0 a.e. in R , a ( p , X(Y* -
@)I + ( 6 7 ( Y * - @ ) x )= 0 B*p
E
dh(u*),
a ( P , P ) + ( 6 , P ) I 0. Zf 1 I N
I3,
vx E C ' ( h ) ,
(2.87) (2.88) (2.89) (2.90) (2.91)
then Eq. (2.87) reduces to ( A p + 6 ) ( y * - @)
=
Oin R .
(2.92)
177
3.2. Optimal Control of Elliptic Variational Inequalities
We have denoted by A the operator A, with the domain D ( A ) = H,'(R) n H 2 ( C l ) and by ( A p ) , the absolutely continuous part of A*p. If N I3, then y* E H2(R) c C(n) and so A p ( y * - #) is well-defined as an element of (L"(R))*. In particular, (2.92) implies that Ap* = in [ x ; y * ( x ) > +(x)l. The system (2.871, (2.89) represents a quasivariational inequality of elliptic type.
Proof of Theorem 2.2. Consider the penalized problem (2.93) subject to Ay
+ P E ( Y - S)
=
Bu
(2.94)
+f,
where
In other'words, p" is defined by formula (2.741, where p is the graph (1.35). By Proposition 2.1, problem (2.93) has at least one optimal pair (u,,y , ) E ux ( ~ ~ n ( H~(R)). 0 ) Arguing as in the proof of Theorem 2.1 (see problem (2.72)), it follows by Proposition 1.2 that
u,
+
u*
stronglyin H,'(R),weaklyin H 2 ( R ) ,
y, + y *
p " ( y , - 9)
strongly in U ,
+ Bu* - Ay*
weakly in L 2 ( n ) .
+f
Moreover, we have for (2.93) the optimality system (see (2.771, (2.78)) -AP, - P ( Y & - S I P , B*p,
E
=
W(Y,)
+
d h ( ~ , ) U, - u*.
in
(2.96) (2.97)
3. Controlled Elliptic Variational Inequalities
178
Then multiplying Eq. (2.96) first by p, and then by sign p , , we get the estimate
c.
IcIIvg"(y,)lL2(n) I
Hence, there is a subsequence, again denoted P, Vg"(y,) Letting
E
+
-+
such that
P
weakly in H i ( R),strongly in L2(R ) ,
6 E dg(y*)
weakly in L 2 ( R ) .
tend to zero in (2.97), we get B*p
Now, let bY
E,
(2.98)
&: R
.+
R and 7,:R
On the other hand, we have
--j
E
dh(u*).
R be the measurable functions defined
179
3.2. Optimal Control of Elliptic Variational Inequalities
This yields IP&P"(Y&-
$11 I EIP&PE(Y&- $ W - ' l Y & - $16, +E-'ly& -
+ 2clp,l
$177,)
a.e. in R . (2.101)
We note that p " ( y , - $)q& = &-'(ye - $177, + C q , remain in a bounded subset of L2(R), whilst by the definition of 6, wee see that ~-'ly~(x )$(x)l&.(x)
a.e. x
IE
E
R.
Since ( p , P " ( y , - #)) is bounded in L'(R), it follows by (2.99) and (2.101) that, for some subsequence E,, + 0, p & , ( x ) p E f l ( Y & , ( x-) $ ( x ) )
+
0
ax. x
E
a,
(2.102)
whilst p,, p " n ( y E n- $)
+p ( f
+ Bu* - A y * )
weakly in L'( R).
Together with (2.102) and the Egorov theorem, the latter yields a.e.inR,
p(f+Bu* - A y * ) = O and therefore
stronglyin L'(S1). (2.103)
p E a p e n ( y e n- $) -+p(f+ Bu* - A y * )
Then, by (2.1001, we see that ( y E n- $ ) P " n ( y , , - $ ) p , ,
Inasmuch as I p " ( y , - $) (2.103), and (2.104) that
+
0
+ &-'(y8 - $)-I
(Y&n- $)+ P Y Y , , - $ ) P & n
+
Since (yen +&JY&"
-
$)+E
0
stronglyin L'(fl). (2.104) I C E , it follows by (2.99),
strongly in
Jw).
Hi(R), applying Green's formula in (2.96) yields
$I+ x ) + ( v g Y Y , , ) , ( Y & " - @ I +x )
-+
0
vx E c l m
Since p,, + p weakly in Hd(R) and (yen - $ ) + + y * - $ strongly in H'(R), we get (2.89). Regarding inequality (2.91), it is an immediate consequence of Eq. (2.96) because we have
(4+, V g & ( Y & ) , P &I) 0
V& > 0-
180
3. Controlled Elliptic Variational Inequalities
Now, selecting a further subsequence, if necessary, we may assume that y,,(x)
+
a.e. x
y*(x)
E
R.
On the other hand, by estimate (2.98) it follows that there is p of E, such that and a generalized subsequence p"~(y,,- $ ) p ,
weakstar in
-+ p
E
(L"(R))*
(,!,"(a))*.
This implies that Ap admits an extension as element of (L"(R))*, and we have -A*p - p
=
5 E dg(y*).
Now, by Egorov's theorem, for every 6 > 0 there is a measurable subset E, of R such that rn(E,) I 6, y* - $ is bounded on R \ E, = R, and
y,"
- $ -+
y*
- $
uniformly on R,.
Then, by (2.104), it follows that p(y* - $)
=
0 in R,, i.e.,
On the other hand, there is an increasing sequence and ps = 0 on L"(Rk). Hence,
rn(n \ Rk)I k-'
(ak)such
that
Thus, ( y * - $ ) p a = 0 a.e. in R,, and letting 6 tend to zero we infer that ( y * - $ ) p a = 0. Hence,
-(Ap), If 1 s N
=
5E
a.e. in [ y * >
dg(y*)
$1.
s 3, then H2(R) c C ( n ) and so y,(x)
-+ y
*(x)
uniformly on
a.
Since $ E H2(R) c C(n), it follows by (2.104) that ( y * - $ ) p
(Y*
-
$)(AP + 5 )
This completes the proof of Theorem 2.2.
=
0.
=
0, i.e.,
181
3.2. Optimal Control of Elliptic Variational Inequalities
Remark 2.1. Theorems 2.1 and 2.2 remain valid if one assumes that ( y * , u * ) is merely local optimal in problem (2.42) (respectively, (2.841, i.e., g(y*) + h(u*)I g ( Y ) + h(u)
for all ( y , u ) satisfying (2.43)) (respectively, (2.85)) and such that lu - u*Iu < r. Indeed, in problem (2.72) (respectively, (2.94)) replace the cost functional by g"(y)
+ h ( u ) + aIu - u*1;,
where a is sufficiently large that Iim sup g"( y,"') &+
0
+ h( u * ) I ar'.
Then Iu, - u*Iu I r for all E > 0 and this implies as before that u, strongly in U.The rest of the proof remains unchanged.
+
u*
Problems of the form (2.84) arise in a large variety of situations, and we now pause briefly to present on such an example. Consider the model, already described in Section 2.2, of an elastic plane membrane clamped along the boundary an, inflated from above by a vertical field of forces with density u and limited from below by a rigid obstacle y = $ ( x ) < 0, Vx E n (see Fig. 1.1). We have a desired shape of the membrane, given by the distribution y = yo(x> of the deflection, and we look for a control parameter u subject to constraints. lu(x)l I p
a.e. x
E
n,
(2.105)
such that the system response y" has a minimum deviation of y o . For instance, we may consider the problem of minimizing the functional / , ( y ( x ) - y0(x))' dw on all ( y , u ) E (H,'(R) n H 2 ( n ) )X L'(fl), subject to control constraint (2.105) and to state equation (2.861, where f = u. This is a problem of the form (2.84), where A, = - A , B = I , U = L'(n), f = 0, and
h(u) =
if lu(x)l I p a.e. x otherwise.
E
n,
182
3. Controlled Elliptic Variational Inequalities
By Proposition 2.1, this problem has at least one solution ( y * , u * ) whilst by Theorem 2.2 such a solution must satisfy the optimality system Ay*
+ u*
=
y*
=
0
+,
y* = O
in R + = ( x
R; y*(x) > + ( x ) } , a.e. in R, = R \ R', A + + u* I0 indR,
Ap =y* -yo
p(u*
+ Ay*) = 0
E
a.e. in R + ,
a.e. in R ,
u* = p sign p
(2.106) p
(2.107)
=
in d R ,
0
(2.108)
a.e. in R .
(2.109)
Assume that IA+I # p a.e. in 0. Then, by Eq. (2.108), we see that p R,. Hence, p is the solution to boundary value problem in R + , A p = y* - y o p=O inR,, p=O
indR.
=
0 in
(2.110)
This system could be solved numerically using an algorithm of the following type: yi 2
+,
( A y , + u i ) ( y i - +) = 0 Ay, + u , I0 a.e. in R
Ap, = y i - y o
ui+
a.e. in 0, y,
=
0
in d R ,
in Ri = ( x E R ; y i ( x ) > + ( x ) } , pi = 0 in d R i , =
p sign p i
a.e. in R i .
Let us assume now that y o E HdCCl) n H2(R> and A y o ( x ) 2 max(A+(x), p ) a.e. x E 0. Then, by Eqs. (2.106), we see that A(y* - y o ) I0 a.e. in R and, therefore, by the maximum principle y* - y o 2 0 in a. Then, assuming that R + is smooth enough, we deduce by (2.110) and by virtue of the maximum principle that p < 0 a.e. in a+,and therefore, by (2.1091, u* = - p
inR+
Hence, y* E H,'(R) n H2(R) satisfies the variational inequality Ay* = p
inat,
Ay* < p
+
y*
=J!,I
in d R + ,
y* >
y*
=
in R,
y* = O
+
inR, inR+, indR,
(2.111)
183
3.2. Optimal Control of Elliptic Variational Inequalities
from which we may determine R+. For instance, if R = ( 0 , l ) and = - 1, then clearly the solution to (2.111) is convex and so it is of the following form: y * ( x ) = -1
forO
y*(O) = y * ( l ) = 0
-1
<0
forx
E
(0,a) u ( b , l )
=
R+.
Hence for x
E
[a, b),
for x
E
[a,b],
and one determines constants a, b, c, and C , from the continuity conditions y*(a) =y*,(b)
=
-1
(y*)’(a)
=
(y*)\(b)
=
0.
A problem of great interest in the study of a physical system modeled by the obstacle problem is that of controlling the incidence set R, = ( x E R, y ( x ) = $ ( X I } or its boundary dR,. For instance, in the case of the contact problem (1.46) recalled in the preceding a problem of interest would be to find the force field f (viewed as a control parameter) such that the set of all points where the membrane is in contact with the obstacle be as close as possible (in a certain acceptable sense) to a given measurable subset D of 0. The least squares approach to this problem leads us to consider the optimal control problem: Minimize (2.112)
on all ( y , u ) E H,’(R) X U subject to (2.85). Here, ,yo is the characteristic function of D and xy is that of the set R; y ( x ) = $ ( X I } . However, since the function y + xy is not locally Lipschitz on L2(R) we shall replace it by g,(y) = A / ( ( y - $I++ A), A > 0, and so we shall consider the problem:
{x E
184
3. Controlled Elliptic Variational Inequalities
To be more specific, we shall assume that A, is given by h: L2(a> + h(u)
=
{L
=
-A, B
if lu(x)l I p a.e. in otherwise.
= I,
a,
This problem has at least one solution (y, , u,) and since, for A g,(y)
+
x,
f = 0, and
+
0,
strongly in L2(a )
for every y E L 2 ( a ) ,it is readily seen that for A + 0, u, u* weak star in Lm(a), y, + y* weakly in H2(a), where (y*, u * ) is an optimal pair for problem (2.112). On the other hand, the optimality system of (2.87H2.90) is in this case Ay,
+ u, = 0
y, 2 $, Ay,
pIp,I +p, A $
+ u,
I0
=
at = [ y , = $1, a.e. in a.
a.e. in
0
=
$1
a:, a.e. in a ,
a.e. in [y, >
uA = p sign p,
This problem can be treated as in the previous examples. Now we shall study problem (2.84) in the case where g is a continuous for instance, convex function on
~(a),
g ( y ) = IlY -Yollccn,, The subdifferential d g : C ( a ) Chapter 1, Section 1)
a;
+
Y
E
cm.
(2.113)
M ( a ) of g is given by (see Example 4 in
where M , = (xo E Iz(x,)l = Ilzll~~n,}. is a bounded, open, and smooth For simplicity, we shall assume that domain of R3, and A, = -A.
185
3.2. Optimal Control of Elliptic Variational Inequalities
Theorem 2.3. Let ( y * ,u * ) be optimal in problem (2.84) where g is given by and (2.113). Then there existsp E Wdvq(R),1 < q < 3/2, with A p E p E dg(y*) such that
Ap
=
p ( x)( Ay*( x)
p in { x E
R;y * ( x ) > @ ( x ) ) ,
+ Bu*( x ) + f( x)) B*p
E
=
dh(u*).
0
a.e. x
E
R,
Proog Since the proof is essentially the same as that of Theorem 2.2, it will be sketched only. We consider the approximating control problem:
Minimizeg,(y) + h ( u ) + n~ ~ ( x 0 Usatisfy ) ) -Ay
31.
- u*It,
+ p " ( y - @)
subject to having all ( y , u ) E (H,'(R) = Bu
+f
in R .
Here, g , is the usual convex regularization of g ( z ) = llz - y o l l ~ - ( ni.e., ),
Let (y,, u,) be optimal in the preceding problem. Then we have
p"(y,
-
u,
+
Y,
+
Y* @) + Bu*
and there are p,
Ap,
-
E
strongly in U ,
u*
+ by* +f
strongly in H,'(R) n ~
'(n),
weakly in L2(R),
H,'(R) n H'(R) such that
B E ( y , - @ ) p ,= Vg,(y,)
in ~ , B * P E , dh(u,) + u,
-
u*.
Since sup{ll,$ IIL~(n);5 E d g ( y ) } I1, we have
l l v g & ( y & ) l l LI ~ (c~ )
V& > 0.
Then, multiplying the latter equation by sign p , and integrating on R,we get
186
3. Controlled Elliptic Variational Inequalities
Now, let hi E La(R),i = 0,1,2 (see Section 3.21, the problem
has a unique solution 8
E
CY
> 3. Then, as mentioned in Chapter 2
H,'(R) n LYR) and
and, therefore,
Ilp&llw;4(n)Ic where l/q quence,
+ l/a
=
V,F
> 0,
1, i.e., 1 < q < 3/2. Hence, on a generalized subsep,
+
Q&(Y&) j 6 ( y &- $ ) p ,
p
weakly in W , . q (R ) ,
CL
vaguely in M ( Q ,
v
vaguely in
+
-+
~(fi).
We have, therefore, in g'(fl),
Ap
=
v+ p
B*p
E
dh(u*).
Since y, + y * uniformly on R (because H2(R) c C ( 1 ) compactly), we have v
=
0
in { x
E
R; y*(x) > $ ( x ) } .
Then, arguing as in the proof of Theorem 2.2, we get
p(x)(Bu*(x) Finally, letting
E
+ Ay*(x) + f ( x ) ) = 0
a.e. x
tend to zero in the obvious inequality
E
R.
187
3.2. Optimal Control of Elliptic Variational Inequalities
we infer that
Remark 2.2. In applications, the function g : L2(R) + R that occurs in the payoff of problem (P) and subsequent optimal control problems considered here is usually an integral functional of the form
where g o :R x R -+ R is measurable in x and locally Lipschitz in y , whilst condition (i) or (a) requires that go be global Lipschitz in y. However, it turns out that most of the optimality results established here remain valid if instead of (i), one merely assumes that (i)' go 2 0 on R X R, go(.,O) E L'(R), and for each r > 0 there is h, E L ' ( 0 ) such that l g o ( x , y ) - g o ( x , z)l I h,(x)ly - zI
a.e. x
E
a,
for all y , z E R such that lyl, IzI 5 r. (ii)' There exists some positive constants a, C , , C , and p such that
a.e. x
E
E
L'(R)
R,y
E
R.
For the form and the proof of the optimality conditions under these general assumptions on g we refer to author's work [3].
3.2.5. Elliptic Control Problems with Nonlinear Boundary Conditions We will study here the following problem:
(2.114)
188
3. Controlled Elliptic Variational Inequalities
on all ( y , u ) E H2(R) X U, subject to
y - Ay
=
f + Bu
dY
+p(y)3 0 dU
a.e. in R, (2.1 15)
a.e.in d R ,
where B E L(U, L2(R)), f E L2(R), and p c R X R is a maximal monotone graph. The functions g: L2(R) + R and h: U + R satisfy assumptions (a), (b) of Section 2.3. We know by Proposition 2.1 that if h satisfies the coercivity condition (2.5) then this problem admits at least one solution. Let ( y * , u * ) be any optimal pair for problem (2.114). Then, using the standard approach, we associate with (2.114) the adapted penalized problem: Minimize
g"( y )
+ h( u ) + $1.
(2.116)
- u*I21,
on all ( y , u ) E H 2 ( n ) x U, subject to
y - Ay
=
f + Bu
dY
+ P"(Y> 3 0 dU
a.e. in R, (2.117)
a.e. in d R ,
where p" is defined by (2.74). Let (y,*,u,*)be an optimal pair for problem (2.116). Then, by the inequality
g"( y , )
+ h( U & ) + ;Iu& - u*I2 Ig"( y,"') + h( u * )
(y," is the solution to (2.117)), we deduce as in the previous cases that u,
+
u*
y, + y *
strongly in U weaklyin H2(R),stronglyin H ' ( R ) .
Indeed, by Proposition 1.3 if u, + ii weakly in U then y , H2(R), where y" is the solution to (2.115). This yields limsup &+
0
IU, -
u*12 = o
-+
(2.118)
y" weakly in
3.2. Optimal Control of Elliptic Variational Inequalities
189
because lim inf, h(u,) 2 h(E) and g " ( y , ) + g ( y " ) . Now the optimal pair ( y , , u,) satisfies, along with some p , E H2(R),the first order optimality system ~
+ Ap, = Vg"(y,)
-p,
dP&
-+ dV
PYYJP, B*p,
E
=
a.e. in R , a.e. in d R ,
0
+
(2.1 19)
d h ( ~ , ) U, - u*.
Now, multiplying Eq. (2.119) first by p , and then by sign p , , we get the estimate (2.120) Hence, on a subsequence, again denoted
-+
we have
weakly in H'(
P, + P Vg"(y,)
E,
a),
weaklyin L ' ( S Z ) ,
6 E dg(y*)
strongly in L ~ ( R )weakly , in H ' ( R ) , (2.121)
P,+P and, by (2.118), P"(Y,)
+
weakly in L'(
770
where q o ( x ) E p ( y * ( x ) ) a.e. x Note also that, by (2.120),
E
an),
dSZ.
weak star in (L"( all))*,
P e ~ ( y S , ) p s+ , p
(2.122)
where ( E J is a generalized subsequence (directed subset) of ( E } . Now, letting E tend to zero in Eq. (2.1191, we see that p satisfies the system -p A p E dg(y*) a.e. in R , (2.123) dP - + p = o in d R ,
+
dV
B*p
E
(2.124)
dh(u*).
We note that since p E H'(SZ) and A p well-defined by the formula
E
L2(R), d p / d v
E
H-'/'(dR) is
190
3. Controlled Elliptic Variational Inequalities
where yocp is the trace of cp to dR. The boundary condition in (2.123) means of course that dP
-(
dV
cp)
+ p( cp)
4
vcp E L"( an) n ~ l / ~ ( d R ) ,
o
and in particular it makes sense in 9 ' ( d R ) . These equations can be made more explicit in some specific situations.
Theorem 2.4. Let ( y * , u * ) be optimal for problem (2.114), where p is monotonically increasing and locally Lipschitz. Then there are p E H'(R), q E L'(dR), and 6 E L 2 ( R ) such that p - A p E L2(R>, dp/dv E (L"(dR))*and inR, -p+Ap=( ((x) E dg( y*( x)) a.e. x E 0, (2.125)
(
+q =0
T(X) E
a.e. in d a ,
P(Y*(X>)P(X) ax. x B*p E dh(u*).
E
(2.126) (2.127)
dR,
If either 1 I N I 3 or p satisfies condition (2.52) then d p / d v and Eq. (2.126) becomes dP
dV
+ d p ( y * )p
3
0
E L'(dR)
a.e. in d R .
(2.128)
Here, ( d p / d v ) , is the absolutely continuous part of dp/dv. &oo$ By the Egorov theorem, for every 6 > 0 there is E, c dR such that m(E,) I 6, p,, y , are uniformly bounded on dR \ E,, and
y, p,
y* +p +
and, on a subsequence
{E,)
PYY,") +fo
uniformly on dR uniformly on dR
\
\
E6, E,
c {E}, weak star in L'( dR
\
Eo).
Then, by Lemma 2.5, we infer that f o ( x ) E dp(y*(x>>a.e. x and so, by (2.122) (see the proof of Theorem 2.1), pa(x)
E
d p ( y * ( x)) p ( x)
a.e. x
E
dR
\
E, .
E
dR
\
E,
3.2. Optimal Control of Elliptic Variational Inequalities
191
Since 6 is arbitrary, Eq. (2.126) follows. Now, if 1 I N I 3 then H2(R) c C(n), and so ( y J is bounded in C(fi). We may conclude, therefore, that
l p " ( y & ) lI c
vx
E
dR.
This implies that { b"(y,)) is weak star compact in L Y R ) and so 77 = = - d p / d u E LYdR). If condition (2.52) holds then we derive as in the ] proof of Theorem 2.1, via the Dunford-Pettis criterion, that ( b E ( y E ) is weakly compact in L ' ( R ) . Hence p E L'(dR), and the proof of Theorem 2.4 is complete. Now we shall consider the case where p is defined by (1.71). Then, Eq. (2.13) reduces to the Signorini problem
y
-
Ay
=
Bu
+f
a.e. in R , y dY
dY -20,
0,
a.e. in d R , (2.129)
y-=o
dU
2
dU
which models the equilibrium of an elastic body in contact with a rigid supporting body. The control of displacement y is achieved through a distributed field of forces with density Bu. Theorem 2.5. Let ( y * , u * ) E H2(R) X U be an optimal pair for problem (2.114) governed by Signorini system (2.129). Then there exist functions p E H'(R) and 6 E L 2 ( R ) such that d p / d u E (L"(dR))*, 6 E dg(y*), and
-p
+ Ap = 6 B*p
E
a.e. in R,
(2.130)
(2.132)
h( u * ) .
If 1 I N I 3, then y * -JP dU
=o
a.e. in d R .
(2.133)
Prooj The proof is almost identical with that of Theorem 2.2. However, we sketch it for reader's convenience.
192
3. Controlled Elliptic Variational Inequalities
Note that in this case P" is given by formula (2.95). Let A,: dR + R be the measurable functions
A&(X) =
We have
i
0
if y , ( x ) >
- E ~ ,
1
if y , ( x ) 4
--E
2
6,: dR
and
.
I P&( x 1P "( Y , ( x 11I I E I P&( x 1B ( Y , ( x 11I( E - ' I Y&( x ) I 6, ( x 1
+ 2~lp,(x)l
+~-'Iy,(x)A,(x)l)
+R
a.e. x
E
dR.
Since { P"(y,)) is bounded in L2(dR) and P " ( Y & ) A & ( x )= E - ' Y & ( x ) A & ( x ) +
EA&(X) /l
0
we infer that { ~ - l y , A , ) is bounded in L2(dR).Note also that ~-'Iy,l 6, I E a.e. in dR and, since ( b " ( y , ) p , ) is bounded in L'(dR), there is a sequence E,, + 0 such that p,,(x)pEn(yE,(x))
+
0
a.e. x
E
dR.
Then, by (2.120, we conclude that
P&,P Y Y & , )
+
-P
dY *
-= O dV
strongly in L'( an).
Finally, by (2.134) we see that yEnBCn(yEn)pEn +o
strongly in L ' ( ~ R ) .
(2.135)
Now, by the Egorov theorem, for every 6 > 0 there exists E, c dR such that rn(E,) I 8, ye" are uniformly bounded on dR \ E, and yCn+ y*
uniformly on dR
\
E,.
3.2. Optimal Control of Elliptic Variational Inequalities
193
By virtue of (2.fZZl this impfies that y*p
=
in dR
0
\
Es
and, arguing as in the proof of Theorem 2.2, we see that y * p , dR, which along with (2.123) yields (2.131). If 1 I N I 3, then H2(R) c C(n) and y,
+ y*
uniformly on
=
0 a.e. in
a.
Then, by (2.122) and (2.13.9, we deduce that Y*P
=
0,
where y * p is the product of the measure p y * E C(dR). H
E
(L”(dR))* with the function
3.2.6. Control and Observation on the Boundary We consider here the following problem: Minimize
(2.136)
in R ,
Y-AY=f
-+p(y) dY
dv
3
B,u
in
rl,
y
=
0
in
r,,
(2.137)
where p c R X R is maximal monotone, B, E L(U, L2(rl)), f E L2(R), and aR = rl u r,, where rl and I’, are smooth disjoint parts of dR. The functions g, : L2(R) + R and h : U + R satisfy conditions (a), (b) of Section 2.3, whilst g o : rl x R + R is measurable in x , differentiable in y , and
194
3. Controlled Elliptic Variational Inequalities
As seen in Section 1.4, for each u E U, Eq. (2.137) has a unique solution y E H'(R) (if n F2 = 0, then y E H2(fl)).As a matter of fact (2.136) is a problem of the form (PI considered in Section 2.1, where
r,
g(Y)
= gdx)
so(x,Y(x))
+
VY
E
I
and V ,A are defined as in Section 1.4 (see (1.62)). Let ( y * , u * ) be an arbitrary optimal pair for problem (2.136). Then, following the general approach developed in Section 2.3, consider the penalized problem:
subject to in 0
Y-AY=f dY
dU
+ p & ( y )= Bou
in
rl,
y
=
in
0
r2.
Here, g ; is defined by (2.15) and p" by (2.74).
By a standard argument, it follows that, for
E
-+
u,
+
u*
strongly in U ,
y,
-+
y*
weakly in
y*
strongly in L ~d(a ) ,
y, p"(y,)
--f
-+
fo
HI(
0,
a),strongly in L'( a),
weakly in L2(an),
(2.140)
where f o ( x ) E P ( y * ( x ) ) a.e. x E dR. On the other hand, the optimality principle for problem (2.138) has the form dP,
-+
dv
P, - AP€ = v g ; ( Y & )
in
a,
B"(Y,>P&= - v g o ( x Y Y € )
in
rl,
0
in
r2,
P, B*p,
E
=
dh(u,)
+ U, - u*.
( 2.141a)
(2.14lb)
3.2. Optimal Control of Elliptic Variational Inequalities
195
Now, multiplying Eq. (2.141a) by p , and sign p , , (more precisely, with [ ( p e l , where 5 is a smooth approximation of sign), we find the estimates
and so on a subsequence, again denoted
Letting
E
dP
we have
tend to zero in Eqs. (2.141a), we get
dU
E,
+ p = -Vgo(x,y*) B*p
E
in
rl,
p
=
0
in
r,,
(2.142) (2.143)
dh(u*),
where p E (L"(T,))*. One can give explicit forms for these equations if p is locally Lipschitz or if p is the graph of the form (1.71). Since the proofs are identical with that of Theorem 2.2 and 2.4 respectively, we only mention the results. Theorem 2.6. Let ( y * ,u * ) be optimal in problem (2.136), (2.137), where /3 is monotonically increasing and satisfies condition (2.52). Then there exists p E H'(R) such that A p E L 2 ( R ) ,d p / d u E L2(rl), -p
+ Ap E dgl(y*) B,*p E d h ( u * ) .
a.e. in R ,
(2.144)
196
3. Controlled Elliptic Variational Inequalities
Consider now the case where /3 is given by (1.71). In this case, the state equation (2.137) reduces to the unilateral problem inR,
y-Ay=f
dY
LO, - - B o u > O , dV
y
=
0
in I‘,.
(2.145)
Theorem 2.7. Let ( y * ,u * ) be optimal forproblem (2.136) governed by state equations (2.145). Then there is p E H’(R) such that A p E L2(R), d p / d u E (L”(I‘,))*and -p
+ Ap E
-)
p ( B o u * - dY*
dgl(y*)
a.e. in R , a.e. in ( x E
=
0
a.e. in
rl ; y * ( x ) > 0},
rl,
B,*p E d h ( u * ) . Similar results can be obtained if one considers problems of the form (2.135) governed by variational inequalities on R with Dirichlet boundary control (see V. Barbu [71, p. 107). Bibliographical Notes and Remarks Section 1. The results of this section are classical and can be found in standard texts and monographs devoted to variational inequalities (see, for instance, J. L. Lions [l], Kinderlehrer and G. Stampacchia [l], A. Friedman [l], and C. M. Elliott and J. R. Ockendon [l]). For other recent results on analysis and shape of free boundary in elliptic variational inequalities and nonlinear elliptic boundary value problems, we refer to J. Diaz [l]. Section 2. Most of the results presented in this section rely on author’s work [3,7]. For other related results, we refer the reader to the works of A. Friedman [2], V. Barbu and D. Tiba [l],V. Barbu and Ph. Korman [l],D. Tiba [l], G. MoroSanu and Zheng-Xu He [l], and L. Nicolaescu [l]. An attractive feature of the approach used here is that it allows the treatment of more general problems such as optimal control problems governed by
Bibliographical Notes and Remarks
197
not well-posed systems (J. L. Lions [2]) or by hemivariational inequalities (Haslinger and Panagiotopoulos [ 11, Panagiotopoulos [2]). A different approach to first order necessary conditions for optimal control problem governed by elliptic variational inequalities is due to F. Mignot [l] (see also Mignot and Puel [l]), and relies on the concept of conical derivative of the map u + y". Let us briefly describe such a result, for the optimal control problem (2.84), where J!,I = 0, U = L2(R), B = I , and
= {rp E H,'(R), rp 2 0 in 2,; uf) = O}, where y" is the solution to (2.85), then ( y * , u * ) is optimal if and only if there is p E -Sue such that u* = p and
If Z,
=
( x E R, y " ( x ) = 0} and S,
(rp, Aoy" -
(Mignot and Puel [l]). A different approach related to the method developed in Section 2.2 (see problem Q,) has been used by Bermudez and Saguez ([1-4]). In a few words, the idea is to transform the original problem in a linear optimal control problem with nonconvex state constraints and to apply to this problem the abstract Lagrange multiplier rule in infinite dimensional spaces. A different approach involving Eckeland variational principle was used by J. Yong [4]. Optimal controllers for the dam problem were studied by A. Friedman et' a f . [ 11. Optimality conditions for problems governed by general variational inequalities in infinite dimensional spaces were obtained in the work of Shuzong Shi [l] and Barbu and Tiba [l]. For some earlier results on optimal control problems governed by variational inequalities and nonlinear partial differential equations, we refer to J. L. Lions [4] (see also [2]). There is an extensive literature on optimal control of free boundary in elliptic variational inequalities containing control parameters on variable domains (shape optimization). We mention in this direction the works of Ch. Saguez [l], J. Haslinger and P. Neittaanmaki [l], P. Neittaanmaki, et a f . [l], J. P. Zolesio [l], V. Barbu and A. Friedman [l], W. B. Lui and J. E. Rubio [l], Hlavacek, et al. [l], and Hoffman and Haslinger [l]. A standard shape optimization problem involving free boundaries is the following: Let R, be a domain in R N that depends upon a control variable u E U and let its boundary dR, = ro u r,, To n T, = 0, where r0 is prescribed inde-
198
3. Controlled Elliptic Variational Inequalities
pendently of u. The problem is to find u E U such that robecomes the free boundary and R, the noncoincidence set of a given obstacle problem in a domain R 3 R,, for instance: A y = f in [ y > 01, A y s f in R,, y > 0 in R, , y = 1 in r, . Such a problem is studied by the methods of this chapter in the work of Barbu and Friedman [l] and an explicit form of the solution is found in a particular case. A different approach has been developed in Barbu and Tiba [2] and Barbu and Stojanovic [l]. The idea is to reduce the problem to a linear control problem of the following type: Find u E U such that d y / d v = 0 in r0and y > 0 in R,, where y is the solution to Dirichlet problem A z = f in r, , z = 0 in r0,z = 1 in r, .
First order necessary conditions for state constraint optimal control problems governed by semilinear elliptic problems have been obtained by Bonnans and Casas [l] using methods of convex analysis (see also Bonnans and Tiba [l]).