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Chapter VIII Existence of Solutions for Variational Problems
CHAPTER VIII
Existence of Solutions for Variational Problems
Orientation
In this chapter, we shall study non-convex problems. In Section 1, we shall introduce normal (not necessarily convex) integrands, an important class of functions of two variables without any convexity; we shall establish their main properties, including a measurable selection theorem, and we shall recall the characterization of weakly relatively compact subsets of L 1. In Section 2, these results will be applied to the study of a non-convex optimization problem, and a sufficient condition for the existence of solutions will be given. The final sections will show that a number of problems in the calculus of variations (Section 3) and in optimal control (Section 4) can be put into the above form and from this we deduce theorems on the existence of solutions. 1. NON-CONVEX NORMAL INTEGRANDS 1.1. Definition and main property
We recall that the Borel e-algebra of a topological space is the e-algebra generated by the closed subsets; in other words, the Borel subsets are the sets obtained from open and closed subsets by denumerable union, denumerable intersection, complementation and by any denumerable combination of such operations. A mapping f into R will be called a Borel function if f-l(F) is Borel for every closed set F. For instance, continuous functions are Borel. Let Q be an open subset of R" provided with the Lebesgue measure. All the Borel subsets of Q, especially the open and closed sets, are measurable. We recall that a mapping f: Q -+ R is measurable if the inverse image under f of every closed subset of R is measurable. The Borel functions are measurable and, in general, we have at our disposal the following criterion for measur› ability: Lusin Theorem. A function f: Q -+ R is measurable if and only if, for every compact set K c Q and all s > 0, there exists a compact set K. c K such that meas(K - K.)::; e for which the restriction off to K. is continuous. 231
232
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
The integrands used in the calculus of variations are functions of several variables playing different roles; by and large, they are measurable with respect to some variables and l.s.c. with respect to others.
Definition 1.1. If B is a Borel subset of RP, a mapping f of Q x B into R is termed a normal integrand if: (1.1)
for almost all x
(1.2)
there exists a Borel function f(x, .)for almost all x E Q.
E
Q,j(x,.) is l.s.c. on B
f
Qx B-
R such that lex, .) =
A first consequence of this definition is that for all a E B,j(. ,a) is measurable on Q. Better still, if u is a measurable mapping of Q into B, the function x 1-+ f(x, u(x)) is measurable on Q. Indeed, it is almost everywhere equal to the function x -l(x,u(x)), which is measurable sincefis Borel. We note that this property is no longer satisfied if, instead of assuming that f is a normal inte› grand, we merely assume it to be measurable in x and l.s.c. in a. Later on, we shall see that functions measurable in x and continuous in a are normal inte› grands. Furthermore, it follows from the definition that: iffis a normal integrand, Afis a normal integrand for all A E R; iffand g are normal integrands, (f + g) and inf(j,g) are normal integrands; if (f,,)neN is a denumerable family of normal integrands, sup, eN In is a normal integrand. The study of normal integrands depends on the following characterization, which is a Lusin theorem "uniform" in the second variable:
Theorem 1.1. Let B be a Borel subset ofRP. For f: Q x B - R to be a normal integrand, it is necessary and sufficient thatfor every compact set K. c Q and all 8> 0, there exists a compact set K. c K such that meas(K - K.) ~ e for which the restriction off to K. x B is l.s.c. Proof The sufficient condition is obvious: we take 8 = lin and let K’ = UneN Kiln’ Then meas(K - K’) = 0, K’ is Borel,jis Borel on K’ x B andf(x, .) is l.s.c. for all x E K’. To show the necessary condition, we begin by modifyingf on a Borel set of Q with null measure so thatf(x,.) becomes l.s.c. for all x E Q andfbecomes Borel over all of Q x B. Using if necessary an isomorphism of R onto [0, I], we may assume thatftakes its values in [0,1]. Since B is a subspace of RP, it possesses a denumerable basis dlt of open sub› sets. Let us denote by
lui U E dlt, k E Q;
: :;
k ::::; 1 }, where Q is the set of rationals.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
233
Clearly tP is denumerable and for all l.s.c. functions h: B -+ [0,1] we have h = sup{
En
=
{x
E
QI f(x,.)
G; = {(x,a)EQ
X
~
Sincefis Borelian and
f(x, a) = sup
If the compact set K c Q and the number e > are given, we choose for all n EN, by virtue of Lusin’s Theorem, a compact space K; c K such that meas(K - K n) ::::; e2-(n+!l and that the restriction of IE. to K; be continuous. Let K. = UnEN K; Then meas(K - K.) ::::; e and the restriction of
f(x, a)
=
f(x, a)
=
if (x, a)
+
E
C,
if (x, a) ¢ C,
00
is a positive normal integrand since it is Borelian and trivially satisfies (1.1). In particular, for all measurable mappings u of Q into B the following three conditions are equivalent to each other:
(1.3)
(1.4) (1.5)
(x, u(x))
E
C
U(X) E C, Lf(x, u(x)) dx <
a.e, a.e.
+
00.
(1) We admit here the fact that the projection of a Borel set is measurable. This is a difficult result; it is a consequence of, e.g., Choquet’s capacity theorem.
234
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
1.3. Second example: Caratheodory functions Definition 1.2. Let B be a Borel subset of RP. A mapping f: said to be a Caratheodory function if (1.6) (1.7)
for almost all x for all a
E
E
Q
x B -+
R is
Q,f(x,.) is continuous on B,
B,j(. ,a) is measurable on Q.
Proposition 1.1. Every Caratheodory function is a normal integrand. Proof By modifyingf on a Borel set of Q with null measure and by using an isomorphism ofR onto [0,1], we may assume thatfis measurable in x for all a, is continuous in a for all x, and takes values in [0, I]. Once again we introduce a denumerable base of open sets tJIt of B and the family rp of l.s.c. functions of B into [0,1] defined by rp = {k1 u luE tJIt, k E Q, 0 ~ k ~ I}. For all l.s.c. functions h of A into [0,1] we have h = sup{cp E rpl cP ~ h}. We now introduce a dense denumerable family 88. Enumerating rp = {CPn}neN and for all n E N and a E 88let us set: E n. a = {
If(x, a)
X E Q
~ CPn(a) }.
Since f(. ,a) is measurable, En. a is measurable and hence En = measurable
naeB8
En. a is
Now f(x, .) is continuous, CPn is l.s.c. and 88is everywhere dense. We deduce that
En = {
X E
Q I f(x, a) ~ CPn(a) Va
E
B }.
And by definition of the family rp:
f(x, a) = sup CPn(a) Idx). neN
For each n EN, there exists a Borel subset C; of almost everywhere. Let:
Q
such that IE
=
Ie
](x, a) = sup CPn(a)ldx). neN
The function/is Borel on Q x B as the least upper bound of Borel functions, andf(x, .) = J(x, .) for almost all x. Hypothesis (1.2) is thus verified; hypothesis (1.1) is covered by (1.6).
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
235
In this particular case Theorem 1.1 takes the following form, known as the Scorza-Dragoni theorem:
Scorza-Dragoni Theorem. A mapping f: Q x B --+ R is a Caratheodory function if and only iffor all compact sets K c Q and all 8 > 0, there exists a compact set K. c K such that meas(K - K.) ::;; 8 for which the restriction of fto K. x B is continuous. Proof Let the compact subset K and 8 > 0 be given. Since f is a normal integrand, there exists a compact set K+ c K such that meas(K - K+) ::;; 8/2 and that the restriction off to K+ x B is l.s.c. But -fis also a normal integrand and we can thus find a compact set K_ c K such that meas(K - K_) ::;; 8/2 and for which the restriction of -fto K_ x B is l.s.c. If K. = K+ n K_,fwill be l.s.c. and U.S.c. and hence continuous on K. x A and meas(K - K.) ::;; 8. The converse is an easy consequence of Theorem 1.1.
1.4. A measurable selection theorem We shall now assume that B is a compact subset of RP. For almost all x E Q, there thus exists an a(x) E B wheref(x,.) attains its minimum. We will show that we can choose a(x) in such a way that the mapping a defined from Q into B is measurable. Q
Lemma 1.1. Let B be a compact subset of RP and g a normal integrand of x B. We set
(1.8)
go(x, a) = 0 gO(X’ a) =
+
if g(x, a) = min { g(x, b) } beB
00
if g(x, a) > min { g(x, b) }. bEB
Then go is a normal integrand. Proof Using an isomorphism, we may assume that g takes its values in [-1,1]. For every 8> 0 and every compact K c Q we can find a compact subset K. c K such that meas(K - K.) ::;; 8 and that the restriction of g to K. x B is l.s.c. Define sp : K. --+ [-1,1] by:
cp(x) = min {g(x, b) }. bEB
Let us show that cp is l.s.c. Let (Xn)nEN be a sequence of K. converging to a subsequence X n’ such that:
x. We extract (1.9)
236
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Since B is compact, there always exists an’ E B such that
lim g(x n", an") ~ g(x, a)
11" -. 00
and afortiori lim
(1.10)
n"
~
--+ 00
Whence by comparing (1.9) and (1.10), the lower semi-continuity of
E
K. x B I g(x, a) =
This is the set of points where two I.s.c. functions coincide and is thus a Borel subset of K. x B. Moreover, for all x E K.. C; = {a E BI(x,a) E C} is closed since it is the set of points where an I.s.c. function attains its minimum. This means that go, which is none other than the indicator function of C, is a normal integrand of K. x B. In particular, we can find a compact subset K 2 c K. such that meas(K. - K 2 . ) ::;; 28 and for which the restriction of go to K 2 is I.s.c. Let us collect our results together: for all 8 > 0 and all compact sets K c Q, we have found a compact set K 2 c K such that meas(K - K 2 . ) ::;; 28 and for which the restriction of go to K2 X B is I.s.c. Hence go is a normal integrand of
QxB. Q
Theorem 1.2. Let B be a compact subset of RP and g a normal integrand of x B. Then there exists a measurable mapping it: Q -+ B such that for all
XE
Q:
(1.11)
g(x, u(x)) = min { g(x, a) }. aeB
Proof We define go by formulae (1.8). We then take a sequence am n ~ 1, which is dense in B. We now define by induction a sequence g., n ~ 1, of normal integrands in the following way: hn(x, a) = gn(x, a)
+
la -
ani
gn+ l(X, a) = 0 if hn(x, a) = min { hn(x, b)} gn+ l(X, a) =
+
beB
00
if hix, a) > min { hn(x, b) }. beB
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
237
Successive applications of Lemma 1.1 show that the gno n EN, are all normal integrands. Thus g = sUPneNgn is also a normal integrand. But it can be easily verified that domg(x, .) has been reduced to a point for all x E Q, i.e. g is ofthe form:
g(x, a) = 0 if a = u(x)
Ig(x, a) =
+
00
if a "# u(x).
The function u thus defined is measurable, since g is a normal integrand and satisfies (1.11) since g ~ go. In particular we deduce that the mapping x f-+ minaeB{g(x,a)} of Q into R is measurable. Theorem 1.2 includes as a special case the following measurable selection theorem.
if C is a Borel set of Q x B
Corollary 1.1. If B is a compact subset of RP, whose sections:
C,
=
{a
E
B I(x, a)
E
C}
are closed and non-emptyfor almost all x, then there exists a measurable map› ping u: Q ~ B such that for almost all x
We say that u is a measurable selection of C. This corollary is proved by applying Theorem 1.2 to the indicator function of C. 1.5. Polars and bipolars of normal integrands Let us now take B = RP, and consider a normal integrandf: Q x RP --+ R. For all fixed x E Q, the polar of the function f(x, .) will be a mapping of RP into R denoted by: ~*
1-+
f*(x ; C)•
Proposition 1.2. Iff is a normal integrand of Q x RP, then f* is a normal integrand of Q x RP.
Proof For all x E Q, f*(x,.) is a convex I.s.c. function. It only remains to verify hypothesis (1.2) for f*. To do this we define: fn(x, ~) =
+
fn(x, ~) = f(x,~)
00
if I~I if I~I
> n ~ n.
238
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
We havef = infneNf., hence (by 1(4.6)), for all x
(1.12)
E
Q:
f*(x;.) = sup f:(x; .).
neN
By definition
(1.13)
f:(x; ~*)
=
<~,~*
sup {
I~I.,
n
) - fn(x, ~)}.
For all x E Q, f:(x;.) will be either identically equal to -00, or a finite convex function of RP into R. Let us take ~* E RP, and any compact set K c Q. For all s > 0 there exists a compact subset K. c K such that meas(K - K.) ~ e and for which the re› striction of J,. to K. x RP is l.s.c. Since the balls of RP are compact, the family of mappings -J,.(’,~) + <~,~*>, for I~I ~ n, will be equi-l.s.c, on K . From equation (1.13) we also deduce that f.*(. ; ~*) is U.S.c. on K . Since e > 0 is arbitrary, fn*(’ ; ~*) is measurable on K. Since the compact set K is arbitrary f,,*(. ; ~*) is measurable on Q. For all n E N,fis therefore a Caratheodory function, and afortiori, a normal integrand. We thus haveJ,.*(x;.) = In*(x; .), for almost all x, wherefn* is Borel. From (1.12) we deduce that f*(x; .) = l*(x; .) for almost all x, where 1* = sUPneNl: is Borel. Whence (1.2) forf*. By repeating this operation, we arrive at the r-regularization of the function f(x, .) which will be denoted by: ~ t--* f**(x; O. We at once obtain the following corollary of Proposition 1.2:
Proposition 1.3. Iff is a normal integrand of Q x RP, then f** is a normal integrand of Q x RP. 1.6. Lower semi-continuity of integrals Let us now pass from the questions of measurability to those of integrability. Here we give an easy consequence of Fatou’s lemma which will be extremely useful to us in the future:
Proposition 1.4. Let f be a normal positive integrand of Q x RP and (Un)neN a sequence ofmeasurable mappings of Q into RP, converging almost everywhere to iI. Then we have:
(1.14)
f Q
f(x, ii(x)) dx
~
lim
"~OO
f Q
f(x, un(x)) dx.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
239
Proof We have a sequence of positive measurable functions to which we can apply Fatou’s Lemma: (1.15)
But sincef(x,.) is l.s.c. for almost all x (1.16)
E Q:
f(x, u(x)) ~ lim f(x, un(x)) n- 00
a.e.
Whence the result, on substituting (1.16) into (1.15).
Corollary 1.2. Let f be a normal positive integrand. Thefunction F :u
>--+
Lf(X, u(x)) dx,
is positive and l.s.c. ofL"(Q) into H,for all
IX,
1~
IX ~ 00.
1.7. Weak compactness inL 1(n) To conclude these preliminaries, let us recall the characterization of the weakly relatively compact subsets of L 1 (Q).
Theorem 1.3. Let!F c L 1(Q). The following statements are then equivalent
to one another:
(a) from any sequence (UJneN of !F, we can extract a subsequence which is weakly convergent in V ; (b) for all e > 0, there exists A > such that:
f
Vu E!F,
(lui" i.)
lu(x)1 dx
~
s;
(c) for all s > 0, there exists 0 > such that we have fB lu(x)ldx ~ e for all u E !F and all measurable B ofmeasure ~ 0; (d) there exists a positive Borel function iP: [0, 00 [ -+ H+ such that lil!4t"’( iP(t)/t = +00 and
sup SiPo lui < lIE§
+
00.
240
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Condition (b) is called equi-integrability. The equivalence (a) <0> (b) <0> (c) is Dunford-Pettis’ compactness criterion. The equivalence (b)-e- (d) is due to de la Vallee-Poussin.
Lemma 1.2. The function l.s.c. and increasing.
(]1
introduced in (d) can be assumed to be convex,
Proof To say that lim t (]1(t)/t = +00 means that for all mER, (]1 admits an affine minorant with slope m. Then so does (]1**, and hence (]1**(t)/t-+ +00 when t -+ +00. The function (]1** is convex and l.s.c. and attains its mini› mum ii at i: -> ",
a=
inf
(]1 =
min
= (]1**(I).
(]1**
It decreases on [O,i], then increases over [i,+oo[. Let us define a convex function increasing and l.s.c. on [O,+oo[ by
s.
~(t)
=
a on [0, t]
$(t) = (]1**(t) We have $ <
(]1, and
on
[t, + 00[.
thus
J 0 lui
sup ~ uefF
~ sup uefF
J(]10 lui
< 00.
Theorem 1.3 is a very deep result. We are going to use it to prove a generaliz› ation of Lebesgue’s theorem which will be extremely useful in what follows:
Corollary 1.3. Let (Un)neN be an equi-integrable sequence of L 1(Q) such that uix) -+ u(x) almost everywhere. Then u is integrable, and u; -+ U in £1(Q). Proof We begin by showing that, under the given hypotheses, the sequence u; converges weakly to u in L 1 (Q) . For this it is sufficient to show that we can extract from it a subsequence converging weakly in L 1(Q) to u. Now there exists a subsequence Un’ converging weakly in £1(Q) to a function v (Theorem 1.3, (b)=:>(a»). From Mazur’s lemma, we can find a sequence of convex com› binations Vn ’ E co Up’;;’n’ {up’} converging to v in£1(Q), and so we can extract a sequence Vn" converging to v almost everywhere:
(I.l7)
a.e.
But by hypothesis:
(I.l8)
Vn,(X) E co
U
p’",n’
{up,(x)} - u(x)
a.e.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
241
Comparing (1.17) and (1.18), we obtain v = U almost everywhere, and the sequence u; thus converges weakly to u. We immediately deduce from this that it also converges in the norm: it is sufficient to apply the preceding result to the sequence offunctions JU n - ul. We can easily verify that it is equi-integrable and that it converges almost everywhere to zero. It thus converges weakly to zero, and in particular:
If for example there exists a E V(Q) such that for all n, u.(x) :::; a(x) almost everywhere, the sequence (Un)neN will then be equi-integrable. We thus recover Lebesgue’s theorem.
2. AN OPTIMIZATION PROBLEM 2.1. The integrandf: definition, first properties
Let P: [0,+00[ --+ R U {+oo} be a non-negative increasing, convex, l.s.c. function such that lim P(t) =
(2.1)
1~
t
+
00.
Let us consider a mappingf of Q x (Rt X R’") into R. We shall assume that it is a normal integrand, i.e. that it satisfies (1.1) and (1.2) with B = Rt X Rm, and that we have the estimate
(2.2)
P( I~I
)~
f(x, s, ~)
In particular, f is non-negative, f(x, ., .) is l.s.c. for almost all x E Q, and f(. ,s,~) is measurable for all (s,~) E Rt X R", By epij(x,s) we shall denote the epigraph of the function j(x,s,.) in R" x R. We shall now prove a continuity property of the multi-valued mapping (x,s) --+ coepif(x,s). Lemma 2.1. Let E be a metric space and
R, satisfying (2.3)
Then,for all
(2.4)
qJ
a l.s.c. mapping of E x Rm into
’ieEE,
e E E, we have
n co £>0
U
Ie-iii.,;’
epi P(e,.) = co epi lP(e, .).
242
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Proof It is trivial that the right-hand side is included in the left-hand side. For the converse, take any:
(2.5)
(~, a) 1= co epi cp(e, .).
There exists an affine hyperplane of Rm x R which strictly separates (~,a) from co epicp(e, .). Ifthis hyperplane is non-vertical, it is the graph of an affine function ( over R" such that:
(2.6)
ii < f’(~)
(2.7)
f(e) < cp(e, e).
Ve E R",
If the hyperplane in question, denoted by :¥t, is vertical, there exists an affine function (’ over Rm such that (fee) = 0 for (e,a) E :¥t,ne) > 0 and (IW < 0 for E domcp(e, .). But, from (2.3), cp is non-negative; the function (= c/", for c > 0 sufficiently large, will thus also satisfy (2.6) and (2.7). From (2.3) we deduce that there exists M > 0 such that
e
(2.8)
e
Since the balls, of Rm are compact, the family IP(., e), for I I ~ M, of mappings from E into R, is equi-l.s.c, Let p be an increasing homeomorphism ofR onto [-1,1].(1) We then set:
(2.9)
m = min {p I~I"
0
M
cp(e, e) - p
0
fm }.
It is the minimum of a l.s.c. function on a compact set. It is thus attained and we have m > 0 from (2.7). By equi-lower-semi-continuity, there exists e > 0 such that Ie - el ~ e and I I ~ M imply that:
e
(2.10)
p
0
cp(e, e) ~ p
0
cp(e, e) - m
po CP(e, e) ~ po
(2.11)
rm
by (2.9). Finally, let:
Ie -
(2.12)
el
~ e and
lei
~ M => cp(e,
e)
~
(m.
Grouping together (2.8) and (2.12) and taking (2.3) into account, we see that: (2.13) (1)
For example p(s) = (2/n) arctan s.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
243
But (2.6) means that (e,ii) belongs to the lower open half-space determined by t, and (2.13) means that Ule-el". epiqJ(e,.) belongs to the upper closed half-space. But then co Ule-ell>. epiqJ(e, .) also belongs to the upper closed half-space, and hence: a)¢ co
(~,
A fortiori:
(2.14)
(~, a)¢
U
Ie -el<> "
n co
,>0
epi qJ(e .).
U
le-el,,’
epi ot«, .).
The fact that (2.5) implies (2.14) gives us the desired inclusion:
n co Ie-eke n epi qJ(e,
co epi qJ(e .) c
,>0
Corollary 2.1. If f is a normal integrand of D x Rt we have for almost all xED, (2.15)
n co ,>0
U
IS-51,;;’
.). X
Rm satisfying (2.2),
epi I(x, s) = co epi I(x, s).
Proof It is sufficient to apply Lemma 2.1 to the function f(x .. ,.) over R1 X R", at any point xED which makes it l.s.c.
2.2. A lower semi-continuity result We are now in a position to state the fundamental result of this chapter which yields a property of lower semi-continuity:
Theorem 2.1. Let f be a normal integrand of D x (Rt (2.2)
ep( I~I)
X
R"), such that:
~ I(x, s, ~)
where ep: R+ f-+ R+ is a convex increasing l.s.c. function satisfying (2.1) and,
(2.16)
Vex, s) E
Q X
Rf ,
I(x, s,.) is convex over R".
Let (Pn)neN be a sequence converging weakly to ft in V(Q)m and (Un)neN a sequence ofmeasurable functions converging almost everywhere to ii. Then: (2.17)
LI(X, u(x),
p(x)) dx
~ !~~ LI(X, un(x), Pn(x)) dx.
244
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Proof Since f is a normal non-negative integrand, all the written integrals have the same sign. If the right-hand side of (2.17) takes the value +00, the in› equality is trivial. Otherwise, by extracting a subsequence, we may assume that:
(2.18)
lim
n-+
00
f
Q
+
f(x, un(x), Pn(x)) dx = c <
00.
We now apply Mazur’s lemma (cf Chap. I, Section 1) to the sequencep .. which is weakly convergent in V. There exists a sequence of convex combinations LZ=n’ akPk, with ak > 0 and L~=n’ ak = 1, converging to pin V. We can thus extract a subsequence L:=n akPk converging almost everywhere to p: l
N
(2.19)
k
L
akPk(x)
=n’
Let us take a point x
u.(x) -+ u(x) as n -+ for all n’, we have: (2.20)
N
(L
k=n’
p(x)
n’
a.e. as
-+ 00.
where the convergence (2.19) occurs and where A fortiori, Un’ (x) -+ u(x) when n’ -+ 00. In Rm x R,
E Q
00.
akPk(x),
-+
N
2:
k=n’
ad(x, uk(x), Pk(X)))
co
E
N
U epi
f(x, uk(x))
k=n’
and afortiori:
(2.21)
N
(2:
k= n
akPk(x),
N
2:
k = n’
ad(x, uk(x), Pk(X)))
E
co
U epi
q ;;., n’
f(x, uq(x)).
no
We now take e > O. From (2.13), there exists sufficiently large to give us /U2(X) - u(x)1 ~ e for all q ~ n~. Hence for all n’ ~ n~, we have:
(2.22)
N
(2:
k=n’
N
2:
akPk(x),
k=n’
ad(x, uk(x), Pk(X)))
E
U
co
epi f(x, s).
Is-u(x)/,;;,
By making n’ go to infinity, we obtain from (2.19):
(2.23)
(p(x), lim
N
n’r-r o:
2:
k=n’
o:d(x, uk(x), Pk(X)))
E
co
U
Is-u(xll,;;,
epi f(x, s)
and as this is true for all e > 0: N
(2.24) (p(x), lim 2: ad(x, uk(x), Pk(X))) n’-+OOk=n’
E
n co
U
’>Ols-u(x)I,,’
epi f(x, s).
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
245
From Corollary 2.1, we have: (2.25)
U
e> 0
U
co
Is - u(xl! ", e
=
epi f(x, s)
co epi f(x, u(x)).
But because of hypothesis (2.16), the function !(x,u(x), .) is convex and I.s.c., and its epigraph is therefore closed and convex. Finally (2.24) can be written as:
(2.26)
(p(x), lim n'
N
-+ 00
L
k:;;; n’
ad(x, uk(x), Pk(X)))
E
epi f(x, u(x))
which by definition means that:
(2.27)
f(x, u(x), p(x))
~
lim n’
-+
N
L ' ad(x, uk(x), Pk(X)).
ro k =
n
We now integrate both sides over Q:
But all the integrands are positive, which allows us to apply Fatou’s lemma: (2.29)
But from (2.18) it is easy to deduce that the right-hand side of (2.30) is equal to c. Indeed, for all s > 0, we can find no such that (2.31)
(2.32)
c - s
C -
f.
~
~
Lf(X, un(x), Pn(x)) dx
ktn. a k Lf(X, uk(x), Pk(X)) dx
~
~ c+e
c + e,
246
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
whence finally
(2.33)
}i~C()
kt (Xk Lf(X, uk(x), Pk(X)) dx = c. n
Returning to (2.30) and (2.18), we obtain the desired result:
Lf(X, u(x), p(x)) dx
~
c =
}~n;,
f
f(x, uk(x), Pk(X)) dx.
2.3. The case wherefis not convex in ~ We now return to the general case wherefis a normal integrand satisfying (2.2) but no longer satisfying (2.16). Then, of course, we no longer have Theorem 2.1 at our disposal. It is natural to introduce f**(x,s; .), the r› regularization of the functionf(x,s, .). From the results of Section 1, this is a normal integrand of (D x Rt) X R", but the question arises as whether it is a normal integrand of D x (Rt X R"). (l)
Proposition 2.1. Iffis a normal integrand of D x (Rt
(2.2)
cI>( 1~1
)~
X
Rm) satisfying
f(’x, s, 0
where cI> answers to the above description, thenf** is also a normal integrand of D x (Rt X Rm) and satisfies
(2.34)
cI>( I~I
)~
f**(x, s; ~).
Proof By (2.2), the convex l.s.c. function cI>(j.l) is everywhere less than f(x,s, .), Taking the r-regularization of both sides, we obtain (2.34). Let us now take any compact subset KeD, and 8> O. Since/is a normal integrand, we can find a compact subset K. c K such that meas(K - K) ~ 8 and such that the restriction of f to K. X Rt X Rm is l.s.c. (Theorem 1.1). Moreover we know that epif**(x,s) = coepif(x,s). From Lemma 2.1 applied to / over (K. x Rt) X R", we have
(2.35)
epi f**(x, s) =
n co Ix U
s> 0
-xl,;:.
epi f(x, s).
S-Sl~E
(1) We already know thatf**(x, s; .) is l.s.c, in thatf**(x, .; .) is l.s,c. in (s, e).
e, but we are not yet in a position to state
247
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
But it is clear that: (2.36)
co
U
ls-sl .. e
x - xl.. s
epi f(x, s):::>
[x -
U
xl .. s
co epi f(x, s)
Is-sl,;; e
which enables us to write (2.35) in the form: (2.37)
epi f**(x, s)
:::>
n
.>0
U co epi
lx-xl . Is-sl
f(x, s)
.
or again, replacing coepij(x,s) by epij**(x,s), (2.38)
epi f**(x, s):::>
n
£>0
U
lx-x!,;,
epi f**(x, s).
Is -51,;;’
Let us now take a sequence (XmSm’n), n E N, converging to
K. x Rt x R", We obtain:
(2.39)
(x, s,~,
lim f**(x n, Sn; ~n))
E
n
.>0
U
lx-xl ...
(x,s,~)
in
epi f**(x, s),
Is-sl,;;’
and by virtue of (2.38): (2.40)
(x, s,~,
lim f**(x n, Sn; ~n))
E
epi f**(x, s)
which by definition means that: (2.41 )
Thus we have shown that for any compact set K c {J and for all B> 0, we can find a compact subset K. c K such that meas(K - K) ::;; B for which the restriction ofj** to K. x Rt x Rm is l.s.c. By Theorem 1.1,/** is thus a normal integrand of {J x (Rt X R"), We can now have a partial extension of Theorem 2.1 to the non-convex case, making use ofj**. The result is as follows:
Proposition 2.2. Let j be a normal integrand of {J x (Rt x R") satisfying (2.2)
248
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Let (Pn)neN be a sequence converging weakly topin Lt",(O) and (Un)neN a sequence ofmeasurable functions converging almost everywhere to u. Then (2.42)
f
Q
f**(x, u(x); p(x)) dx
~
lim
n--+
00
f
Q
f(x, un(x), Pn(x)) dx.
Proof It is sufficient to apply Theorem 2.1 to f**, and to make use of the inequality f** «t. We obtain: Lf**(X, u(x); p(x)) dx
~ !~~
Lf**(X, un(x); Pn(x)) dx
~ !~~
Lf(X, un(x), Pn(x)) dx.
2.4. Calculus of variations: existence of solutions by convexity We shall now formulate an optimization problem, which embodies a large class of problems in the calculus of variations, and apply to it the preceding results. As before, we are given a l.s.c., convex, increasing function rp: [0, +00 [ -i>› R+ which satisfies:
(2.1)
lim rp(t) dt = t
1--+ 00
+
00.
By L~ we shall denote those (classes of) measurable mappings P from Q into Rm (modulo equality almost everywhere) for which fn rp 0 !pl < +00.0> From (2.1) it is clear that L~ c V(Q)m. If, for instarice we took for rp the function t 1-+ t", with 1 < a < 00, the set L~ would be none other than L""(Q)m which will also be denoted by n. If we take for rp the indicator function of the interval [0,1], L% coincides with the unit ball of Loo(Q)m (or
r:
L~).
We are given a normal integrand f of Q(R"" x R") into il, such that there exists a function a E L I( 0) which satisfies:
(2.43)
a(x) + rp(IW ~ f(x, s, ~).
(I) The L~ are Orlicz classes. For the theory of Orlicz spaces see M. A. Krasnosel’skii and Ruticki [1], A. Fougeres [1].
249
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
We are also given a weakly closed subset 0If of L~, and a mapping r:fJ of 0If into with 1 ~ fJ ~ 0Ci, which satisfy the following property:
u,
n
L~
if a sequence (Pn) of 0If converges weakly to p in L~ and if sup III cP 0 IPnl < +0Ci, then we can extract from the sequence (r:fJPn) a subsequence which converges almost everywhere to r:fJp
(2.44)
Assumption (2.44) is satisfied if, for example, r:fJ maps the sequences (Pn) which converge weakly to p in L~ and are such that I cP 0 Pn ~ constant ’tin, into sequences (r:fJPn) which converge strongly to r:fJp in Lg: we then say that r:fJ is a (cP, fJ)-compactifier. If cP(t) = t", I < a < 0Ci, this amounts to saying that r:fJ maps the bounded and weakly convergent sequences of L~ into strongly con› vergent sequences of Y; indeed, the topologies oil: l,L 00) and u(La,L a’) coincide I/a + I/a’ = 1. We say on the bounded subsets of L~, since L~ is dense inL~, that r:fJ is an (a,fJ)-compactifier. Here are the main examples:
Proposition 2.3. Let t < a < 0Ci. Ifr:fJ is a compact continuous linear mapping of L~ into LV, with t ~ fJ ~ 0Ci, then r:fJ is an (a, fJ)-compactifier. We recall that, by definition, a continuous linear mapping of L::’ into is called compact if it maps the bounded subsets of L’:" into relatively compact subsets of Lf. Proposition 2.3 follows directly.
Lf
Proposition 2.4. If r:fJ is a continuous linear mapping of
1 < fJ
~
0Ci,
L~
into Lf, with
then r:fJ satisfies (2.44).
Proof Since Q is bounded, Lf(Q) c LHQ), and the bounded sets of Lf(Q) are weakly relatively compact in L}(Q). The mapping r:fJ maps the bounded sets of L~ into weakly relatively compact sets of L ~; by Grothendieck [1], theorem V.4.2 it will map the weakly compact subsets of L~ into compact subsets of L~. Let us now take cP into account. If the I cP 0 /Pn I’s are uniformly bounded, the (Pn)neN form a weakly relatively compact subset of Lt", (Theorem 1.3, de la Vallee-Poussin’s criterion). The (r:fJPn)neN thus form a relatively compact subset of L\, and we can extract a subsequence (r:fJPnk)keN converging in L~. Since the Pn converge weakly to p, the r:fJPn converge weakly to rip. The limit of the sequence (r:fJpnk)keN in L~ can therefore only be r:fJp. Finally, we can extract from (r:fJPnk)keN a subsequence which converges almost everywhere to r:fJp.
At last we are in a position to state the optimization problem:
(&»
pe~!L:
Lf(X, r:fJ p(x), p(x)) dx
250
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
which can be put into the equivalent form:
r [(x, u(x), p(x)) dx.
~~f
(&’)
pe"llnL~
u= ’i#P
JQ
From Theorem 2.1, we immediately deduce an existence criterion for solu› tions to problem (&’):
Theorem 2.2. Let f be a normal integrand of Q x (Rt
a(x)
(2.43)
(2.45)
+
V(x, s) E Q
~
[(x, s,
R f,
X
~),
X
R") satisfying
with a E L1(Q)
[(x, s,.) is convex on R".
Let t’§ be a mapping ofL~ intoL~ and qt a weakly closed subset ofL1satisfying
(2.44). Problem (&’) admits at least one solution.
Proof We set g(x, s,~)
=
that
f(x, s,~)
(2.46) (2.47)
- a(x). This is a normal integrand such ~ g(x, s, ~)
V(x, s) E
Q X
R ’,
g(x, s,.) is convex on R",
It is clear that:
(2.48)
L[(X, u(x), p(x)) = Lg(x, u(x), p(x))
+ L a(x) dx.
The last term is a constant. Let us therefore take a minimizing sequence of problem (&’); and let us set u; = t’§Pn’ By definition, P« E rJU for all n, and:
(Pn)neN
(2.49)
Lg(X, un(x), Pn(x)) dx
-+
inf(&’) - L a(x) dx.
From (2.46) we deduce that:
(2.50)
L
0
IPnl
~
constant.
From Theorem 1.3, we can extract from (Pn)neN a subsequence Pn’ which is weakly convergent to p in n; Since t’§ satisfies (2.44) we can extract a sub-
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
251
sequence Pn" such that Pn" converges weakly to p in L~ and C§Pn" converges to C§p almost everywhere. Applying Theorem 2.1 :
(2.51) Adding
(2.52)
Lg(X, u(x), p(x)) dx
~ !~~
f
~ !~~
Lf(X, un(x), Pn(x)) dx.
g(x, un(x), Pn(x)) dx.
In a(x)dx to both sides: Lf(X, u(x), p(x)) dx
That is, as the sequence (Pn) is minimizing:
(2.53)
Lf(X, u(x), p(x)) dx
But P E dJj is the weak limit of the Pn" fore that Ii = C§p is the solution of (9).
E
~
inf(9).
dJj’s. From (2.53) we conclude there›
2.5. Calculus of variations: relaxation In the case where we no longer assume f(x,s,.) to be convex, problem (9) in general no longer has a solution. We shall see that it is natural to associ› ate with problem (&’) the following problem, termed the relaxed problem,
(&’~)
Inf
pe"llnL!.
fn f**(x,
C§ p(x); p(x)) dx
or again
r f**(x,
~~!
ped/uinL~=
’§ p
In
u(x); p(x)) dx.
From Theorem 2.2 we immediately deduce Proposition 2.5. Let f be a normal integrand of D(pt x R’"), satisfying
(2.43)
a(x) +
~
f(x, s, ~),
witha E D(D).
Let C§ be a mapping which is a (
252
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Remark 2.1. Since f**
3. EXAMPLES We shall describe several examples of mappings rg, satisfying (2.44), and arising naturally in variational problems. We shall use these examples in the following chapter, where the relationship between problems (9) and (9~) will be stated more precisely. Example 1 Let 0 be a very regular open subset of R", and A the Laplace operator:
(3.1 )
Au
= flu.
For p given in U(O), there exists a unique u in HMO) such that:
(3.2)
Au=p
a.e.
and the mapping p 1-+ U is linear and continuous from L 2 (0) into HMO). Since the injection of HMO) into L 2(0) is compact (cf Lions and Magenes, [1 D, the mapping p 1-+ U is linear and compact from L 2(0) into itself. If we call this operator rg, it then satisfies property (2.44) trivially with lP(s) = S2: it is a (2,2)-compactifier. Example 2 In a much more general way, rg can be the Green operator of any regular elliptic problem. Let A be a differential operator of order 2m in a very regular open subset o < R":
(3.3)
Au = A(x, D)u =
L (-
lal,IPI’;m
1) laIDa(aap(x )DPu)
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
253
with
(3.4) and
(3.5)
the operator A is uniformly elliptic in Q.
We denote by oj u] onJ thej-th normal derivative of a function u on the bound› ary oQ, as it is defined by the usual trace theorems (oJujon J E Hr-J-’!{oQ) if u E H’ (Q)). Under those assumptions, for p given in £l(Q), there exists a unique u in H 2 m(Q) satisfying:
(3.6) (3.7)
(
Au=p
a.e. in Q
oJ u =0 on
on oQ, forO~j~m-1
-J
and the mapping I:§:p ~ u is linear and continuous from £2(Q) into H 2m(Q), and thus linear and compact from £l(Q) into itself. It is a (2,2)-compactifier.
Example 3 A being defined as in the preceding example, denote by Q the cylinder T a positive real number, and consider the parabolic
Q x (0, T) of Rn+l, with
equation: (3.8)
auat + Au = p
(3.9)
oJ u onJ =0
(3.10)
u(x,O) =0
a.e, in Q
on oQ x (O,T), for 0 ~j~
m-l
a.e. in Q.
For every p E £2(Q) thereisa unique solution u E H 2m.l (Q), and the mapping I:§:p ~ u is linear and compact from £l(Q) into itself. Example 4 Assume moreover that A is coercive on H’8(Q), and symmetric:
3c> 0 =
cllcpllHg’
254
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Consider the hyperbolic equation: 02 U 2
-
ot + Au = p
(3.11) (3.12)
Oi U -
oni
(3.13) .
a.e. in Q
on oD x (0, T), for 0 ::::;j::::; m - 1
= 0
u(x, 0)
=
~~
(x,O)
= 0 a.e. in D.
n
For every p E L2(Q) there is a unique solution u E m,l(Q), and the mapping C§:p 1-+ u is linear and compact from L2(Q) into itself. Example 5 Let us now consider cases where C§ is non-linear. If D is bounded and if 1 < Y < 00, forp given inV’ (D), where I/y + l/y’ ~ I, we verify with the help of Theorem 3.1 and Remark 3.4 from Chapter II that there exists a unique u in WA,1 (0) which satisfies: (3.14)
Au =
–~(1~11-2~)
i=
lOX;
OX;
OX;
=
p.
We term C§ the non-linear mapping which results from this and which sends V’ (0) into V (D). We now have the following result: Lemma 3.1. The mapping C§ defined by (3.14) is a (y’, y)-compactijier.
Proof We note that:
whence by Poincare’s inequality it follows that:
If now a sequence Pm converges top weakly inV’ (0), it is bounded inV’ (0) and the sequence of Um= C§Pm’s is bounded in W~,1 (0); by extracting a subse-
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
quence we can assume that Um converges to a limit hence strongly in V (Q). Then:
U
weakly in
255 W~•1
(Q) and
which obviously implies that:
and we deduce, as in Lemma II.3.3 that: Au =
Thus lemma.
Um
= ’§Pm
converges to
U
= ’§P
strongly in V (Q), which proves the
We can generalize this to the more general situation of Theorem II.3.!. Taking all the hypotheses of this theorem, together with (3.26), we assume moreover that V c V (Q) with compact injection and dense image, so that the dual V’ of V contains V’ (Q). In this case, for P given in V’ (Q) there exists a unique u in V satisfying equation (3.2) of Chapter II, and the mapping ’§:/= pI-? U from V’ (Q) into V (Q) is a (y’, y)-compactifier. Remark 3.1. We could give a great many more examples of operators ’§ which are (lX,p)-compactifiers by considering evolution equations, as in Lions [1], and inhomogeneous boundary-value problems as in Lions and Magenes [1]: a large number of examples from this work would allow us to define in a similar way the operators ’§.
4. OPTIMAL CONTROL We shall now apply Theorem 2.2 to the optimal control of systems governed by ordinary differential equations. In this example we shall show how to pass from a formulation of the "optimal control" type to a formulation of the "calculus of variations" type.
256
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
4.1. Theoptimal control problem The evolution equation. We take a number T> 0, a metrizable compact set K and a continuous mapping qJ from [0, T] x R" x K into R" such that: (denoting the unit-ball of R" by B) for all p
(4.1)
~ 0, there exists k ~ 0 such that,for 0 ~ t ~ jqJ(t, y, co) - qJ(t, y’, co)1
I’Vy, y’ E t’B,
there exists a constant
(4.2)
I’V(t, y, co)
E
t
~ 0 such that
[0, T] x R x K, n
~
T and co
k Iy - y’l ;
E
K:
Iy’ qJ(t, y, co)1 ~ (’(1 + Iyll).
Let us choose Yo ERn. The system is governed by the ordinary differential equation:
dy(t)/dt = qJ(t, y(t), CO(t))
(4.3)
Iy(O) =
Yo’
Lemma 4.1. For any measurable mapping co: [0, T] ~ K, the differential equation (4.3) has a unique solution y: [O,T] ~ R", and we have,for 0 ~ t ~ T: (4.4) Proof Since tp has been assumed to be continuous there exists ’r > 0 suffi› ciently small for the equation (4.3) to have a solution defined on [O,«], By virtue of the inequality (4.2), we now have, for 0 ~ t ~ ’r
~
=
dt
ly(tW
2y(t)• dy(t) ~ 2f(1 + ly(tW) dt "’"
~ IYol2
+ 2ft + {21!y(sW
ds.
By Gronwall’s inequality, for 0 ~ t ~ r:
ly(tW ~
(IYoI2 +
2ft) e 2 f r ~
(IYoI2 +
2fT) e 2f t
Set p = (lyl2 + 2tT)e U T It can be seen that the solution lies in the bounded set pB, independently of r, We know that in this case the solution can be ex› tended to the whole of [0, T]. If there were two solutions of (4.3) on [0, T], they would both have values in pB, and by applying the Lipschitz condition (4.1), we could show them to coincide.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
257
We term control any measurable mapping of [0, T] into K. Once the control has been chosen, the unique solution y of (4.3) is the trajectory of the system, and yet) is its state at the instant t.
Lemma 4.2. The set oj trajectories oj the system is relatively compact in ~([O,T];Rn).
Proof It is sufficient to note that the estimate (4.4) is independent of the control w: along all the trajectories, ly(t)1 is bounded above by a constant p, Let us now denote by p. the maximum of the continuous function qJ over the compact set [O,T] x pB x K. By equation (4.3) we have:
Id~~t) I ~
(4.5)
u.
This means that all the trajectories are p.-Lipschitzian. In particular, they are equicontinuous and so form a relatively compact subset of ~([o, T]; R") (Ascoli’s theorem). _ Constraints. We take a closed subset E of [0, T] x R" x K. For (t,y) given in [O,T] x R", we denote by Et,y the section:
(4.6)
E"y
= {wEKI(t,y,W)EE},
We term a control wand its corresponding trajectory y admissible if they are linked by:
(4.7)
Vt E [0, T],
(t, y(t), w(t)) E E.
Cost. We take a Caratheodory functionJof [O,T] x (Rn x K) into [0,+00[. We associate with an admissible control wand its corresponding trajectory y the cost function
(4.8)
IT f(t, y(t), w(t)) dt.
An admissible control wwill be called optimal if it minimizes (4.8); we shall also call the corresponding trajectory ji optimal. Let us gather all the data: to minimize
(&»
IT f(t, y(t), w(t)) dt
dy(t)/dt = qJ(t, y(t), w(dt)) dt (r, y(t), w(t)) E E y(O) = Yo
258
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
4.2. Compactness of the set of trajectories For ~ t ~ T and Y E R", we denote by at the instant t at the point y:
r;,y
rt,y
the set of admissible speeds
= { cp{t, y, w) I(t, y, w) E E }
= cp(t, y, E
1)
.
This is a compact set in R". If moreover it is convex and non-empty it can be shown that the set of admissible trajectories is compact and non-empty in ’tt([O,T];R"). We shall here be content with part of this result.
Proposition 4.1. If rt,y is convex for all t E [0, T] and all Y admissible trajectories is compact in ’tt([0, T]; R").
E
R", the set of
Proof In ’tt([0, T]; R"), the set of trajectories is relatively compact (Lemma 4.2), and it suffices therefore to show that the set of admissible trajectories is
closed. Hence, let (Yk)keN be a sequence of admissible trajectories which converges uniformly to a continuous function y. For all kEN we have
(4.9) (4.10) for ~ t ~ T, and Yk(O) = Yo. We must show that y also is an admissible trajectory; we have immediately that yeO) = Yo. From the estimate (4.5), IldYkjdt II ~ Jl for kEN. By extracting a subsequence we can thus assume that dYkjdt converges to dY/dt in the topology U(L2,L2). By Mazur’s lemma there exists a sequence of convex combinations L~=k lX"dy"jdt which converges to dyjdt in L~(O,T) as k ~ 00. We can there› fore extract a subsequence which converges simply to dyjdt on [O,T]. We thus have for t E [0, T]:
(4.11)
VkEN,
But we can summarize (4.9) and (4.10) by: (4.12)
k}
dy (t) E co {dy,,(t) In >dt dt,...•
259
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
and on substituting into (4.1 1):
(4.13)
dy(t)
Vk EN,
_
-d- E co t
U r;,Yn(t)
for
n:;;ok
~ t ~ T.
Let us fix t E [0, T] and take a > O. Since cp is continuous and K is compact there exists 17 > 0 such that:
Iy - Y(t)1 ~ 17
=>
Icp(t, y, w) - cp(t, y(t), w)1 ~ a
VWEK.
Since Yk converges uniformly to ji, we can take kEN large enough for: "In ?: k,
U
n ";;?;k
r;,Yn(t)
r;,Yn(t) c
+
r;,)i(t)
c
r;,y(t)
+ BB,
es.
The right-hand side is the sum of two convex compact sets. lt is thus a convex compact set and:
(4.14)
co
U r;,Yn(t)
n;;.k
c
r;,Y(t)
+
BB.
Substituting this into (4.13):
dY(t) (ftE
r;,y(,)
+
lt only remains to let a tend to zero. Since limit for 0 ::::; t ::::; T:
(4.15)
dY(t) ( f t E r;,y(t)’
(4.16)
d~~t)
E {
BB. rr,y(t)
is closed, we obtain in the
cp(t, Y(t), w) I(t, Y(t), w) E E}.
There exists a Borel subset N C [0, T], with null measure, such that the restriction of dy/dt to N is Borel. We can then define a Borel subset of xK by:
eN
(4.17)
G
=
{
(t, w) I(t, Y(t), w) E E and cp(t, y(t), w) E dY(t)jdt }.
By Corollary 1.7 there exists a measurable selection wof G. In suitably extend› ing it over N, we obtain a measurable mapping w from [0, T] into A which satisfies
(4.18)
dy(t)jdt = cp(t, y(t), w(t))
260
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
(t, y(t), w(t)) E E
(4.19) and hence
y is indeed an admissible trajectory.
_
4.3. Existence of optimal controls Let us define a function g from [0, T] x R" x R" into R by:
(4.20)
g(t, y, f/) = min { f(t, y, w) I(t, y, w) E E and
Clearly, g(t,y, f/)
E
R+ if f/
E
r t ."
and g(t,y, f/) =
+00
otherwise.
Lemma 4.3. g is a positive normal integrand of [0, T] x (R" x R") into R.
Proof Let us take 8 > 0. From the Scorza-Dragoni theorem, there exists a compact subset C. c [O,T] such that meas([O,T] - C.) ~ 8 and for which the restriction of fto C. x R" x K is continuous. We now show that the restriction of g to C. x R" x R" is l.s.c., which will prove that g is a normal integrand (Theorem 1.2). Let (tmYm fin) be a sequence of C. x R" x R" which converges to (i, y, ij). Set: (4.21 ) We wish to show that g(i,y,ii) ~ t. If t = +00, this is trivial. If t is finite, we may assume thatg(tmYm fin) is finite for all n EN and converges to t. From (4.20), there exists co; such that: (4.22) (4.23) Since K is compact, we can extract from the sequence Wm n EN, a subsequence converging to a ill E K, We can then pass to the limit in (4.22) and (4.23):
Wn’
(4.24) (4.25)
(t,
y, w) E
E and
f(t,
y, w)
= f.
Whence necessarily, by (4.20): (4.26)
g(i,
y, w)
y, w)
~ f.
=
iT
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
261
We can now state a sufficient condition for the existence of optimal controls:
Proposition 4.2. We assume the previous hypotheses. Iffor all (t,y) E [0, T] x R", the function g(t,y, .) is convex from R" into R, then there exists at least one
admissible optimal control.
Let us explain this condition. To say that g(t,y, .) is convex means that its epigraph is convex, i.e. that for 0 :::;; t :::;; T and y ERn, the set (4.27) {
(1],
a) ERn x R
13 W E E t y : cp(t, Y, w)
= 1] and
a ;;:: f(t, y, w) }
is convex in R" x R. This implies that its horizontal projection r t ,y is convex, i.e. the set of admissible speeds is convex and compact. Proof Let
Wn>
n E N, be a minimizing sequence of admissible controls and
n EN, their corresponding trajectories. By Lemma 4.2, there exists a con› stant it such that for all n EN,
Yn>
(4.28) From Proposition 4.1, possibly by extracting a subsequence, we may assume that there exists an admissible trajectory y such that:
yn
(4.29)
dyjdt
(4.30)
-+
-+
Y
uniformly
dY/dt for (J(L"o, U).
It only remains to apply Theorem 2.1 to the integrand
We obtain: (4.31)
Tg(t, jl(t), ddYt (t)) dt ~ Io
lim
n--+
00
ITii(t, y,,(t), ddYnt (t)) dt. 0
Taking (4.28) and (4.20) into account: (4.32)
I
Tg(t, y(t), ddYv (t) ~
o
t
lim
a-e co
ITg(t, Yn(t), ddY" (t)) dt 0
t
~ !~~ foT f(t, Y,,(t), wn(t)) dr.
262
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
The right-hand side is equal to inf(8l’). By the measurable selection theorem (Cor. 1.7), it is easily shown that there exists a measurable mapping w: [0, T] ...-+ K such that for 0 ~ t ~ T:
(t, Y(t), w(t)) E E and cp(t, y(t), w(t)) =
~~
f(t, y(t), w(t)) = y(t, Y(t), Thus
(4.33) and
f:
wis an admissible
wis an admissible
~~
(t)
(t)).
control and on substituting into (4.32) we obtain:
f(t, y(t), w(t)) dt
optimal control.
~
inf (ffl!)
Existence of Solutions for Variational Problems
Orientation
In this chapter, we shall study non-convex problems. In Section 1, we shall introduce normal (not necessarily convex) integrands, an important class of functions of two variables without any convexity; we shall establish their main properties, including a measurable selection theorem, and we shall recall the characterization of weakly relatively compact subsets of L 1. In Section 2, these results will be applied to the study of a non-convex optimization problem, and a sufficient condition for the existence of solutions will be given. The final sections will show that a number of problems in the calculus of variations (Section 3) and in optimal control (Section 4) can be put into the above form and from this we deduce theorems on the existence of solutions. 1. NON-CONVEX NORMAL INTEGRANDS 1.1. Definition and main property
We recall that the Borel e-algebra of a topological space is the e-algebra generated by the closed subsets; in other words, the Borel subsets are the sets obtained from open and closed subsets by denumerable union, denumerable intersection, complementation and by any denumerable combination of such operations. A mapping f into R will be called a Borel function if f-l(F) is Borel for every closed set F. For instance, continuous functions are Borel. Let Q be an open subset of R" provided with the Lebesgue measure. All the Borel subsets of Q, especially the open and closed sets, are measurable. We recall that a mapping f: Q -+ R is measurable if the inverse image under f of every closed subset of R is measurable. The Borel functions are measurable and, in general, we have at our disposal the following criterion for measur› ability: Lusin Theorem. A function f: Q -+ R is measurable if and only if, for every compact set K c Q and all s > 0, there exists a compact set K. c K such that meas(K - K.)::; e for which the restriction off to K. is continuous. 231
232
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
The integrands used in the calculus of variations are functions of several variables playing different roles; by and large, they are measurable with respect to some variables and l.s.c. with respect to others.
Definition 1.1. If B is a Borel subset of RP, a mapping f of Q x B into R is termed a normal integrand if: (1.1)
for almost all x
(1.2)
there exists a Borel function f(x, .)for almost all x E Q.
E
Q,j(x,.) is l.s.c. on B
f
Qx B-
R such that lex, .) =
A first consequence of this definition is that for all a E B,j(. ,a) is measurable on Q. Better still, if u is a measurable mapping of Q into B, the function x 1-+ f(x, u(x)) is measurable on Q. Indeed, it is almost everywhere equal to the function x -l(x,u(x)), which is measurable sincefis Borel. We note that this property is no longer satisfied if, instead of assuming that f is a normal inte› grand, we merely assume it to be measurable in x and l.s.c. in a. Later on, we shall see that functions measurable in x and continuous in a are normal inte› grands. Furthermore, it follows from the definition that: iffis a normal integrand, Afis a normal integrand for all A E R; iffand g are normal integrands, (f + g) and inf(j,g) are normal integrands; if (f,,)neN is a denumerable family of normal integrands, sup, eN In is a normal integrand. The study of normal integrands depends on the following characterization, which is a Lusin theorem "uniform" in the second variable:
Theorem 1.1. Let B be a Borel subset ofRP. For f: Q x B - R to be a normal integrand, it is necessary and sufficient thatfor every compact set K. c Q and all 8> 0, there exists a compact set K. c K such that meas(K - K.) ~ e for which the restriction off to K. x B is l.s.c. Proof The sufficient condition is obvious: we take 8 = lin and let K’ = UneN Kiln’ Then meas(K - K’) = 0, K’ is Borel,jis Borel on K’ x B andf(x, .) is l.s.c. for all x E K’. To show the necessary condition, we begin by modifyingf on a Borel set of Q with null measure so thatf(x,.) becomes l.s.c. for all x E Q andfbecomes Borel over all of Q x B. Using if necessary an isomorphism of R onto [0, I], we may assume thatftakes its values in [0,1]. Since B is a subspace of RP, it possesses a denumerable basis dlt of open sub› sets. Let us denote by
lui U E dlt, k E Q;
: :;
k ::::; 1 }, where Q is the set of rationals.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
233
Clearly tP is denumerable and for all l.s.c. functions h: B -+ [0,1] we have h = sup{
En
=
{x
E
QI f(x,.)
G; = {(x,a)EQ
X
~
Sincefis Borelian and
f(x, a) = sup
If the compact set K c Q and the number e > are given, we choose for all n EN, by virtue of Lusin’s Theorem, a compact space K; c K such that meas(K - K n) ::::; e2-(n+!l and that the restriction of IE. to K; be continuous. Let K. = UnEN K; Then meas(K - K.) ::::; e and the restriction of
f(x, a)
=
f(x, a)
=
if (x, a)
+
E
C,
if (x, a) ¢ C,
00
is a positive normal integrand since it is Borelian and trivially satisfies (1.1). In particular, for all measurable mappings u of Q into B the following three conditions are equivalent to each other:
(1.3)
(1.4) (1.5)
(x, u(x))
E
C
U(X) E C, Lf(x, u(x)) dx <
a.e, a.e.
+
00.
(1) We admit here the fact that the projection of a Borel set is measurable. This is a difficult result; it is a consequence of, e.g., Choquet’s capacity theorem.
234
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
1.3. Second example: Caratheodory functions Definition 1.2. Let B be a Borel subset of RP. A mapping f: said to be a Caratheodory function if (1.6) (1.7)
for almost all x for all a
E
E
Q
x B -+
R is
Q,f(x,.) is continuous on B,
B,j(. ,a) is measurable on Q.
Proposition 1.1. Every Caratheodory function is a normal integrand. Proof By modifyingf on a Borel set of Q with null measure and by using an isomorphism ofR onto [0,1], we may assume thatfis measurable in x for all a, is continuous in a for all x, and takes values in [0, I]. Once again we introduce a denumerable base of open sets tJIt of B and the family rp of l.s.c. functions of B into [0,1] defined by rp = {k1 u luE tJIt, k E Q, 0 ~ k ~ I}. For all l.s.c. functions h of A into [0,1] we have h = sup{cp E rpl cP ~ h}. We now introduce a dense denumerable family 88. Enumerating rp = {CPn}neN and for all n E N and a E 88let us set: E n. a = {
If(x, a)
X E Q
~ CPn(a) }.
Since f(. ,a) is measurable, En. a is measurable and hence En = measurable
naeB8
En. a is
Now f(x, .) is continuous, CPn is l.s.c. and 88is everywhere dense. We deduce that
En = {
X E
Q I f(x, a) ~ CPn(a) Va
E
B }.
And by definition of the family rp:
f(x, a) = sup CPn(a) Idx). neN
For each n EN, there exists a Borel subset C; of almost everywhere. Let:
Q
such that IE
=
Ie
](x, a) = sup CPn(a)ldx). neN
The function/is Borel on Q x B as the least upper bound of Borel functions, andf(x, .) = J(x, .) for almost all x. Hypothesis (1.2) is thus verified; hypothesis (1.1) is covered by (1.6).
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
235
In this particular case Theorem 1.1 takes the following form, known as the Scorza-Dragoni theorem:
Scorza-Dragoni Theorem. A mapping f: Q x B --+ R is a Caratheodory function if and only iffor all compact sets K c Q and all 8 > 0, there exists a compact set K. c K such that meas(K - K.) ::;; 8 for which the restriction of fto K. x B is continuous. Proof Let the compact subset K and 8 > 0 be given. Since f is a normal integrand, there exists a compact set K+ c K such that meas(K - K+) ::;; 8/2 and that the restriction off to K+ x B is l.s.c. But -fis also a normal integrand and we can thus find a compact set K_ c K such that meas(K - K_) ::;; 8/2 and for which the restriction of -fto K_ x B is l.s.c. If K. = K+ n K_,fwill be l.s.c. and U.S.c. and hence continuous on K. x A and meas(K - K.) ::;; 8. The converse is an easy consequence of Theorem 1.1.
1.4. A measurable selection theorem We shall now assume that B is a compact subset of RP. For almost all x E Q, there thus exists an a(x) E B wheref(x,.) attains its minimum. We will show that we can choose a(x) in such a way that the mapping a defined from Q into B is measurable. Q
Lemma 1.1. Let B be a compact subset of RP and g a normal integrand of x B. We set
(1.8)
go(x, a) = 0 gO(X’ a) =
+
if g(x, a) = min { g(x, b) } beB
00
if g(x, a) > min { g(x, b) }. bEB
Then go is a normal integrand. Proof Using an isomorphism, we may assume that g takes its values in [-1,1]. For every 8> 0 and every compact K c Q we can find a compact subset K. c K such that meas(K - K.) ::;; 8 and that the restriction of g to K. x B is l.s.c. Define sp : K. --+ [-1,1] by:
cp(x) = min {g(x, b) }. bEB
Let us show that cp is l.s.c. Let (Xn)nEN be a sequence of K. converging to a subsequence X n’ such that:
x. We extract (1.9)
236
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Since B is compact, there always exists an’ E B such that
lim g(x n", an") ~ g(x, a)
11" -. 00
and afortiori lim
(1.10)
n"
~
--+ 00
Whence by comparing (1.9) and (1.10), the lower semi-continuity of
E
K. x B I g(x, a) =
This is the set of points where two I.s.c. functions coincide and is thus a Borel subset of K. x B. Moreover, for all x E K.. C; = {a E BI(x,a) E C} is closed since it is the set of points where an I.s.c. function attains its minimum. This means that go, which is none other than the indicator function of C, is a normal integrand of K. x B. In particular, we can find a compact subset K 2 c K. such that meas(K. - K 2 . ) ::;; 28 and for which the restriction of go to K 2 is I.s.c. Let us collect our results together: for all 8 > 0 and all compact sets K c Q, we have found a compact set K 2 c K such that meas(K - K 2 . ) ::;; 28 and for which the restriction of go to K2 X B is I.s.c. Hence go is a normal integrand of
QxB. Q
Theorem 1.2. Let B be a compact subset of RP and g a normal integrand of x B. Then there exists a measurable mapping it: Q -+ B such that for all
XE
Q:
(1.11)
g(x, u(x)) = min { g(x, a) }. aeB
Proof We define go by formulae (1.8). We then take a sequence am n ~ 1, which is dense in B. We now define by induction a sequence g., n ~ 1, of normal integrands in the following way: hn(x, a) = gn(x, a)
+
la -
ani
gn+ l(X, a) = 0 if hn(x, a) = min { hn(x, b)} gn+ l(X, a) =
+
beB
00
if hix, a) > min { hn(x, b) }. beB
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
237
Successive applications of Lemma 1.1 show that the gno n EN, are all normal integrands. Thus g = sUPneNgn is also a normal integrand. But it can be easily verified that domg(x, .) has been reduced to a point for all x E Q, i.e. g is ofthe form:
g(x, a) = 0 if a = u(x)
Ig(x, a) =
+
00
if a "# u(x).
The function u thus defined is measurable, since g is a normal integrand and satisfies (1.11) since g ~ go. In particular we deduce that the mapping x f-+ minaeB{g(x,a)} of Q into R is measurable. Theorem 1.2 includes as a special case the following measurable selection theorem.
if C is a Borel set of Q x B
Corollary 1.1. If B is a compact subset of RP, whose sections:
C,
=
{a
E
B I(x, a)
E
C}
are closed and non-emptyfor almost all x, then there exists a measurable map› ping u: Q ~ B such that for almost all x
We say that u is a measurable selection of C. This corollary is proved by applying Theorem 1.2 to the indicator function of C. 1.5. Polars and bipolars of normal integrands Let us now take B = RP, and consider a normal integrandf: Q x RP --+ R. For all fixed x E Q, the polar of the function f(x, .) will be a mapping of RP into R denoted by: ~*
1-+
f*(x ; C)•
Proposition 1.2. Iff is a normal integrand of Q x RP, then f* is a normal integrand of Q x RP.
Proof For all x E Q, f*(x,.) is a convex I.s.c. function. It only remains to verify hypothesis (1.2) for f*. To do this we define: fn(x, ~) =
+
fn(x, ~) = f(x,~)
00
if I~I if I~I
> n ~ n.
238
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
We havef = infneNf., hence (by 1(4.6)), for all x
(1.12)
E
Q:
f*(x;.) = sup f:(x; .).
neN
By definition
(1.13)
f:(x; ~*)
=
<~,~*
sup {
I~I.,
n
) - fn(x, ~)}.
For all x E Q, f:(x;.) will be either identically equal to -00, or a finite convex function of RP into R. Let us take ~* E RP, and any compact set K c Q. For all s > 0 there exists a compact subset K. c K such that meas(K - K.) ~ e and for which the re› striction of J,. to K. x RP is l.s.c. Since the balls of RP are compact, the family of mappings -J,.(’,~) + <~,~*>, for I~I ~ n, will be equi-l.s.c, on K . From equation (1.13) we also deduce that f.*(. ; ~*) is U.S.c. on K . Since e > 0 is arbitrary, fn*(’ ; ~*) is measurable on K. Since the compact set K is arbitrary f,,*(. ; ~*) is measurable on Q. For all n E N,fis therefore a Caratheodory function, and afortiori, a normal integrand. We thus haveJ,.*(x;.) = In*(x; .), for almost all x, wherefn* is Borel. From (1.12) we deduce that f*(x; .) = l*(x; .) for almost all x, where 1* = sUPneNl: is Borel. Whence (1.2) forf*. By repeating this operation, we arrive at the r-regularization of the function f(x, .) which will be denoted by: ~ t--* f**(x; O. We at once obtain the following corollary of Proposition 1.2:
Proposition 1.3. Iff is a normal integrand of Q x RP, then f** is a normal integrand of Q x RP. 1.6. Lower semi-continuity of integrals Let us now pass from the questions of measurability to those of integrability. Here we give an easy consequence of Fatou’s lemma which will be extremely useful to us in the future:
Proposition 1.4. Let f be a normal positive integrand of Q x RP and (Un)neN a sequence ofmeasurable mappings of Q into RP, converging almost everywhere to iI. Then we have:
(1.14)
f Q
f(x, ii(x)) dx
~
lim
"~OO
f Q
f(x, un(x)) dx.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
239
Proof We have a sequence of positive measurable functions to which we can apply Fatou’s Lemma: (1.15)
But sincef(x,.) is l.s.c. for almost all x (1.16)
E Q:
f(x, u(x)) ~ lim f(x, un(x)) n- 00
a.e.
Whence the result, on substituting (1.16) into (1.15).
Corollary 1.2. Let f be a normal positive integrand. Thefunction F :u
>--+
Lf(X, u(x)) dx,
is positive and l.s.c. ofL"(Q) into H,for all
IX,
1~
IX ~ 00.
1.7. Weak compactness inL 1(n) To conclude these preliminaries, let us recall the characterization of the weakly relatively compact subsets of L 1 (Q).
Theorem 1.3. Let!F c L 1(Q). The following statements are then equivalent
to one another:
(a) from any sequence (UJneN of !F, we can extract a subsequence which is weakly convergent in V ; (b) for all e > 0, there exists A > such that:
f
Vu E!F,
(lui" i.)
lu(x)1 dx
~
s;
(c) for all s > 0, there exists 0 > such that we have fB lu(x)ldx ~ e for all u E !F and all measurable B ofmeasure ~ 0; (d) there exists a positive Borel function iP: [0, 00 [ -+ H+ such that lil!4t"’( iP(t)/t = +00 and
sup SiPo lui < lIE§
+
00.
240
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Condition (b) is called equi-integrability. The equivalence (a) <0> (b) <0> (c) is Dunford-Pettis’ compactness criterion. The equivalence (b)-e- (d) is due to de la Vallee-Poussin.
Lemma 1.2. The function l.s.c. and increasing.
(]1
introduced in (d) can be assumed to be convex,
Proof To say that lim t (]1(t)/t = +00 means that for all mER, (]1 admits an affine minorant with slope m. Then so does (]1**, and hence (]1**(t)/t-+ +00 when t -+ +00. The function (]1** is convex and l.s.c. and attains its mini› mum ii at i: -> ",
a=
inf
(]1 =
min
= (]1**(I).
(]1**
It decreases on [O,i], then increases over [i,+oo[. Let us define a convex function increasing and l.s.c. on [O,+oo[ by
s.
~(t)
=
a on [0, t]
$(t) = (]1**(t) We have $ <
(]1, and
on
[t, + 00[.
thus
J 0 lui
sup ~ uefF
~ sup uefF
J(]10 lui
< 00.
Theorem 1.3 is a very deep result. We are going to use it to prove a generaliz› ation of Lebesgue’s theorem which will be extremely useful in what follows:
Corollary 1.3. Let (Un)neN be an equi-integrable sequence of L 1(Q) such that uix) -+ u(x) almost everywhere. Then u is integrable, and u; -+ U in £1(Q). Proof We begin by showing that, under the given hypotheses, the sequence u; converges weakly to u in L 1 (Q) . For this it is sufficient to show that we can extract from it a subsequence converging weakly in L 1(Q) to u. Now there exists a subsequence Un’ converging weakly in £1(Q) to a function v (Theorem 1.3, (b)=:>(a»). From Mazur’s lemma, we can find a sequence of convex com› binations Vn ’ E co Up’;;’n’ {up’} converging to v in£1(Q), and so we can extract a sequence Vn" converging to v almost everywhere:
(I.l7)
a.e.
But by hypothesis:
(I.l8)
Vn,(X) E co
U
p’",n’
{up,(x)} - u(x)
a.e.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
241
Comparing (1.17) and (1.18), we obtain v = U almost everywhere, and the sequence u; thus converges weakly to u. We immediately deduce from this that it also converges in the norm: it is sufficient to apply the preceding result to the sequence offunctions JU n - ul. We can easily verify that it is equi-integrable and that it converges almost everywhere to zero. It thus converges weakly to zero, and in particular:
If for example there exists a E V(Q) such that for all n, u.(x) :::; a(x) almost everywhere, the sequence (Un)neN will then be equi-integrable. We thus recover Lebesgue’s theorem.
2. AN OPTIMIZATION PROBLEM 2.1. The integrandf: definition, first properties
Let P: [0,+00[ --+ R U {+oo} be a non-negative increasing, convex, l.s.c. function such that lim P(t) =
(2.1)
1~
t
+
00.
Let us consider a mappingf of Q x (Rt X R’") into R. We shall assume that it is a normal integrand, i.e. that it satisfies (1.1) and (1.2) with B = Rt X Rm, and that we have the estimate
(2.2)
P( I~I
)~
f(x, s, ~)
In particular, f is non-negative, f(x, ., .) is l.s.c. for almost all x E Q, and f(. ,s,~) is measurable for all (s,~) E Rt X R", By epij(x,s) we shall denote the epigraph of the function j(x,s,.) in R" x R. We shall now prove a continuity property of the multi-valued mapping (x,s) --+ coepif(x,s). Lemma 2.1. Let E be a metric space and
R, satisfying (2.3)
Then,for all
(2.4)
qJ
a l.s.c. mapping of E x Rm into
’ieEE,
e E E, we have
n co £>0
U
Ie-iii.,;’
epi P(e,.) = co epi lP(e, .).
242
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Proof It is trivial that the right-hand side is included in the left-hand side. For the converse, take any:
(2.5)
(~, a) 1= co epi cp(e, .).
There exists an affine hyperplane of Rm x R which strictly separates (~,a) from co epicp(e, .). Ifthis hyperplane is non-vertical, it is the graph of an affine function ( over R" such that:
(2.6)
ii < f’(~)
(2.7)
f(e) < cp(e, e).
Ve E R",
If the hyperplane in question, denoted by :¥t, is vertical, there exists an affine function (’ over Rm such that (fee) = 0 for (e,a) E :¥t,ne) > 0 and (IW < 0 for E domcp(e, .). But, from (2.3), cp is non-negative; the function (= c/", for c > 0 sufficiently large, will thus also satisfy (2.6) and (2.7). From (2.3) we deduce that there exists M > 0 such that
e
(2.8)
e
Since the balls, of Rm are compact, the family IP(., e), for I I ~ M, of mappings from E into R, is equi-l.s.c, Let p be an increasing homeomorphism ofR onto [-1,1].(1) We then set:
(2.9)
m = min {p I~I"
0
M
cp(e, e) - p
0
fm }.
It is the minimum of a l.s.c. function on a compact set. It is thus attained and we have m > 0 from (2.7). By equi-lower-semi-continuity, there exists e > 0 such that Ie - el ~ e and I I ~ M imply that:
e
(2.10)
p
0
cp(e, e) ~ p
0
cp(e, e) - m
po CP(e, e) ~ po
(2.11)
rm
by (2.9). Finally, let:
Ie -
(2.12)
el
~ e and
lei
~ M => cp(e,
e)
~
(m.
Grouping together (2.8) and (2.12) and taking (2.3) into account, we see that: (2.13) (1)
For example p(s) = (2/n) arctan s.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
243
But (2.6) means that (e,ii) belongs to the lower open half-space determined by t, and (2.13) means that Ule-el". epiqJ(e,.) belongs to the upper closed half-space. But then co Ule-ell>. epiqJ(e, .) also belongs to the upper closed half-space, and hence: a)¢ co
(~,
A fortiori:
(2.14)
(~, a)¢
U
Ie -el<> "
n co
,>0
epi qJ(e .).
U
le-el,,’
epi ot«, .).
The fact that (2.5) implies (2.14) gives us the desired inclusion:
n co Ie-eke n epi qJ(e,
co epi qJ(e .) c
,>0
Corollary 2.1. If f is a normal integrand of D x Rt we have for almost all xED, (2.15)
n co ,>0
U
IS-51,;;’
.). X
Rm satisfying (2.2),
epi I(x, s) = co epi I(x, s).
Proof It is sufficient to apply Lemma 2.1 to the function f(x .. ,.) over R1 X R", at any point xED which makes it l.s.c.
2.2. A lower semi-continuity result We are now in a position to state the fundamental result of this chapter which yields a property of lower semi-continuity:
Theorem 2.1. Let f be a normal integrand of D x (Rt (2.2)
ep( I~I)
X
R"), such that:
~ I(x, s, ~)
where ep: R+ f-+ R+ is a convex increasing l.s.c. function satisfying (2.1) and,
(2.16)
Vex, s) E
Q X
Rf ,
I(x, s,.) is convex over R".
Let (Pn)neN be a sequence converging weakly to ft in V(Q)m and (Un)neN a sequence ofmeasurable functions converging almost everywhere to ii. Then: (2.17)
LI(X, u(x),
p(x)) dx
~ !~~ LI(X, un(x), Pn(x)) dx.
244
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Proof Since f is a normal non-negative integrand, all the written integrals have the same sign. If the right-hand side of (2.17) takes the value +00, the in› equality is trivial. Otherwise, by extracting a subsequence, we may assume that:
(2.18)
lim
n-+
00
f
Q
+
f(x, un(x), Pn(x)) dx = c <
00.
We now apply Mazur’s lemma (cf Chap. I, Section 1) to the sequencep .. which is weakly convergent in V. There exists a sequence of convex combinations LZ=n’ akPk, with ak > 0 and L~=n’ ak = 1, converging to pin V. We can thus extract a subsequence L:=n akPk converging almost everywhere to p: l
N
(2.19)
k
L
akPk(x)
=n’
Let us take a point x
u.(x) -+ u(x) as n -+ for all n’, we have: (2.20)
N
(L
k=n’
p(x)
n’
a.e. as
-+ 00.
where the convergence (2.19) occurs and where A fortiori, Un’ (x) -+ u(x) when n’ -+ 00. In Rm x R,
E Q
00.
akPk(x),
-+
N
2:
k=n’
ad(x, uk(x), Pk(X)))
co
E
N
U epi
f(x, uk(x))
k=n’
and afortiori:
(2.21)
N
(2:
k= n
akPk(x),
N
2:
k = n’
ad(x, uk(x), Pk(X)))
E
co
U epi
q ;;., n’
f(x, uq(x)).
no
We now take e > O. From (2.13), there exists sufficiently large to give us /U2(X) - u(x)1 ~ e for all q ~ n~. Hence for all n’ ~ n~, we have:
(2.22)
N
(2:
k=n’
N
2:
akPk(x),
k=n’
ad(x, uk(x), Pk(X)))
E
U
co
epi f(x, s).
Is-u(x)/,;;,
By making n’ go to infinity, we obtain from (2.19):
(2.23)
(p(x), lim
N
n’r-r o:
2:
k=n’
o:d(x, uk(x), Pk(X)))
E
co
U
Is-u(xll,;;,
epi f(x, s)
and as this is true for all e > 0: N
(2.24) (p(x), lim 2: ad(x, uk(x), Pk(X))) n’-+OOk=n’
E
n co
U
’>Ols-u(x)I,,’
epi f(x, s).
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
245
From Corollary 2.1, we have: (2.25)
U
e> 0
U
co
Is - u(xl! ", e
=
epi f(x, s)
co epi f(x, u(x)).
But because of hypothesis (2.16), the function !(x,u(x), .) is convex and I.s.c., and its epigraph is therefore closed and convex. Finally (2.24) can be written as:
(2.26)
(p(x), lim n'
N
-+ 00
L
k:;;; n’
ad(x, uk(x), Pk(X)))
E
epi f(x, u(x))
which by definition means that:
(2.27)
f(x, u(x), p(x))
~
lim n’
-+
N
L ' ad(x, uk(x), Pk(X)).
ro k =
n
We now integrate both sides over Q:
But all the integrands are positive, which allows us to apply Fatou’s lemma: (2.29)
But from (2.18) it is easy to deduce that the right-hand side of (2.30) is equal to c. Indeed, for all s > 0, we can find no such that (2.31)
(2.32)
c - s
C -
f.
~
~
Lf(X, un(x), Pn(x)) dx
ktn. a k Lf(X, uk(x), Pk(X)) dx
~
~ c+e
c + e,
246
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
whence finally
(2.33)
}i~C()
kt (Xk Lf(X, uk(x), Pk(X)) dx = c. n
Returning to (2.30) and (2.18), we obtain the desired result:
Lf(X, u(x), p(x)) dx
~
c =
}~n;,
f
f(x, uk(x), Pk(X)) dx.
2.3. The case wherefis not convex in ~ We now return to the general case wherefis a normal integrand satisfying (2.2) but no longer satisfying (2.16). Then, of course, we no longer have Theorem 2.1 at our disposal. It is natural to introduce f**(x,s; .), the r› regularization of the functionf(x,s, .). From the results of Section 1, this is a normal integrand of (D x Rt) X R", but the question arises as whether it is a normal integrand of D x (Rt X R"). (l)
Proposition 2.1. Iffis a normal integrand of D x (Rt
(2.2)
cI>( 1~1
)~
X
Rm) satisfying
f(’x, s, 0
where cI> answers to the above description, thenf** is also a normal integrand of D x (Rt X Rm) and satisfies
(2.34)
cI>( I~I
)~
f**(x, s; ~).
Proof By (2.2), the convex l.s.c. function cI>(j.l) is everywhere less than f(x,s, .), Taking the r-regularization of both sides, we obtain (2.34). Let us now take any compact subset KeD, and 8> O. Since/is a normal integrand, we can find a compact subset K. c K such that meas(K - K) ~ 8 and such that the restriction of f to K. X Rt X Rm is l.s.c. (Theorem 1.1). Moreover we know that epif**(x,s) = coepif(x,s). From Lemma 2.1 applied to / over (K. x Rt) X R", we have
(2.35)
epi f**(x, s) =
n co Ix U
s> 0
-xl,;:.
epi f(x, s).
S-Sl~E
(1) We already know thatf**(x, s; .) is l.s.c, in thatf**(x, .; .) is l.s,c. in (s, e).
e, but we are not yet in a position to state
247
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
But it is clear that: (2.36)
co
U
ls-sl .. e
x - xl.. s
epi f(x, s):::>
[x -
U
xl .. s
co epi f(x, s)
Is-sl,;; e
which enables us to write (2.35) in the form: (2.37)
epi f**(x, s)
:::>
n
.>0
U co epi
lx-xl . Is-sl
f(x, s)
.
or again, replacing coepij(x,s) by epij**(x,s), (2.38)
epi f**(x, s):::>
n
£>0
U
lx-x!,;,
epi f**(x, s).
Is -51,;;’
Let us now take a sequence (XmSm’n), n E N, converging to
K. x Rt x R", We obtain:
(2.39)
(x, s,~,
lim f**(x n, Sn; ~n))
E
n
.>0
U
lx-xl ...
(x,s,~)
in
epi f**(x, s),
Is-sl,;;’
and by virtue of (2.38): (2.40)
(x, s,~,
lim f**(x n, Sn; ~n))
E
epi f**(x, s)
which by definition means that: (2.41 )
Thus we have shown that for any compact set K c {J and for all B> 0, we can find a compact subset K. c K such that meas(K - K) ::;; B for which the restriction ofj** to K. x Rt x Rm is l.s.c. By Theorem 1.1,/** is thus a normal integrand of {J x (Rt X R"), We can now have a partial extension of Theorem 2.1 to the non-convex case, making use ofj**. The result is as follows:
Proposition 2.2. Let j be a normal integrand of {J x (Rt x R") satisfying (2.2)
248
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Let (Pn)neN be a sequence converging weakly topin Lt",(O) and (Un)neN a sequence ofmeasurable functions converging almost everywhere to u. Then (2.42)
f
Q
f**(x, u(x); p(x)) dx
~
lim
n--+
00
f
Q
f(x, un(x), Pn(x)) dx.
Proof It is sufficient to apply Theorem 2.1 to f**, and to make use of the inequality f** «t. We obtain: Lf**(X, u(x); p(x)) dx
~ !~~
Lf**(X, un(x); Pn(x)) dx
~ !~~
Lf(X, un(x), Pn(x)) dx.
2.4. Calculus of variations: existence of solutions by convexity We shall now formulate an optimization problem, which embodies a large class of problems in the calculus of variations, and apply to it the preceding results. As before, we are given a l.s.c., convex, increasing function rp: [0, +00 [ -i>› R+ which satisfies:
(2.1)
lim rp(t) dt = t
1--+ 00
+
00.
By L~ we shall denote those (classes of) measurable mappings P from Q into Rm (modulo equality almost everywhere) for which fn rp 0 !pl < +00.0> From (2.1) it is clear that L~ c V(Q)m. If, for instarice we took for rp the function t 1-+ t", with 1 < a < 00, the set L~ would be none other than L""(Q)m which will also be denoted by n. If we take for rp the indicator function of the interval [0,1], L% coincides with the unit ball of Loo(Q)m (or
r:
L~).
We are given a normal integrand f of Q(R"" x R") into il, such that there exists a function a E L I( 0) which satisfies:
(2.43)
a(x) + rp(IW ~ f(x, s, ~).
(I) The L~ are Orlicz classes. For the theory of Orlicz spaces see M. A. Krasnosel’skii and Ruticki [1], A. Fougeres [1].
249
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
We are also given a weakly closed subset 0If of L~, and a mapping r:fJ of 0If into with 1 ~ fJ ~ 0Ci, which satisfy the following property:
u,
n
L~
if a sequence (Pn) of 0If converges weakly to p in L~ and if sup III cP 0 IPnl < +0Ci, then we can extract from the sequence (r:fJPn) a subsequence which converges almost everywhere to r:fJp
(2.44)
Assumption (2.44) is satisfied if, for example, r:fJ maps the sequences (Pn) which converge weakly to p in L~ and are such that I cP 0 Pn ~ constant ’tin, into sequences (r:fJPn) which converge strongly to r:fJp in Lg: we then say that r:fJ is a (cP, fJ)-compactifier. If cP(t) = t", I < a < 0Ci, this amounts to saying that r:fJ maps the bounded and weakly convergent sequences of L~ into strongly con› vergent sequences of Y; indeed, the topologies oil: l,L 00) and u(La,L a’) coincide I/a + I/a’ = 1. We say on the bounded subsets of L~, since L~ is dense inL~, that r:fJ is an (a,fJ)-compactifier. Here are the main examples:
Proposition 2.3. Let t < a < 0Ci. Ifr:fJ is a compact continuous linear mapping of L~ into LV, with t ~ fJ ~ 0Ci, then r:fJ is an (a, fJ)-compactifier. We recall that, by definition, a continuous linear mapping of L::’ into is called compact if it maps the bounded subsets of L’:" into relatively compact subsets of Lf. Proposition 2.3 follows directly.
Lf
Proposition 2.4. If r:fJ is a continuous linear mapping of
1 < fJ
~
0Ci,
L~
into Lf, with
then r:fJ satisfies (2.44).
Proof Since Q is bounded, Lf(Q) c LHQ), and the bounded sets of Lf(Q) are weakly relatively compact in L}(Q). The mapping r:fJ maps the bounded sets of L~ into weakly relatively compact sets of L ~; by Grothendieck [1], theorem V.4.2 it will map the weakly compact subsets of L~ into compact subsets of L~. Let us now take cP into account. If the I cP 0 /Pn I’s are uniformly bounded, the (Pn)neN form a weakly relatively compact subset of Lt", (Theorem 1.3, de la Vallee-Poussin’s criterion). The (r:fJPn)neN thus form a relatively compact subset of L\, and we can extract a subsequence (r:fJPnk)keN converging in L~. Since the Pn converge weakly to p, the r:fJPn converge weakly to rip. The limit of the sequence (r:fJpnk)keN in L~ can therefore only be r:fJp. Finally, we can extract from (r:fJPnk)keN a subsequence which converges almost everywhere to r:fJp.
At last we are in a position to state the optimization problem:
(&»
pe~!L:
Lf(X, r:fJ p(x), p(x)) dx
250
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
which can be put into the equivalent form:
r [(x, u(x), p(x)) dx.
~~f
(&’)
pe"llnL~
u= ’i#P
JQ
From Theorem 2.1, we immediately deduce an existence criterion for solu› tions to problem (&’):
Theorem 2.2. Let f be a normal integrand of Q x (Rt
a(x)
(2.43)
(2.45)
+
V(x, s) E Q
~
[(x, s,
R f,
X
~),
X
R") satisfying
with a E L1(Q)
[(x, s,.) is convex on R".
Let t’§ be a mapping ofL~ intoL~ and qt a weakly closed subset ofL1satisfying
(2.44). Problem (&’) admits at least one solution.
Proof We set g(x, s,~)
=
that
f(x, s,~)
(2.46) (2.47)
- a(x). This is a normal integrand such ~ g(x, s, ~)
V(x, s) E
Q X
R ’,
g(x, s,.) is convex on R",
It is clear that:
(2.48)
L[(X, u(x), p(x)) = Lg(x, u(x), p(x))
+ L a(x) dx.
The last term is a constant. Let us therefore take a minimizing sequence of problem (&’); and let us set u; = t’§Pn’ By definition, P« E rJU for all n, and:
(Pn)neN
(2.49)
Lg(X, un(x), Pn(x)) dx
-+
inf(&’) - L a(x) dx.
From (2.46) we deduce that:
(2.50)
L
0
IPnl
~
constant.
From Theorem 1.3, we can extract from (Pn)neN a subsequence Pn’ which is weakly convergent to p in n; Since t’§ satisfies (2.44) we can extract a sub-
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
251
sequence Pn" such that Pn" converges weakly to p in L~ and C§Pn" converges to C§p almost everywhere. Applying Theorem 2.1 :
(2.51) Adding
(2.52)
Lg(X, u(x), p(x)) dx
~ !~~
f
~ !~~
Lf(X, un(x), Pn(x)) dx.
g(x, un(x), Pn(x)) dx.
In a(x)dx to both sides: Lf(X, u(x), p(x)) dx
That is, as the sequence (Pn) is minimizing:
(2.53)
Lf(X, u(x), p(x)) dx
But P E dJj is the weak limit of the Pn" fore that Ii = C§p is the solution of (9).
E
~
inf(9).
dJj’s. From (2.53) we conclude there›
2.5. Calculus of variations: relaxation In the case where we no longer assume f(x,s,.) to be convex, problem (9) in general no longer has a solution. We shall see that it is natural to associ› ate with problem (&’) the following problem, termed the relaxed problem,
(&’~)
Inf
pe"llnL!.
fn f**(x,
C§ p(x); p(x)) dx
or again
r f**(x,
~~!
ped/uinL~=
’§ p
In
u(x); p(x)) dx.
From Theorem 2.2 we immediately deduce Proposition 2.5. Let f be a normal integrand of D(pt x R’"), satisfying
(2.43)
a(x) +
~
f(x, s, ~),
witha E D(D).
Let C§ be a mapping which is a (
252
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Remark 2.1. Since f**
3. EXAMPLES We shall describe several examples of mappings rg, satisfying (2.44), and arising naturally in variational problems. We shall use these examples in the following chapter, where the relationship between problems (9) and (9~) will be stated more precisely. Example 1 Let 0 be a very regular open subset of R", and A the Laplace operator:
(3.1 )
Au
= flu.
For p given in U(O), there exists a unique u in HMO) such that:
(3.2)
Au=p
a.e.
and the mapping p 1-+ U is linear and continuous from L 2 (0) into HMO). Since the injection of HMO) into L 2(0) is compact (cf Lions and Magenes, [1 D, the mapping p 1-+ U is linear and compact from L 2(0) into itself. If we call this operator rg, it then satisfies property (2.44) trivially with lP(s) = S2: it is a (2,2)-compactifier. Example 2 In a much more general way, rg can be the Green operator of any regular elliptic problem. Let A be a differential operator of order 2m in a very regular open subset o < R":
(3.3)
Au = A(x, D)u =
L (-
lal,IPI’;m
1) laIDa(aap(x )DPu)
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
253
with
(3.4) and
(3.5)
the operator A is uniformly elliptic in Q.
We denote by oj u] onJ thej-th normal derivative of a function u on the bound› ary oQ, as it is defined by the usual trace theorems (oJujon J E Hr-J-’!{oQ) if u E H’ (Q)). Under those assumptions, for p given in £l(Q), there exists a unique u in H 2 m(Q) satisfying:
(3.6) (3.7)
(
Au=p
a.e. in Q
oJ u =0 on
on oQ, forO~j~m-1
-J
and the mapping I:§:p ~ u is linear and continuous from £2(Q) into H 2m(Q), and thus linear and compact from £l(Q) into itself. It is a (2,2)-compactifier.
Example 3 A being defined as in the preceding example, denote by Q the cylinder T a positive real number, and consider the parabolic
Q x (0, T) of Rn+l, with
equation: (3.8)
auat + Au = p
(3.9)
oJ u onJ =0
(3.10)
u(x,O) =0
a.e, in Q
on oQ x (O,T), for 0 ~j~
m-l
a.e. in Q.
For every p E £2(Q) thereisa unique solution u E H 2m.l (Q), and the mapping I:§:p ~ u is linear and compact from £l(Q) into itself. Example 4 Assume moreover that A is coercive on H’8(Q), and symmetric:
3c> 0 =
cllcpllHg’
254
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
Consider the hyperbolic equation: 02 U 2
-
ot + Au = p
(3.11) (3.12)
Oi U -
oni
(3.13) .
a.e. in Q
on oD x (0, T), for 0 ::::;j::::; m - 1
= 0
u(x, 0)
=
~~
(x,O)
= 0 a.e. in D.
n
For every p E L2(Q) there is a unique solution u E m,l(Q), and the mapping C§:p 1-+ u is linear and compact from L2(Q) into itself. Example 5 Let us now consider cases where C§ is non-linear. If D is bounded and if 1 < Y < 00, forp given inV’ (D), where I/y + l/y’ ~ I, we verify with the help of Theorem 3.1 and Remark 3.4 from Chapter II that there exists a unique u in WA,1 (0) which satisfies: (3.14)
Au =
–~(1~11-2~)
i=
lOX;
OX;
OX;
=
p.
We term C§ the non-linear mapping which results from this and which sends V’ (0) into V (D). We now have the following result: Lemma 3.1. The mapping C§ defined by (3.14) is a (y’, y)-compactijier.
Proof We note that:
whence by Poincare’s inequality it follows that:
If now a sequence Pm converges top weakly inV’ (0), it is bounded inV’ (0) and the sequence of Um= C§Pm’s is bounded in W~,1 (0); by extracting a subse-
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
quence we can assume that Um converges to a limit hence strongly in V (Q). Then:
U
weakly in
255 W~•1
(Q) and
which obviously implies that:
and we deduce, as in Lemma II.3.3 that: Au =
Thus lemma.
Um
= ’§Pm
converges to
U
= ’§P
strongly in V (Q), which proves the
We can generalize this to the more general situation of Theorem II.3.!. Taking all the hypotheses of this theorem, together with (3.26), we assume moreover that V c V (Q) with compact injection and dense image, so that the dual V’ of V contains V’ (Q). In this case, for P given in V’ (Q) there exists a unique u in V satisfying equation (3.2) of Chapter II, and the mapping ’§:/= pI-? U from V’ (Q) into V (Q) is a (y’, y)-compactifier. Remark 3.1. We could give a great many more examples of operators ’§ which are (lX,p)-compactifiers by considering evolution equations, as in Lions [1], and inhomogeneous boundary-value problems as in Lions and Magenes [1]: a large number of examples from this work would allow us to define in a similar way the operators ’§.
4. OPTIMAL CONTROL We shall now apply Theorem 2.2 to the optimal control of systems governed by ordinary differential equations. In this example we shall show how to pass from a formulation of the "optimal control" type to a formulation of the "calculus of variations" type.
256
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
4.1. Theoptimal control problem The evolution equation. We take a number T> 0, a metrizable compact set K and a continuous mapping qJ from [0, T] x R" x K into R" such that: (denoting the unit-ball of R" by B) for all p
(4.1)
~ 0, there exists k ~ 0 such that,for 0 ~ t ~ jqJ(t, y, co) - qJ(t, y’, co)1
I’Vy, y’ E t’B,
there exists a constant
(4.2)
I’V(t, y, co)
E
t
~ 0 such that
[0, T] x R x K, n
~
T and co
k Iy - y’l ;
E
K:
Iy’ qJ(t, y, co)1 ~ (’(1 + Iyll).
Let us choose Yo ERn. The system is governed by the ordinary differential equation:
dy(t)/dt = qJ(t, y(t), CO(t))
(4.3)
Iy(O) =
Yo’
Lemma 4.1. For any measurable mapping co: [0, T] ~ K, the differential equation (4.3) has a unique solution y: [O,T] ~ R", and we have,for 0 ~ t ~ T: (4.4) Proof Since tp has been assumed to be continuous there exists ’r > 0 suffi› ciently small for the equation (4.3) to have a solution defined on [O,«], By virtue of the inequality (4.2), we now have, for 0 ~ t ~ ’r
~
=
dt
ly(tW
2y(t)• dy(t) ~ 2f(1 + ly(tW) dt "’"
~ IYol2
+ 2ft + {21!y(sW
ds.
By Gronwall’s inequality, for 0 ~ t ~ r:
ly(tW ~
(IYoI2 +
2ft) e 2 f r ~
(IYoI2 +
2fT) e 2f t
Set p = (lyl2 + 2tT)e U T It can be seen that the solution lies in the bounded set pB, independently of r, We know that in this case the solution can be ex› tended to the whole of [0, T]. If there were two solutions of (4.3) on [0, T], they would both have values in pB, and by applying the Lipschitz condition (4.1), we could show them to coincide.
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
257
We term control any measurable mapping of [0, T] into K. Once the control has been chosen, the unique solution y of (4.3) is the trajectory of the system, and yet) is its state at the instant t.
Lemma 4.2. The set oj trajectories oj the system is relatively compact in ~([O,T];Rn).
Proof It is sufficient to note that the estimate (4.4) is independent of the control w: along all the trajectories, ly(t)1 is bounded above by a constant p, Let us now denote by p. the maximum of the continuous function qJ over the compact set [O,T] x pB x K. By equation (4.3) we have:
Id~~t) I ~
(4.5)
u.
This means that all the trajectories are p.-Lipschitzian. In particular, they are equicontinuous and so form a relatively compact subset of ~([o, T]; R") (Ascoli’s theorem). _ Constraints. We take a closed subset E of [0, T] x R" x K. For (t,y) given in [O,T] x R", we denote by Et,y the section:
(4.6)
E"y
= {wEKI(t,y,W)EE},
We term a control wand its corresponding trajectory y admissible if they are linked by:
(4.7)
Vt E [0, T],
(t, y(t), w(t)) E E.
Cost. We take a Caratheodory functionJof [O,T] x (Rn x K) into [0,+00[. We associate with an admissible control wand its corresponding trajectory y the cost function
(4.8)
IT f(t, y(t), w(t)) dt.
An admissible control wwill be called optimal if it minimizes (4.8); we shall also call the corresponding trajectory ji optimal. Let us gather all the data: to minimize
(&»
IT f(t, y(t), w(t)) dt
dy(t)/dt = qJ(t, y(t), w(dt)) dt (r, y(t), w(t)) E E y(O) = Yo
258
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
4.2. Compactness of the set of trajectories For ~ t ~ T and Y E R", we denote by at the instant t at the point y:
r;,y
rt,y
the set of admissible speeds
= { cp{t, y, w) I(t, y, w) E E }
= cp(t, y, E
1)
.
This is a compact set in R". If moreover it is convex and non-empty it can be shown that the set of admissible trajectories is compact and non-empty in ’tt([O,T];R"). We shall here be content with part of this result.
Proposition 4.1. If rt,y is convex for all t E [0, T] and all Y admissible trajectories is compact in ’tt([0, T]; R").
E
R", the set of
Proof In ’tt([0, T]; R"), the set of trajectories is relatively compact (Lemma 4.2), and it suffices therefore to show that the set of admissible trajectories is
closed. Hence, let (Yk)keN be a sequence of admissible trajectories which converges uniformly to a continuous function y. For all kEN we have
(4.9) (4.10) for ~ t ~ T, and Yk(O) = Yo. We must show that y also is an admissible trajectory; we have immediately that yeO) = Yo. From the estimate (4.5), IldYkjdt II ~ Jl for kEN. By extracting a subsequence we can thus assume that dYkjdt converges to dY/dt in the topology U(L2,L2). By Mazur’s lemma there exists a sequence of convex combinations L~=k lX"dy"jdt which converges to dyjdt in L~(O,T) as k ~ 00. We can there› fore extract a subsequence which converges simply to dyjdt on [O,T]. We thus have for t E [0, T]:
(4.11)
VkEN,
But we can summarize (4.9) and (4.10) by: (4.12)
k}
dy (t) E co {dy,,(t) In >dt dt,...•
259
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
and on substituting into (4.1 1):
(4.13)
dy(t)
Vk EN,
_
-d- E co t
U r;,Yn(t)
for
n:;;ok
~ t ~ T.
Let us fix t E [0, T] and take a > O. Since cp is continuous and K is compact there exists 17 > 0 such that:
Iy - Y(t)1 ~ 17
=>
Icp(t, y, w) - cp(t, y(t), w)1 ~ a
VWEK.
Since Yk converges uniformly to ji, we can take kEN large enough for: "In ?: k,
U
n ";;?;k
r;,Yn(t)
r;,Yn(t) c
+
r;,)i(t)
c
r;,y(t)
+ BB,
es.
The right-hand side is the sum of two convex compact sets. lt is thus a convex compact set and:
(4.14)
co
U r;,Yn(t)
n;;.k
c
r;,Y(t)
+
BB.
Substituting this into (4.13):
dY(t) (ftE
r;,y(,)
+
lt only remains to let a tend to zero. Since limit for 0 ::::; t ::::; T:
(4.15)
dY(t) ( f t E r;,y(t)’
(4.16)
d~~t)
E {
BB. rr,y(t)
is closed, we obtain in the
cp(t, Y(t), w) I(t, Y(t), w) E E}.
There exists a Borel subset N C [0, T], with null measure, such that the restriction of dy/dt to N is Borel. We can then define a Borel subset of xK by:
eN
(4.17)
G
=
{
(t, w) I(t, Y(t), w) E E and cp(t, y(t), w) E dY(t)jdt }.
By Corollary 1.7 there exists a measurable selection wof G. In suitably extend› ing it over N, we obtain a measurable mapping w from [0, T] into A which satisfies
(4.18)
dy(t)jdt = cp(t, y(t), w(t))
260
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
(t, y(t), w(t)) E E
(4.19) and hence
y is indeed an admissible trajectory.
_
4.3. Existence of optimal controls Let us define a function g from [0, T] x R" x R" into R by:
(4.20)
g(t, y, f/) = min { f(t, y, w) I(t, y, w) E E and
Clearly, g(t,y, f/)
E
R+ if f/
E
r t ."
and g(t,y, f/) =
+00
otherwise.
Lemma 4.3. g is a positive normal integrand of [0, T] x (R" x R") into R.
Proof Let us take 8 > 0. From the Scorza-Dragoni theorem, there exists a compact subset C. c [O,T] such that meas([O,T] - C.) ~ 8 and for which the restriction of fto C. x R" x K is continuous. We now show that the restriction of g to C. x R" x R" is l.s.c., which will prove that g is a normal integrand (Theorem 1.2). Let (tmYm fin) be a sequence of C. x R" x R" which converges to (i, y, ij). Set: (4.21 ) We wish to show that g(i,y,ii) ~ t. If t = +00, this is trivial. If t is finite, we may assume thatg(tmYm fin) is finite for all n EN and converges to t. From (4.20), there exists co; such that: (4.22) (4.23) Since K is compact, we can extract from the sequence Wm n EN, a subsequence converging to a ill E K, We can then pass to the limit in (4.22) and (4.23):
Wn’
(4.24) (4.25)
(t,
y, w) E
E and
f(t,
y, w)
= f.
Whence necessarily, by (4.20): (4.26)
g(i,
y, w)
y, w)
~ f.
=
iT
EXISTENCE OF SOLUTIONS FOR VARIATIONAL PROBLEMS
261
We can now state a sufficient condition for the existence of optimal controls:
Proposition 4.2. We assume the previous hypotheses. Iffor all (t,y) E [0, T] x R", the function g(t,y, .) is convex from R" into R, then there exists at least one
admissible optimal control.
Let us explain this condition. To say that g(t,y, .) is convex means that its epigraph is convex, i.e. that for 0 :::;; t :::;; T and y ERn, the set (4.27) {
(1],
a) ERn x R
13 W E E t y : cp(t, Y, w)
= 1] and
a ;;:: f(t, y, w) }
is convex in R" x R. This implies that its horizontal projection r t ,y is convex, i.e. the set of admissible speeds is convex and compact. Proof Let
Wn>
n E N, be a minimizing sequence of admissible controls and
n EN, their corresponding trajectories. By Lemma 4.2, there exists a con› stant it such that for all n EN,
Yn>
(4.28) From Proposition 4.1, possibly by extracting a subsequence, we may assume that there exists an admissible trajectory y such that:
yn
(4.29)
dyjdt
(4.30)
-+
-+
Y
uniformly
dY/dt for (J(L"o, U).
It only remains to apply Theorem 2.1 to the integrand
We obtain: (4.31)
Tg(t, jl(t), ddYt (t)) dt ~ Io
lim
n--+
00
ITii(t, y,,(t), ddYnt (t)) dt. 0
Taking (4.28) and (4.20) into account: (4.32)
I
Tg(t, y(t), ddYv (t) ~
o
t
lim
a-e co
ITg(t, Yn(t), ddY" (t)) dt 0
t
~ !~~ foT f(t, Y,,(t), wn(t)) dr.
262
RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS
The right-hand side is equal to inf(8l’). By the measurable selection theorem (Cor. 1.7), it is easily shown that there exists a measurable mapping w: [0, T] ...-+ K such that for 0 ~ t ~ T:
(t, Y(t), w(t)) E E and cp(t, y(t), w(t)) =
~~
f(t, y(t), w(t)) = y(t, Y(t), Thus
(4.33) and
f:
wis an admissible
wis an admissible
~~
(t)
(t)).
control and on substituting into (4.32) we obtain:
f(t, y(t), w(t)) dt
optimal control.
~
inf (ffl!)