Optimal control problems governed by some semilinear parabolic equations

Nonlinear Analysis 56 (2004) 241 – 252

www.elsevier.com/locate/na

Optimal control problems governed by some semilinear parabolic equations Sang-Uk Ryu Department of Mathematics, Cheju National University, Jeju 690-756, South Korea Received 3 September 2002; accepted 23 September 2003

Abstract This paper is concerned with the optimal control problem for the Keller–Segel equations. That is, as a generalization of Ryu and Yagi (J. Math. Anal. Appl. 256 (2001) 45–66) we derive the optimality conditions for the optimal control problem governed by a semilinear abstract equation of nonmonotone type. Moreover, we prove the uniqueness of the optimal control under some conditions. ? 2003 Elsevier Ltd. All rights reserved. MSC: 49J20; 49K20 Keywords: Optimal control; Keller–Segel equations; Optimality conditions; Uniqueness

1. Introduction Many papers have already been published to study the control problems for nonlinear parabolic equations. In the books by Ahmed and Teo [1] and Barbu [2], some general frameworks are given for handling the semilinear parabolic equations with monotone perturbations. In [1] the nonlinear terms are monotone functions with linear growth, and in [2] they are generalized to the multivalued maximal monotone operators determined by lower semicontinuous convex functions. Papageorgiou [7] and Casas et al. [3] have studied some quasilinear parabolic equations of monotone type. However, since Keller– Segel equations are not of monotone type in any sense, it seems that there is no known theoretical framework of controls which cover them. Recently, Ryu and Yagi [8] studied the optimal control problem for the Keller–Segel equations. In that paper they showed the existence of optimal control and optimality conditions by formulating the E-mail address: [email protected] (S.-U. Ryu). 0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.09.013

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S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

Keller–Segel equations as a semilinear abstract equation of nonmonotone type. Under mild assumptions, this paper derives the optimality conditions characterizing the optimal control and obtains the uniqueness of the optimal control. This paper is organized as follows: In Section 2, we recall some known results. Section 3 is devoted to derive a characterization for the optimal control. In Section 4, some uniqueness of the optimal control is shown by the strict convexity of the functional. Finally, we apply the previous results to the optimal control problem for the Keller–Segel equations. Notations: R denotes the sets of real numbers. For a region
⊂ R2 , the usual p L space of real-valued functions in
is denoted by Lp (
), 1 6 p 6 ∞. The real Sobolev space in
with an exponent s ¿ 0 is denoted by H s (
). Let I be an interval in R. Lp (I ; H), 1 6 p 6 ∞, denotes the Lp space of measurable functions in I with values in a Hilbert space H. C(I ; H) denotes the space of continuous functions in I with values in H. For simplicity, we shall use a universal constant C to denote various constants which are determined in each occurrence in a speciFc way by ; M; N , and so forth. In a case when C depends also on some parameter, say , it will be denoted by C . 2. Formulation of problem Let V and H be two separable real Hilbert spaces with dense and compact embedding V ,→ H. Identifying H and its dual H
and denoting the dual space of V by V
, we have V ,→ H ,→ V
. We denote the scalar product of H by (·; ·) and the norm by | · |. The duality product between V
and V which coincides with the scalar product of H on H × H is denoted by ·; ·, and the norms of V and V
by · and · ∗ , respectively. U = L2 (0; T ; V
) and Uad is closed, bounded and convex subset of U. We consider the following Cauchy problem: dY + AY = F(Y ) + U (t); 0 ¡ t 6 T; dt (2.1)

Y (0) = Y0

in the space V
. Here, A is the positive deFnite self-adjoint operator on H, deFned by a symmetric sesquilinear form a(Y; Y˜ ) on V, AY; Y˜  = a(Y; Y˜ ), which satisFes |a(Y; Y˜ )| 6 M Y Y˜ ; a(Y; Y ) ¿  Y 2 ;

Y; Y˜ ∈ V;

Y ∈V

(2.2) (2.3)

with some  and M ¿ 0. A is also a bounded operator from V to V
. F(·) is a given continuous function from V to V
satisfying (f.i) For each  ¿ 0, there exists an increasing continuous function  : [0; ∞) → [0; ∞) such that F(Y ) ∗ 6  Y +  (|Y |);

Y ∈ V;

S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

(f.ii) For each  ¿ 0, there exists an increasing continuous function [0; ∞) such that

243 

: [0; ∞) →

F(Y˜ ) − F(Y ) ∗ 6  Y˜ − Y + ( Y˜ + Y + 1)  (|Y˜ | + |Y |) × |Y˜ − Y |;

Y˜ ; Y ∈ V:

U (·) ∈ L2 (0; T ; V
) is a given function and Y0 ∈ H is an initial value. We then obtain the following result (For the proof, see Ryu and Yagi [8].) Theorem 2.1. Let (a.i), (a.ii), (f.i), and (f.ii) be satis2ed. Then, for any U ∈ L2 (0; T ; V
) and Y0 ∈ H, there exists a unique weak solution Y ∈ H 1 (0; T (Y0 ; U ); V
) ∩ C([0; T (Y0 ; U )]; H) ∩ L2 (0; T (Y0 ; U ); V) to (2.1), the number T (Y0 ; U ) ¿ 0 is determined by the norms U L2 (0; T ;V
) and |Y0 |. Now, let S ¿ 0 be such that for each U ∈ Uad , (2.1) has a unique weak solution Y (U ) ∈ H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V). Thus we may formulate the optimal control problem as follows: (P)

minimize J (U );

where the cost functional J (U ) is of the form
S
S J (U ) = DY (U ) − Yd 2 dt +  U 2∗ dt; 0

0

U ∈ Uad :

Here, D is a bounded operator from V into V and Yd is a Fxed element of L2 (0; S; V).  ¿ 0 is a positive constant. Theorem 2.2. There exists an optimal control UK ∈ Uad for (P) such that J (UK ) = min J (U ): U ∈Uad

Proof. The proof is quite standard, so it will be only sketched (cf. [2, Chapter 5, Proposition 1.1] and [6, Chapter III, Theorem 15.1]). Let {Un } ⊂ Uad be a minimizing sequence such that limn→∞ J (Un ) = minU ∈Uad J (U ). Since {Un } is bounded, we can assume that Un → UK weakly in L2 (0; S; V
). For simplicity, we will write Yn instead of the solution Y (Un ) of (2.1) corresponding to Un . Using the similar estimate of the solution Yn , we see as in the proof of [8, Theorem 2.1] that    dYn   Yn L∞ (0; S;H) 6 C; Yn L2 (0; S;V) 6 C;  6 C:  dt  2 L (0; S;V
) Therefore, on subsequences, again denoted by n, we have Yn → YK

weakly in L2 (0; S; V);

Yn → YK

in weak star topology of L∞ (0; S; H);

d YK dYn → dt dt

weakly in L2 (0; S; V
):

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S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

Since V is compactly embedded in H, we can conclude that Yn → YK strongly in L2 (0; S; H). Hence, by the uniqueness, YK is the weak solution of (2.1) corresponding to UK (i.e. YK = Y (UK )). Since Y (Un ) − Yd is weakly convergent to Y (UK ) − Yd in L2 (0; S; V), we have min J (U ) 6 J (UK ) 6 lim inf J (Un ) = min J (U ):

U ∈Uad

n→∞

U ∈Uad

Hence, minU ∈Uad J (U ) = J (UK ). 3. Optimality conditions In this section, we show the optimality conditions for the Problem (P). We denote the scalar products in V and V
by ·; ·V and ·; ·V
, respectively. In order to the optimality conditions, we need some additional assumptions: (f.iii) The mapping F(·) : V → V
is FrLechet diMerentiable and for each  ¿ 0, there exists an increasing continuous functions ! ; " : [0; ∞) → [0; ∞) such that   Z P + ( Y + 1)! (|Y |)|Z| P ; Y; Z; P ∈ V;    |F
(Y )Z; P| 6  Z P + ( Y + 1)! (|Y |) Z |P|; Y; Z; P ∈ V;    "(|Y |) Z P ; Y; Z; P ∈ V: (f.iv) F
(·) is continuous from H into L(V; V
). Proposition 3.1. Let (a.i), (a.ii), (f.i), (f.ii), (f.iii), and (f.iv) be satis2ed. The mapping Y : Uad → H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V) is Gˆateaux di:erentiable with respect to U . For V ∈ Uad , Y
(U )V = Z is the unique solution in H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V) of the problem dZ + AZ − F
(Y )Z = V (t); dt

0 ¡ t 6 S;

Z(0) = 0:

(3.1)

Proof. Let U; V ∈ Uad and 0 6 h 6 1. Let Yh and Y be the solutions of (2.1) corresponding to U + hV and U , respectively. Step 1: Yh → Y strongly in C([0; S]; H) as h → 0. Let W = Yh − Y . Obviously, W satisFes dW + AW − (F(Yh (t)) − F(Y (t))) = hV (t); dt

0 ¡ t 6 S;

W (0) = 0:

(3.2)

Taking the scalar product of Eq. (3.2) with W , we obtain 1 d |W (t)|2 + AW (t); W (t) = F(Yh (t)) − F(Y (t)); W (t) + hV (t); W (t): 2 dt

S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

245

Using (2.3) and (f.ii), we have 1 d |W (t)|2 +  W (t) 2 2 dt  W (t) 2 + C=4 ( Yh (t) 2 + Y (t) 2 + 1) 2

6

=4 (|Yh (t)|

+ |Y (t)|)2 |W (t)|2

+ 4h2 −1 V (t) 2∗ : Using Gronwall’s lemma, we obtain |W (t)|2 6 Ch2 V 2L2 (0; S;V
) e

S 0

C=4 (Yh (s)2 +Y (s)2 +1)

=4 (|Yh (s)|+|Y (s)|)

2

ds

for all t ∈ [0; S]. Hence, Yh → Y strongly in C([0; S]; H) as h → 0. Step 2: (Yh − Y )=h → Z strongly in H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V) as h → 0. We rewrite the problem (3.2) in the form d Yh − Y Yh − Y F(Yh ) − F(Y ) +A − = V (t); 0 ¡ t 6 S; dt h h h Yh − Y (0) = 0: (3.3) h On the other hand, we consider the linear problem (3.1). From (a.i), (a.ii), (f.i), (f.ii), and (f.iii), we can easily verify that (3.1) possesses a unique weak solution Z ∈ H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V) on [0; S] (cf. [4, Chapter XVIII, 1 Theorem 2]). DeFne Fh
= 0 F
(Y + (Yh − Y )) d . Then W˜ = (Yh − Y )=h − Z satisFes d W˜ (t) + AW˜ (t) − Fh
W˜ (t) = (Fh
− F0
)Z(t); 0 ¡ t 6 S; dt W˜ (0) = 0:

(3.4)

Taking the scalar product of the equation of (3.4) with W˜ , we obtain 1 d ˜ |W (t)|2 + AW˜ (t); W˜ (t) 2 dt = Fh
W˜ (t); W˜ (t) + (Fh
− F0
)Z(t); W˜ (t)  6 W˜ (t) 2 + ( Y (t) 2 + Yh (t) − Y (t) 2 + 1)!(|Y ˜ h |2 + |Y |2 )|W˜ (t)|2 2 4 + (Fh
− F0
)Z(t) 2∗ ;  where !˜ : [0; ∞) → [0; ∞) is some increasing continuous function. Therefore,
t W˜ (s) 2 ds |W˜ (t)|2 + 
6

0

0

t

( Y (s) 2 + Yh (s) 2 + 1)!(|Y ˜ h |2 + |Y |2 )|W˜ (s)|2 ds

8 + (Fh
− F0
)Z(t) 2L2 (0; S;V
) : 

(3.5)

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S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

From (f.iii), we have Fh
Z(t) ∗ 6 C Z(t) , t ∈ [0; S]. Since Yh → Y strongly in H, it follows from (f.iv) that Fh
Z(t) → F0
Z(t)

strongly in V
a:e:

By the dominated convergence theorem, we have (Fh
− F0
)Z(t) 2L2 (0; S;V
) → 0

as h → 0:

Using Gronwall’s lemma, it follows from (3.5) that (Yh − Y )=h is strongly convergent to Z in H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V). With the aid of this proposition, we can easily show the optimality conditions for (P). Theorem 3.2. Let UK be an optimal control of (P) and let YK ∈ H 1 (0; S; V
) ∩ C([0; S]; H)∩L2 (0; S; V) be the optimal state, that is YK is the solution to (2.1) with the control UK (t). Then, there exists a unique solution P ∈ H 1 (0; S; V
)∩C([0; S]; H)∩L2 (0; S; V) to the linear problem −

dP + AP − F
(YK )∗ P = D∗ )(DYK − Yd ); dt

0 6 t ¡ S;

P(S) = 0

(3.6)

in V
, where ) : V → V
is a canonical isomorphism; moreover,
S )P + UK ; V − UK V
dt ¿ 0 for all V ∈ Uad : 0

Proof. Since J is Gˆateaux diMerentiable at UK and Uad is convex, it is seen that J
(UK )(V − UK ) ¿ 0

for all V ∈ Uad :

On the other hand, we verify that
S
S J
(UK )(V − UK ) = DY (UK ) − Yd ; DZV dt +  UK ; V − UK V
dt 0

0

(3.7)

with Z = Y
(UK )(V − UK ). Let P be the unique solution of (3.6) in H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V). From [4, Chapter XVIII, Theorem 2], we can guarantee that such a solution P exists. Thus, in view of Proposition 3.1 the Frst integral on the right-hand side of (3.7) is shown to be
S
S K DY (U ) − Yd ; DZV dt = D∗ )(Y (UK ) − Yd ); Z dt 0

0


=

0

S

 −

dP + AP − F
(YK )∗ P; Z dt

dt

S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252


=

S

0


= Hence,
0

S

S

0

)P + UK ; V − UK V
dt ¿ 0



dZ P; + AZ − F
(YK )Z dt

247

dt

)P; V − UK V
dt:

for all V ∈ Uad :

4. Uniqueness of the optimal control In this section we obtain the uniqueness of the optimal control for the problem (P). In order to show that, we need additional assumption: (f.v) The mapping F(·) : V → V
is second-order FrLechet diMerentiable and there exist N ¿ 0 such that F

(Y )(Z; Z) ∗ 6 N |Z| Z ;

Y; Z ∈ V:

Proposition 4.1. Let (a.i), (a.ii), (f.i), (f.ii), (f.iii), (f.iv), and (f.v) be satis2ed. The mapping Y : Uad → H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V) is second-order Gˆateaux di:erentiable with respect to U . For V ∈ Uad , Y

(U )(V; V ) = . is the unique solution in H 1 (0; S; V
) ∩ C([0; S]; H) ∩ L2 (0; S; V) of the problem d. + A. − F

(Y )(Z; Z) − F
(Y ). = 0; dt

0 ¡ t 6 S;

.(0) = 0;

(4.1)

where Z = Y
(U )V . Proof. The proof is similar to that of Proposition 3.1. Now, for S suSciently small, we prove the uniqueness of the optimal control. Theorem 4.2. When S is su=ciently small, there is a unique optimal control for (P). Proof. We show uniqueness by showing strict convexity of the following map: U ∈ Uad → J (U ): This convexity follows from showing for all U; V ∈ Uad ; 0 ¡ h ¡ 1, g

(h) ¿ 0; where g(h) = J (U + h(V − U )) (cf. [10, Section 25.5]).

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S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

To calculate J (U + (h + l)(V − U )) − J (U + h(V − U )) ; l→0 l

g
(h) = lim

we denote Y˜ h = Y˜ (U + h(V − U )) and Y˜ h+l = Y˜ (U + (h + l)(V − U )). Using the same argument as in Proposition 3.1 Y˜ h+l − Y˜ h → Z˜h l as l → 0 and Z˜h satisFes d Z˜h + AZ˜h − F
(Y˜ h )Z˜h = V − U; dt

0 ¡ t 6 S;

Z˜h (0) = 0: Estimating as in Proposition 3.1 for 0 ¡ t ¡ S, 8 d ˜h 2 ˜ Y˜ h |2 )|Z˜h (t)|2 + V − U 2∗ ; |Z (t)| +  Z˜h (t) 2 6 ( Y˜ h 2 + 1)!(|  dt

(4.2)

where !˜ : [0; ∞) → [0; ∞) is some increasing continuous function. Using Gronwall’s inequality, we obtain |Z˜h (t)|2 6 8−1 V − U 2L2 (0; S;V
) e

S 0

(Y˜ h 2 +1)!(| ˜ Y˜ h |2 ) ds

6 C V − U 2L2 (0; S;V
) :

(4.3)

Using this result in (4.2) and integrating from 0 and t, we have
S Z h (t) 2 dt 6 C V − U 2L2 (0; S;V
) : 0

(4.4)

To calculate g

(h), we need a second derivative of Y with respect to the control. Using the same argument as in Proposition 4.1, we obtain Z˜h+k − Z˜h → .h k as k → 0 and .h satisFes d.h + A.h − F

(Y˜ h )(Z˜h ; Z˜h ) − F
(Y˜ h ).h = 0; dt

0 ¡ t 6 S;

.h (0) = 0: From the assumptions of Proposition 4.1, we have 1 d h 2 |. (t)| +  .h (t) 2 2 dt  4 6 .h 2 + N 2 Z˜h 2 |Z˜h |2 + ( Y˜ h 2 + 1)!(| ˜ Y˜ h |2 )|.h |2 ; 2 

(4.5)

S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

249

where !˜ : [0; ∞) → [0; ∞) is some increasing continuous function. Therefore, by Gronwall’s inequality and |Y˜ h | 6 C, we obtain
S S h ˜h 2 ˜ Y˜ h (s)|2 ) ds |.h (t)|2 6 8−1 N 2 e 0 (Y  +1)!(| Z˜h (s) 2 |Z˜ (s)|2 ds 0

h 6 C Z˜h 2L2 (0; S;V) Z˜ 2L∞ (0; S;H)

and thus, by (4.3) and (4.4), |.h (t)|2 6 C V − U 4L2 (0; S;V
) : Using this result in (4.5) and integrating from 0 and t, we have
S .h (t) 2 dt 6 C V − U 4L2 (0; S;V
) : 0

For the second derivative, we have g
(h + k) − g
(h) k→0 k
S
S = .h ; D∗ )(DY˜ h − Yd ) dt + DZ˜h ; DZ˜h V dt

g

(h) = lim

0

0




+

S

0


¿−

S

0


+

0

V − U; V − U V
dt

S

h 2

1=2


. dt

0

S

1=2 ˜h

∗ 2

2

D DY − Yd dt

V − U 2∗ dt

¿ ( − C D DY˜ h − Yd L2 (0; S;V) ) ∗


0

S

V − U 2∗ dt:

Noting that DY˜ h − Yd L2 (0; S;V) → 0 as S → 0, we obtain the desired result for S suSciently small. Remark. We can also obtain the strict convexity of the cost functional and the uniqueness of the optimal control if one can assume that  is suSciently large. 5. Applications We consider the following optimal control problem: K (P)

Minimize J (u) u

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S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

with the cost functional J (u) of the form
J (u) =

0

T

y(u) −

yd 2H 1 (
)


dt + 

T

0

u 2H 3 (
) dt;

u ∈ L2 (0; T ; H 3 (
));

where y = y(u) is governed by Keller–Segel equations [5] @y = aTy − b∇{y∇6} @t

in
× (0; T ];

@6 = dT6 + fy − g6 + "u @t @y @6 = =0 @n @n y(x; 0) = y0 (x);

in
× (0; T ];

on @
× (0; T ]; 6(x; 0) = 60 (x)

in
:

(5.1)

Here,
is a bounded region in R2 of C3 class. a; b; d; f; g; ";  ¿ 0 are given positive numbers. u ¿ 0 is a control function varying in some bounded subset Uad of L2 (0; T ; H 3 (
)); 3 being some Fxed exponent such that 0 ¡ 3 ¡ 12 . n = n(x) is the outer normal vector at a boundary point x ∈ @
and @=@n denotes the diMerentiation along the vector n. y0 (x), 60 (x) ¿ 0 are nonnegative initial functions in L2 (
) and in H 1+3 (
), respectively. The Keller–Segel equations (5.1) was introduced by Keller and Segel [5] to describe the aggregating process of the cellular slime molds by the chemical attraction. Unknown functions y = y(x; t) and 6 = 6(x; t) denote the concentration of amoebae in
at time t and the concentration of the chemical substance in
at time t, respectively. The chemotactic term −b∇{y∇6} indicates that the cells are sensitive to chemicals and are attracted by them, and the production term fy indicates that the chemical substance is itself emitted by cells. Let A1 = −a8 + a and A2 = −d8 + g be the Laplace operators equipped with the Neumann boundary conditions. The part of Ai in L2 (
) is a positive deFnite self-adjoint operator in L2 (
) with the domain D(Ai ) = Hn2 (
) = {y ∈ H 2 (
); @y=@n = 0 on @
}. D(A i ) = H 2 (
) for 0 6 ¡ 34 , and D(A i ) = Hn2 (
) for 34 ¡ 6 32 (see [9]). We introduce two product Hilbert spaces V ⊂ H as ) V = H 1 (
) × D(A1+3=2 2

and

H = L2 (
) × D(A2(1+3)=2 );

respectively, where 3 is some Fxed exponent 3 ∈ (0; 12 ). By the identiFcation of H and its dual H
, we have V ⊂ H=H
⊂ V
. It is then seen that V
=(H 1 (
))
×D(A3=2 2 ) with the duality product .; Y V
×V =:; y(H 1 )
×H 1 +

1+3=2 (A3=2 6)L2 ; 2 ’; A2

.=

: ’




∈ V; Y =

 y 6

∈ V:

S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

We also set a symmetric sesquilinear form on V × V: a(Y; Y˜ ) = (A11=2 y; A11=2 y) ˜ L2

+ (A1+3=2 6; A1+3=2 6) ˜ L2 ; 2 2

Y=

251

 y 6

; Y˜ =

 y˜ 6˜

∈V:

Obviously, the

satisFes (2.2) and (2.3). This form then deFnes a linear isomor form A1 0 phism A = 0 A2 from V to V
, and the part of A in H is a positive deFnite self-adjoint operator in H with the domain D(A) = D(A1 ) × D(A2(3+3)=2 ). Eq. (5.1) is, then, formulated as an abstract equation dY + AY = F(Y ) + U (t); 0 ¡ t 6 T; dt Y (0) = Y0 in the space V
. Here, F(·) : V → V
is the mapping



 −b∇{y∇6} + ay y F(Y ) = ; Y= ∈ V: fy 6

0 U (t) and Y0 are deFned by U (t) = "u(t) and Y0 = y600 , respectively. As veriFed in [8, Section 2], F(·) satisFes (f.i) and (f.ii). Furthermore, F(Y ) is second-order FrLechet diMerentiable with the Frst and second derivative

 −b∇{y∇w} − b∇{z∇6} + az
F (Y )Z = ; fz and





F (Y )(Z; Z) =

−2b∇{z∇w}



0

;

Y=

 y 6

;

Z=

z w

 ∈ V:

As veriFed in [8], we can check (f.iii), (f.iv) and (f.v). If we set U = L2 (0; T ; V
) and    0 2 3 Uad = ∈ U; u ∈ L (0; T ; H (
)); u ¿ 0; u L2 (0; T ;H 3 ) 6 C ; u K is obviously then Uad is closed, bounded and convex subset of U. Problem (P) formulated as follows: (P)

minimize J (U );

where


J (U ) =

Here, D

y 6 1

0

=

L2 (0; S; H (
)).

S

DY (U ) − Yd 2 dt + 

y 0

and Yd =

 yd  0


0

S

U 2∗ dt;

U ∈ Uad :

is a Fxed element of L2 (0; S; V) with yd ∈

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S.-U. Ryu / Nonlinear Analysis 56 (2004) 241 – 252

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