Identification problem of two operators for nonlinear ...

Nonlinear Analysis 70 (2009) 1443–1458 www.elsevier.com/locate/na

Identification problem of two operators for nonlinear systems in Banach spaces Yong Han Kang Department of Mathematics, University of Ulsan, Ulsan 680-749, South Korea Received 29 January 2007; accepted 14 February 2008

Abstract We consider the identification problem of two operators having different properties for the systems governed by nonlinear evolution equations. For the identification problem, we show the existence of optimal solutions and present necessary optimality conditions. We illustrate the approach on two examples. c 2008 Elsevier Ltd. All rights reserved.

MSC: 49J20; 49K20; 49K24 Keywords: Identification problem; Galerkin method; Nonlinear systems; Necessary condition

1. Introduction In this paper, we consider the identification problem (IP) for nonlinear systems in Banach spaces of the form: (d (Eφ) + A(t, φ) + Bφ = g(t, φ) (1.1) dt φ(B, E)(0) = φ0 ∈ H, B ∈ Pa,b , E ∈ M, where A is a nonlinear monotone operator from a Banach space V into its dual V ∗ and g(t, x) is a nonlinear, but not necessary monotone operator. Pa,b is a suitable subset of L(V, V ∗ ) and M is a suitable subset of L+ (H ). Define the error functional J (·, ·) by the form Z J (B, E) = f (t, φ(B, E)(t), B, E)dt, I

where I = [0, T ], T < ∞ and φ(B, E) is a solution of (1.1). The problem is to find (B 0 , E 0 ) ∈ Pa,b ×M (admissible set) so that (IP)

J (B 0 , E 0 ) ≤ J (B, E)

for all (B, E) ∈ Pa,b × M.

E-mail address: [email protected]. c 2008 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2008.02.025

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Since the pioneering work of Ahmed [1], IP problems for distributed systems have been extensively studied in different contexts and functional frameworks in connection with a plethora of physical, engineering, biology, population and economical applications (see Refs. [1–30] and references therein). In this paper, we extend the IP formalism by: (i) setting the IP problem within a general Banach space framework; (ii) dealing simultaneously within this framework with uncertain systems of different nature, in particular two different operators; and (iii) illustrating the approach on two examples that are explicitly shown to satisfy the general assumptions needed to obtain the results. The rest of the paper is organized as follows. In Section 2, we establish the existence of the state solution for problem (1.1). In Section 3, we prove the existence of the optimal pair for the problem (IP). Necessary optimality conditions for the problem (IP) are derived in Section 4. In the last section, we present the examples. 2. Existence and uniqueness of solutions Let H be a separable Hilbert space and let V be a subspace of H having the structure of a reflexive Banach space, with the embedding V ,→ H being compact. Identifying H with its dual, we have V ,→ H ,→ V ∗ , where V ∗ is the topological dual of V . The general framework considered here is based on this evolution triple. Let hφ, ϕi denote the pairing between an element φ ∈ V and an element ϕ ∈ V ∗ . If φ, ϕ ∈ H , then hφ, ϕi = (φ, ϕ) where (, ) denotes the inner product in H . The norm in any Banach space X will be denoted by k · k X . Let {e1 , e2 , . . .} be a basis of V and denote Hn = span{e1 , e2 , . . . , en }. The n-dimensional space Hn is endowed with the scalar product of Hilbert space H . Note that Hn ⊆ V ⊆ H . Let 0 < t ≤ T < ∞, It ≡ [0, t], I ≡ [0, T ], and let p, q ≥ 1 such that 1/ p + 1/q = 1

and

2 ≤ p < ∞.

To simplify notation, we write L rt (X ) ≡ L r (It , X ), L r (X ) ≡ L r (I, X ) for r ≥ 1 and a set X . For p, q satisfying the preceding conditions, it follows from the reflexivity of V that both L tp (V ) and L qt (V ∗ ) are reflexive Banach spaces (see Theorem 1.1.17 of [3]). The duality pairing of L tp (V ) and L qt (V ∗ ) is denoted by hh·, ·iit . In particular, we use hh·, ·ii ≡ hh·, ·iiT . Clearly, for φ, ϕ ∈ L 2 (H ), hhφ, ϕii = ((φ, ϕ)), where ((·, ·)) is the scalar product in Hilbert space L 2 (H ). Let φt = ∂t∂ φ. Define W p,q = {φ : φ ∈ L p (V ), Eφt ∈ L q (V ∗ ), E 1/2 φ ∈ L ∞ (I, H ) ∩ C(I, H )}, kφk2W p,q = kxk2L p (V ) + kφt k2L q (V ∗ ) . Then {W p,q , k · kW p,q } is a Banach space and the embedding W p,q ,→ C(I, H ) is continuous. If V ,→ H is compact, then W p,q ,→ L p (H ) is compact (see Proposition 23.23 of [31]). Let L(X, Z ) denote the space of all bounded linear operators from X to Z and A∗ the dual of the operator A. Let L+ (H ) be the class of all bounded linear positive selfadjoint operators in the Hilbert space H . In general, E is not invertible and E 6= 0. Let M ⊂ L+ (H ) be a compact and convex set in the strong operator topology. Let Pa,b = {B ∈ L(V, V ∗ ) : kBkL(V,V ∗ ) ≤ b, hBξ, ξ i + akξ k2H ≥ 0, ∀ξ ∈ V }. Consider the space of operators L(V, V ∗ ) and suppose that it is given the strong (weak) operator topology which we denote by τso (τwo ). Given this topology, Ls (V, V ∗ ) ≡ (L(V, V ∗ ), τso ) is a locally convex linear topological vector space which is sequentially complete. Similarly, Lw (V, V ∗ ) ≡ (L(V, V ∗ ), τwo ) with the weak operator topology τwo is also a sequentially completely locally convex topological space. We introduce the following assumptions: H(A) The operator A : I × V 7→ V ∗ satisfies (1) t 7→ A(t, φ) is measurable. (2) φ 7→ A(t, φ) is uniformly monotone and hemicontinuous, i.e., there exists a constant c > 0 such that p hA(t, φ1 ) − A(t, φ2 ), φ1 − φ2 i ≥ ckφ1 − φ2 kV , ∀φ, ϕ ∈ V, t ∈ I ; w

A(t, φ + sϕ) → A(t, φ) ∈ V ∗

∀φ, ϕ ∈ V, t ∈ I as s → 0.

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(3) There exist positive constants c1 and c2 such that p hA(t, φ), φi ≥ c1 kφkV − c2 , ∀φ ∈ V, t ∈ I. (4) There exist a positive constant c3 and a function c4 (t) ∈ L q (I, R+ ), such that kA(t, φ)kV ∗ ≤ c4 (t) + p−1 p−1 ∀φ ∈ V, t ∈ I ; kA(t + τ, φ) − A(t, φ)kV ∗ ≤ O(τ )(1 + kφkV ), ∀φ ∈ V and O(τ ) is c3 kφkV independent of φ. H(0) The map (B, E) 7→ φ0 (B, E) is continuous from Pa,b × M into H . H(g) The map g : I × H 7→ V ∗ satisfies (1) t 7→ g(t, ·) is measurable. (2) φ 7→ g(·, φ) is continuous. 2/q (3) There exist α ≥ 0 and h ∈ L q (I, R+ ) such that kg(t, φ)kV ∗ ≤ h(t) + αkφk H ∀φ ∈ V, t ∈ I. (4) g is locally Lipschitz continuous with respect to φ: for any b > 0, there exists L(b) such that φ1 , φ2 ∈ H, kφ1 k H , kφ2 k H ≤ b, kg(t, φ1 ) − g(t, φ2 )kV ∗ ≤ L(b)kφ1 − φ2 k H , ∀t ∈ I. H(f) The function f : I × H × Pa,b × M 7→ R+ satisfies (1) t 7→ f (t, ·, ·, ·) is measurable. (2) φ 7→ f (·, φ, ·, ·) is continuous. (3) B 7→ f (·, ·, B, ·) is continuous. (4) E 7→ f (·, ·, ·, E) is continuous. We consider the state problem (1.1) under the above assumptions. For given φ0 (B, E) ∈ H , we seek a function φ ∈ W p,q such that (1.1) is satisfied in a weak sense. For φ ∈ L p (V ), we set A(φ)(t) = A(t, φ(t)),

G(φ)(t) = g(t, φ(t)),

t ∈ I.

(V ∗ )

Note that A : L p (V ) 7→ L q is bounded, uniformly monotone, hemicontinuous, and coercive and the operator G : L p (V ) 7→ L q (V ∗ ) is bounded. In the remaining part of this section we derive an existence result for the system (1.1) based on Galerkin approximations. First we give an a priori bound and prove the uniqueness of the solution to (1.1). To this end, it is often convenient to write the system (1.1) as an operator equation in 0 (B, E) ≡ {φ ∈ W W p,q p,q ; φ(0) = φ0 (B, E)}:  d (Eφ) + A(φ) + Bφ = G(φ), (2.1) dt φ ∈ W 0 (B, E), B ∈ P , E ∈ M. a,b

p,q

The following lemma is true. Lemma 2.1 ([4]). If φ n → φ 0 weakly in W p,q , then G(φ n ) → G(φ 0 ) in L q (V ∗ ). Lemma 2.2. There exists b > 0 such that kE

1/2

φkC(I,H )∩L ∞ (I,H ) ≤ b,

kφk L p (V ) ≤ b,



d

(Eφ)

dt

for any weak solution φ (if one exists) of the system (1.1). Proof. If φ is a solution of (1.1), then for each t ∈ I ,   d (Eφ), φ + hhA(φ), φiit + hhBφ, φiit = hhG(φ), φiit . dt t Using the assumptions H(A)(3) and the property of Pa,b , we have Z t 1 p (kE 1/2 φ(t)k2H − kE 1/2 φ(0)k2H ) + (c1 kφ(σ )kV − c2 )dσ 2 0 Z t Z t ≤ akφ(σ )k H dσ + kg(σ, φ(σ ))kV ∗ kφ(σ )kV dσ. 0

0

≤ b, L q (V ∗ )



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Using the assumptions H(g)(2) and the property of M, we have Z t p 1/2 2 kφ(σ )kV dσ ≤ 2c2 T + kEkL(H ) kφ(0)k2H kE φ(t)k H + 2c1 0 Z t Z t 2/q 2 + 2|a| kφ(σ )k H dσ + 2 (h(σ ) + αkφ(σ )k H )kφ(σ )kV dσ. 0

0

Using the Cauchy inequality, for any ε > 0, we obtain Z t p kφ(σ )kV dσ ≤ 2c2 T + kEkL(H ) kφ(0)k2H kE 1/2 φ(t)k2H + 2c1 0 Z t Z t 2 h(σ )q dσ + 2|a| kφ(σ )k H dσ + (2/q q ) 0 0 Z t Z t p p + (2(1 + α) / p) kφ(σ )kV dσ + (2α/ q ) kφ(σ )k2H dσ. 0

0

Choosing  > 0 sufficiently small and h ∈ L q (I, R + ), one can easily verify that there exist positive constants c5 , c6 , c7 such that Z t p 1/2 2 kE φ(t)k H + c5 kφk L t (V ) ≤ c6 + c7 kE 1/2 φ(σ )k2H dσ. (2.2) p

0

Using Gronwall’s lemma and inequality (2.2), it is easy to verify that there exist constants c8 and c9 , such that kE 1/2 φ(t)k H ≤ c8

for all t ∈ I

and kφk L p (V ) ≤ c9 for some constants c8 , c9 depending on c6 and c7 . Again, using the assumptions H(A)(3), we get p−1

kA(φ)k L q (V ∗ ) ≤ c10 + c11 kφk L p (V ) .

(2.3)

Hence, using the assumptions H(A)(3), H(g)(2), inequality (2.3), the property of Pa,b , and the continuous embedding L p (V ) ,→ L 2 (V ), we obtain



d

(Eφ) ≤ kA(φ)k L q (V ∗ ) + kBφk L q (V ∗ ) + kg(φ)k L q (V ∗ )

dt L q (V ∗ ) p−1

≤ c12 + c13 kφk L p (V ) + c14 kBφk L q (V ∗ ) .

(2.4)

Using (2.4), it is easy to verify that there exists positive constant c15 such that

d

(Eφ) ≤ c15 .

dt

L q (V ∗ ) Choosing b = max{c8 , c9 , c15 } the assertion follows.



Lemma 2.3. Suppose that the embedding V ,→ H is compact. If a weak solution of the system (1.1) exists, it is unique. 0 (B, E) be two solutions of (2.1). Using integration by parts and the monotonicity of the Proof. Let φ1 , φ2 ∈ W p,q operator A, B ∈ Pa,b , and E ∈ M, we obtain Z t 1 1/2 p kE (φ1 (t) − φ2 (t))k2H + ckφ1 − φ2 k L t (V ) ≤ a kφ1 (σ ) − φ2 (σ )k2H dσ p 2 0 Z t + hg(σ, φ1 (σ )) − g(σ, φ2 (σ )), φ1 (σ ) − φ2 (σ )iV ∗ ,V dσ. 0

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Using assumption H(g), Lemma 2.2, and the Cauchy inequality, for any  > 0, we have Z t 1 1/2 p 2 kφ1 (σ ) − φ2 (σ )k2H dσ kE (φ1 (t) − φ2 (t))k H + ckφ1 − φ2 k L t (V ) ≤ |a| p 2 0 Z t + kg(σ, φ1 (σ )) − g(σ, φ2 (σ ))kV ∗ kφ1 (σ ) − φ2 (σ )kV dσ 0 Z t Z t 2 kφ1 (σ ) − φ2 (σ )k H kφ1 (σ ) − φ2 (σ )kV dσ ≤ |a| kφ1 (σ ) − φ2 (σ )k H dσ + L(b) 0 0 Z t Z t 2 kφ1 (σ ) − φ2 (σ )k2H dσ ≤ |a| kφ1 (σ ) − φ2 (σ )k H dσ + (L(b)/2) 0 0 Z t + (L(b)/2) kφ1 (σ ) − φ2 (σ )k2V dσ. 0

Using the continuous embedding L tp (V ) ,→ L t2 (V ), we obtain p

kE 1/2 (φ1 (t) − φ2 (t))k2H + 2ckφ1 − φ2 k L t (V ) p Z t Z t p 1/2 2 ≤ (2|a| + L(b)/)kEk−1 kE (φ (σ ) − φ (σ ))k dσ + L  kφ1 (σ ) − φ2 (σ )kV dσ, 1 2 1 H L(H ) 0

0

where L 1 is a constant depending on b and the embedding constant. Consequently, for sufficiently small  > 0, there exists a constant c0 > 0 such that Z t p −1 1/2 2 0 kE (φ1 (t) − φ2 (t))k H + c kφ1 − φ2 k L t (V ) ≤ (2|a| + L(b)/)kEkL(H ) kE 1/2 (φ1 (σ ) − φ2 (σ ))k2H dσ. p

0

Using Gronwall’s lemma and the fact that E ∈ M is bounded, uniqueness follows from the above inequality.



The next theorem proves the existence of a state solution using the Galerkin method. Theorem 2.1. Under the assumptions H(A), H(0) and H(g), the system (1.1) has a unique weak solution. Proof. Let the sequence {φ0n } be an approximation of the given initial state φ0 ∈ H , i.e., φ0n ∈ Hn , φ0n → φ0 in H as n → ∞. We consider the sequence φ n (t) =

n X

Ck,n (t)ek

k=1

and seek a function φ n such that   d  n n n n  (Eφ (t)), e  j + hA(t, φ (t)), e j i + hBφ (t), e j i = hg(t, φ (t)), e j i, dt φ n (0) = φ0n ;    n φ ∈ L p (I, Hn ), φtn ∈ L q (I, Hn ).

j = 1, 2, . . . , n; (2.5)

It follows from the existence theorem of Carath´eodory for ordinary differential equation in R n [31,6] and Lemmas 2.2 and 2.3 that, for each n ∈ N , the finite-dimensional system (2.5) has a unique weak solution, φ n . It can be seen from 0 (B, E). Hence, by the above assumptions, {A(φ n )} Lemma 2.2 that {φ n } is contained in a bounded subset of W p,q ∗ n is bounded in L q (V ). Since B ∈ Pa,b , {Bφ } is also bounded in L q (V ∗ ), and also since L p (V ) and L q (V ∗ ) are reflexive Banach spaces, there exists a subsequence, again denoted by {φ n }, an element φ ∈ L p (V ) with its distributional derivative φt ∈ L q (V ∗ ) and W ∈ L q (V ∗ ) such that φn → φ0 E

φtn

weakly in L p (V ),

φ → E 1/2 φ 0

1/2 n

→

φt0

weakly in L ∞ (I, H ) ∩ C(I.H ),

weakly in L q (V ∗ ),

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A(φ n ) → W

weakly in L q (V ∗ ),

Bφ n → Bφ 0

weakly in L q (V ∗ ) as n → ∞.

Combining the assumptions with Lemma 2.1, we have G(φ n ) → G(φ 0 )

in L q (V ∗ ),

φ n (0) → φ0 (B, E) φ (T ) → z n

in H,

weakly in H as n → ∞.

Let ψ ∈ C ∞ (I, R) and v ∈ Hn . Using Eq. (2.5) and integration by parts, one can obtain Z T (φ n (T ), Eψ(T )v) − (φ n (0), Eψ(0)v) = hφ n (t), Eψt (t)vidt 0

Z +

T

hg(t, φ n (t)) − A(t, φ n (t)) − Bφ n (t), ψ(t)vidt.

0

Letting n → ∞, we have (z, Eψ(T )v) − (φ0 , Eψ(0)v) = hhG(φ 0 ) − W − Bφ 0 , ψvii + hhEψt v, φ 0 ii. Using the convergence results above, one can easily verify that the limit elements φ 0 , W, z, Bφ 0 satisfy  d (Eφ 0 ) + W + Bφ 0 = G(φ 0 ), dt φ(0) = φ (B, E), φ 0 (T ) = z, φ 0 ∈ W 0 (B, E), b ∈ P , E ∈ M. 0

p,q

a,b

Again using Eq. (2.5) and integration by parts, we have 1 (kE 1/2 φ n (T )k2H − kE 1/2 φ n (0)k2H ) = hhG(φ n ) − A(φ n ) − Bφ n , φ n ii. 2 By virtue of the fact that lim inf kE 1/2 φ n (T )k H ≥ kE 1/2 φ 0 (T )k H , n→∞

we obtain hhW, φ 0 ii ≤ lim inf hhA(φ n ), φ n ii n→∞

≤ lim sup hhA(φ n ), φ n ii n→∞

1 ≤ hhG(φ 0 ), φ 0 ii − hhBφ 0 , φ 0 ii + (kE 1/2 φ 0 (0)k2H − kE 1/2 φ 0 (T )k2H ) 2 = hhW, φ 0 ii. Since A is monotone and hemicontinuous, W = A(φ 0 ) [31]. Thus the limit element φ 0 satisfies Eq. (2.1) and hence is a solution of (1.1). The uniqueness follows from Lemma 2.3. This finishes the proof of the theorem.  3. Existence of optimal solution In this section, we consider the problem (IP) for nonlinear system (1.1), namely find a pair (B 0 , E 0 ) ∈ Pa,b × M so that J (B 0 , E 0 ) ≤ J (B, E) for all (B, E) ∈ Pa,b × M, where Z J (B, E) = f (t, φ(B, E)(t), B, E)dt. (3.1) I

In the following, we assume that the initial datum φ0 (B, E) ≡ φ0 is fixed.

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Lemma 3.1. Consider the problem (IP). Suppose that the assumptions H(A), H(g), H(0) and H(f) hold. Then the mapping (B, E) → φ(B, E) is continuous from Pa,b × M to C(I, H ) in the sense that whenever {E n , E 0 } ∈ M, {B n , B 0 } ∈ Pa,b and E n → E 0 , B n → B 0 in the strong operator topology, φ(B n , E n ) ≡ φ n → φ 0 ≡ φ(B 0 , E 0 ) weakly in L p (I, V ) and the function (B, E) → J (B, E) is lower semicontinuous on Pa,b × M. Proof. Let {B n } ∈ Pa,b , {E n } ∈ M and suppose B n → B 0 ∈ Pa,b and E n → E 0 ∈ M in the strong operator topology. Let {φ n } and {φ 0 } denote the solutions of the system (1.1) corresponding to (B n , E n ) and (B 0 , E 0 ), respectively. Defining ϕ n = φ n − φ 0 , one observes that ϕ n is the solution of the problem:  d n n  n 0 n n    dt (E ϕ ) + A(t, φ ) − A(t, φ ) + B ϕ d (3.2) = ((E 0 − E n )φ 0 ) + (B 0 − B n )φ 0 + (g(t, φ n ) − g(t, φ 0 ))   dt   n ϕ (0) = 0. Multiplying the evolution equation in (3.2) by ϕ n and using Young’s Inequality, we have, for any ε > 0 1 p k(E n )1/2 ϕ n (t)k2H + ckϕ n k L t (V ) p 2

Z t Z t

d

0 n 0 n

∗ ≤ kϕ n (σ )k2H dσ

dσ ((E − E )φ (σ ))kV kϕ (σ ) dσ + |a| 0 0 V Z t Z t

n n

k(B 0 − B n )φ 0 (σ )kV ∗ kϕ n (σ )kV dσ + L(b) ϕ (σ )k H kϕ (σ ) V dσ + 0

0

2 Z t Z t

d 1

((E 0 − E n )φ 0 (σ )) dσ +  ≤ kϕ n (σ )k2V dσ

∗ 2 0 dσ 2 0 V Z t Z Z L(b) t n L(b) t n n 2 2 + |a| kϕ (σ )k H dσ + kϕ (σ )k H dσ + kϕ (σ )k2V dσ 2 2 0 0 0 Z t Z 1  t n 0 n 0 2 2 + k(B − B )φ (σ )kV ∗ dσ + kϕ (σ )kV dσ 2 0 2 0  Z t Z L(b) (L(b) + 2) t n ≤ |a| + kϕ n (σ )k2H dσ + kϕ (σ )k2V dσ 2 2 0 0

2 Z t Z t

1 d 1 0 n 0 0 n 0 2

+ ((E − E )φ (σ )) k(B − B )φ (σ )kV ∗ dσ +

∗ dσ.

2 2 dσ 0

0

By using the compact embedding

L tp (V ) p

k(E n )1/2 ϕ n (t)k2H + 2ckϕ n k L t

P (V )

,→

with the embedding constant b0 , we obtain  Z t Z t L(b) K n (t) p ≤ + 2|a| + kϕ n (σ )k2H dσ + L 2  kϕ n (σ )kV dσ,   0 0

where L 2 = (L(b) + 2)b0 and Z t Z K n (t) = k(B 0 − B n )φ 0 (σ )k2V ∗ dσ + By taking  =

0 c L2 ,

V

L t2 (V )

0

t

2

d

((E 0 − E n )φ 0 (σ )) dσ.

dσ

∗ V

we have p

k(E n )1/2 ϕ n (t)k2H + ckϕ n k L t (V ) ≤ p

  Z t L 2 K n (t) L 2 L(b) + 2|a| + kE n k−1 k(E n )1/2 ϕ n (σ )k2H dσ. L+ (H ) c c 0

Defining p

Ψ n (t) = k(E n )1/2 ϕ n (t)k2H + ckϕ n k L t (V ) , p

it follows from the above inequality that   Z t L 2 K n (t) L 2 L(b) n Ψ (t) ≤ + 2|a| + kE n k−1 Ψ n (σ )dσ. + L (H ) c c 0

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Using Gronwall’s lemma, one concludes that    L 2 L(b) L 2 K n (t) n n −1 Ψ (t) ≤ exp kE kL+ (H ) T 2|a| + c c

(3.3)

for all t ∈ I . Since B n → B 0 ∈ Pa,b in the strong topology and φ 0 ∈ L p (I, V ), it is clear that



0

(B − B n )x 0 ∗ → 0 V

almost everywhere on I and also there exists a finite number γ1 such that



0

(B − B n )φ 0 (t) ∗ ≤ γ1 kx 0 (t)kV for all t ∈ I. V

From Lemma 2.2, → E 0 ∈ M in the strong topology and φt0 ∈ L q (I, V ∗ ), it is clear that

d

((E 0 − E n )φ 0 ) → 0

dt

∗ V En

almost everywhere on I and also there exists a finite number γ2 such that

d

((E 0 − E n )φ 0 )(t) ≤ γ2 kx 0 (t)kV for all t ∈ I.

dt

∗ V Thus by the Lebesgue dominated convergence theorem, K n (t) → 0 as n → ∞, it follows that Ψ n (t) → 0 as w n → ∞ uniformly on I . Hence one may conclude from (3.3) that (E n )1/2 φ n → (E 0 )1/2 φ 0 in C(I, H ) and w w (E n )1/2 φ n (t) → (E 0 )1/2 φ 0 (t) in H for all t ∈ I . Further φ n → φ 0 in L p (I, V ). This proves the continuity of the map (B, E) → φ(B, E), as desired. Define now Z J (B n , E n ) = f (t, φ(B n , E n )(t), B n , E n )dt I Z 0 0 f (t, φ(B 0 , E 0 )(t), B 0 , E 0 )dt, and J (B , E ) = I

φn

φ0

where and are the solutions of the system (1.1) corresponding to (B n , E n ) and (B 0 , E 0 ), respectively. Since, by assumption H ( f ), for almost all t ∈ I, φ → f (t, φ(t), ·, ·) is lower semicontinuous on H , we have f (t, φ 0 (t), B 0 , E 0 ) ≤ lim inf f (t, φ n (t), B n , E n ) n

almost all on I,

and consequently, by Fatou’s lemma [1] Z Z f (t, φ 0 (t), B 0 , E 0 )dt ≤ lim inf f (t, φ n (t), B n , E n )dt. n

I

(3.4)

I

Using Lemma 2.2, the norm continuity, and the result (3.4), we obtain J (B 0 , E 0 ) ≤ lim inf J (B n , E n ), n

which completes the proof of the lemma.



Lemma 3.2. The set Pa,b considered as a subset of L(V, V ∗ ) is sequentially compact in the strong operator topology τso . Proof. For the proof, see Lemma 1.2 of [3].



Next we prove there exists an optimal operator–operator pair, (B 0 , E 0 ). Theorem 3.1. Suppose that assumptions H(A), H(g), H(0) and H(f) hold. Then there exists (B 0 , E 0 ) ∈ Pa,b × M such that J (B 0 , E 0 ) ≤ J (B, E)

for all (B, E) ∈ Pa,b × M.

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Proof. Define l = inf{J (B, E), (B, E) ∈ Pa,b × M}. Since f (t, φ, B, E) > −∞ for (t, φ, B, E) ∈ I × H × Pa,b × M, the infimum is well defined and l > −∞. Let {B k , E k } be a minimizing sequence from Pa,b × M, i.e., limk J (B k , E k ) = l. By Lemma 3.2 and the fact that the admissible set Pa,b × M is weakly compact, there exist a subsequence {(B ki , E ki )} ⊂ {(B k , E k )} relabeled as {(B k , E k )} and a (B 0 , E 0 ) ∈ Pa,b × M such that (B k , E k ) → (B 0 , E 0 ) in the strong topology. Since (B, E) → J (B, E) is lower semicontinuous with respect to the strong topology (Lemma 3.1) and (B k , E k ) → (B 0 , E 0 ) in the strong topology, we have l ≤ J (B 0 , E 0 ) ≤ lim inf J (B k , E k ) ≤ lim J (B k , E k ) = l. k

k

Hence J (B 0 , E 0 ) = l implies that J (·, ·) attains its infimum on Pa,b × M. This completes the proof.



Remark 3.1. We can also allow the operator B to be a function of time by taking for the admissible class, the set: 0 ≡ {B ∈ L ∞ (I, L(V, V ∗ )) : ess sup{kB(t)kL(V,V ∗ ) , t ∈ I } ≤ b, hB(t)ξ, ξ iv,v ∗ + akξ k2H ≥ 0 a.e. on I }. Pa,b

We obtained the same results as Pa,b . 4. Necessary optimality conditions We consider the necessary optimality conditions for the identification problem (IP). We note that usually the mapping (B, E) → φ(B, E) from Pa,b × M to L p (I, V ) is unique. In this section, we present the results for p = q = 2 but similar results are valid for general p, q. We denote L r+ = {φ(·) ∈ L r |φ(·) ≥ 0}, r = 1, 2, ∞. In order to derive the necessary optimality conditions, we need additional differentiability assumptions H(A)1 : The operator A : I × V → V ∗ satisfies (1) A satisfies condition H(A). (2) φ → A(t, φ) is continuously Fr´echet differentiable and strongly uniformly monotone in t ∈ I . (3) The Fr´echet derivative, A0φ (t, φ), satisfies the inequality kA0φ (t, φ)kV ∗ ≤ a1 (t) + b1 (t)kφkV a.e. with 0 2 a1 (·) ∈ L 2+ , b1 (·) ∈ L ∞ + and hAφ (t, φ)h, hiV,V ∗ ≥ βkhkV a.e. with β > 0, h ∈ V . ∗ H(g)1 : The map g : I × H → V satisfies (1) g satisfies condition H(g). (2) The mapping φ → g(t, φ) is Fr´echet differentiable, hgφ0 (t, φ)h, hiV,V ∗ ≤ 0 a.e. and kgφ0 (t, φ)kV ∗ ≤ a2 (t) + b2 kφk H a.e. with a2 (·) ∈ L 2+ , b2 > 0. (3) The mapping φ → gφ0 (·, φ) is continuous. H(f)1 : The integrable function f : I × H → R + satisfies (1) f satisfies condition H(f). (2) The mapping φ → f (·, φ, ·, ·) is Fr´echet differentiable and φ → f φ0 (·, φ, ·, ·) is continuous and k f φ0 (t, φ, ·, ·)k H ≤ a3 (t) + b3 (t)kφk2H a.e. with a3 (·) ∈ L 1+ , b3 (·) ∈ L ∞ + , for all t ∈ I . (3) The mapping B k f B0 (·, ·, B, ·)k H (4) The mapping E k f E0 (·, ·, ·, E)k H

→ f (·, ·, B, ·) is Fr´echet differentiable and B → f B0 (·, ·, B, ·) is continuous and ≤ a4 for all B ∈ Pa,b , for some a4 > 0. → f (·, ·, ·, E) is Fr´echet differentiable and E → f E0 (·, ·, ·, E) is continuous and ≤ a5 for all E ∈ M, for some a5 > 0.

To derive necessary optimality conditions, we use the Gˆateaux derivative of φ(B, E) with respect to the operator/operator (B, E). Indeed, we show that the Gˆateaux derivative of φ at (B 0 , E 0 ) in the direction (B − B 0 , E − E 0 ), defined by φ(B 0 + ε(B − B 0 ), E 0 + ε(E − E 0 )) − φ(B 0 , E 0 ) ε→0 ε

ψ(B 0 , E 0 ; B − B 0 , E − E 0 ) = w − lim

exists and the sensitivity ψ is the solution of a related differential equation.

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We present in the following lemma the Gˆateaux differentiability of the mapping (B, E) → φ(B, E) in the weak sense. Lemma 4.1. Consider the system (1.1) and suppose that the assumptions H(A)1 , H(g)1 , H(0), and H(f)1 hold. Let φ(B, E) denote the (weak) solution of the problem (1.1) corresponding to (B, E) ∈ Pa,b × M. Then at each point (B, E) ∈ Pa,b ×M the function (B, E) → φ(B, E) has a weak Gˆateaux derivative in the direction (B − B 0 , E − E 0 ), denoted ψ(B 0 , E 0 ; B − B 0 , E − E 0 ), and it is the solution of the Cauchy problem (d d (E 0 ψ) + A0φ (t, φ 0 )ψ + B 0 ψ − gφ0 (t, φ 0 )ψ = − ((E − E 0 )φ 0 ) − (B − B 0 )φ 0 (4.1) dt dt ψ(0) = 0 satisfying ψ ∈ L 2 (I, V ) ∩ L ∞ (I, H ), where φ ε = φ(B ε , E ε ) and φ 0 = φ(B 0 , E 0 ) are the solution of (1.1) corresponding to (B ε , E ε ) and (B 0 , E 0 ), respectively, where B ε = B 0 + ε(B − B 0 ), E ε = E 0 + ε(E − E 0 ) for 0 ≤ ε ≤ 1. The variational operators and maps in (4.1) are defined as   A(t, φ ε ) − A(t, φ 0 ) , A0φ (t, φ 0 ) = w − lim ε→0 φε − φ0 g(t, φ ε ) − g(t, φ 0 ) , ε→0 φε − φ0 φ ε (t) − φ 0 (t) ψ(t) = w − lim . ε→0 ε gφ0 (t, φ 0 ) = w − lim

Proof. Let (B 0 , E 0 ), (B, E) ∈ Pa,b × M. Since Pa,b × M is a sequentially compact and convex set, we have (B ε , E ε ) ∈ Pa,b × M, where B ε = B 0 + ε(B − B 0 ), E ε = E 0 + ε(E − E 0 ), φ ε = φ(B ε , E ε ), and φ 0 = φ(B 0 , E 0 ) for 0 ≤ ε ≤ 1. Using (1.1) and defining ψ ε ≡

φ ε −φ 0 ε ,

we obtain

 d ε ε A(t, φ ε ) − A(t, φ 0 ) g(t, φ ε ) − g(t, φ 0 )   − + Bεψ ε   dt (E ψ ) + ε ε d = − ((E − E 0 )φ 0 ) − (B − B 0 )φ 0 ,    dt  ε ψ (0) = 0.

(4.2)

Multiplying the evolution equation in (4.2) by ψ ε and using the assumptions, we have Z t Z t 1 hA0φ (σ, φ 0 (σ ) + εµ1 (φ ε (σ ) − φ 0 (σ )))ψ ε (σ ), ψ ε (σ )idσ k(E ε )1/2 ψ ε (t)k2H + hB ε ψ ε , ψ ε idσ + 2 0 0  Z t Z t d 0 0 ε h−(B − B 0 )φ 0 (σ ), ψ ε (σ )idσ = − ((E − E )φ )(σ ), ψ (σ ) dσ + dt 0 0 Z t + hgφ0 (σ, φ 0 (σ ) + εµ2 (φ ε (σ ) − φ 0 (σ )))ψ ε (σ ), ψ ε (σ )idσ, (4.3) 0

where µ1 , µ2 ∈ [0, 1]. Using the assumption H(A)1 and H(g)1 in (4.3), we obtain Z t Z t 1 k(E ε )1/2 ψ ε (t)k2H + β kψ ε (σ )k2V dσ − a kψ ε (σ )k2H dσ 2 0 0

Z t Z t

d

0 0 ε

− ((E − E )φ )(σ ) kψ (σ )kV dσ + ≤ k(B − B 0 )φ 0 (σ )kV ∗ kψ ε (σ )kV dσ

dt

∗ 0 0 V Z t + kgφ0 (σ, φ 0 (σ ) + εµ2 (φ ε (σ ) − φ 0 (σ )))ψ ε (σ )kV ∗ kψ ε (σ )kV dσ. 0

Using the inequality ab =

q

β 3a

q

3 βb

≤ 12 ( β3 a 2 + β3 b2 ) in (4.4) implies

(4.4)

Y.H. Kang / Nonlinear Analysis 70 (2009) 1443–1458

k(E ε )1/2 ψ ε (t)k2H + β

Z

Z

t

≤ 2|a|kE ε k−1 L+ (H ) 3 + β

t

Z 0

0

t

0

1453

kψ ε (σ )k2V dσ

k(E ε )1/2 ψ ε (σ )k2H dσ +

k(B − B 0 )φ 0 (σ )k2V ∗ dσ +

3 β

Z 0

t

3 β

2 Z t

d

− ((E − E 0 )φ 0 )(σ ) dσ

dt

∗ 0 V

kgφ0 (σ, φ 0 (σ ) + εµ2 (φ ε (σ ) − φ 0 (σ )))ψ ε (σ )k2V ∗ dσ,

(4.5)

for all ε ∈ [0, 1]. Since εµ2 ∈ [0, 1], it follows from the assumption H(g)1 , Lemma 2.2, using Gronwall’s Lemma, the fact that E ε ∈ M is bounded and the definition of Pa,b that the terms on the right side of (4.5) are well defined. This shows that {(E ε )1/2 ψ ε , 0 ≤ ε ≤ 1} and {ψ ε , 0 ≤ ε ≤ 1} are contained in a bounded subset of L ∞ (I, H ) ∩ L 2 (I, V ). Since L ∞ (I, H ) ∩ L 2 (I, V ) is a reflexive Banach space, we can find a subsequence {(E n )1/2 ψ n } ≡ {(E εn )1/2 ψ εn } ⊂ {ψ ε } with εn ∈ [0, 1] and εn → 0, and a (E 1/2 )ψ ∈ L ∞ (I, H ) ∩ L 2 (I, V ) w such that (E n )1/2 ψ n → E 1/2 ψ in L ∞ (I, H ) ∩ L 2 (I, V ). And also since L 2 (I, V ) is a reflexive Banach space, we w can find a subsequence {ψ n } ≡ {ψ εn } ⊂ {ψ ε } with εn ∈ [0, 1] and εn → 0, and a ψ ∈ L 2 (I, V ) such that ψ n → ψ in L 2 (I, V ). This proves that the Gˆateaux derivative of φ exists and is given by ψ(B 0 , E 0 ; B − B 0 , E − E 0 ) ≡ ψ. It remains to show that ψ is a solution of (4.1). Indeed, since d ε ε w d 0 (E ψ ) → (E ψ) in L 2 (I, V ∗ ), dt dt A(t, φ n ) − A(t, φ 0 ) w 0 → Aφ (t, φ 0 )ψ in L 2 (I, V ∗ ), ε w B ε ψ ε → B 0 ψ in L 2 (I, V ∗ ), g(t, φ n ) − g(t, φ 0 ) w 0 → gφ (t, φ 0 )ψ in L 2 (I, V ∗ ), ε w

it follows from (4.2) that ψt n ∈ L 2 (I, V ∗ ) and ψt n → ηt in L 2 (I, V ∗ ) for a suitable ηt ∈ L 2 (I, V ∗ ), and that η is the distributional derivative of ψ. Hence ψ satisfies the equality d 0 d (E ψ) + A0φ (t, φ 0 )ψ + B 0 ψ − gφ0 (t, φ 0 )ψ = − ((E − E 0 )φ 0 ) − (B − B 0 )φ 0 dt dt in the sense of vector-valued distributions in V ∗ . Since ψ ∈ L 2 (I, V ) and ψt ∈ L 2 (I, V ∗ ), it is clear that ψ ∈ C(I, H ) and ψ(0) is well defined and equals ψ n (0) = 0 for all n. Hence ψ satisfies the differential equation (4.1). This completes the proof.  With the help of Lemma 4.1, we derive the following necessary optimality conditions. Theorem 4.1. Suppose that theR assumptions H(A)1 , H(g)1 , H(0) and H(f)1 hold. Consider the system (1.1) and the problem (IP) with J (B, E) = I f (t, φ(B, E)(t), B, E)dt. Then in order that (B 0 , E 0 ) ∈ Pa,b × M be an optimal pair for the operator/operator, it is necessary that there exists a pair {φ 0 , p 0 } ∈ C(I, H ) × C(I, H ) satisfying the systems:   d (E 0 φ 0 (t)) + A(t, φ 0 (t)) + B 0 φ 0 (t) = g(t, φ 0 (t)) dt  0 φ (0) = φ0 , B 0 ∈ Pa,b , E 0 ∈ M the adjoint system  d 0 0  0 0 ∗ 0 0 ∗ 0  − (E p (t)) + (Aφ (t, φ (t))) p (t) + (B ) p (t) dt −(gφ0 (t, φ 0 (t)))∗ p 0 (t) = f φ0 (t, φ 0 (t)), for all t ∈ (0, T )    0 p (T ) = 0, and the optimality inequality

(4.6)

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Y.H. Kang / Nonlinear Analysis 70 (2009) 1443–1458

Z

f B0 (t, φ 0 (t), B 0 , E 0 ) + f E0 (t, φ 0 (t), B 0 , E 0 )dt

0≤ I

Z

h(B − B 0 )φ 0 , p 0 iV,V ∗ + h(E − E 0 )φ 0 , p 0 i H dt

−

(4.7)

I

for all B ∈ Pa,b , E ∈ M. Here (A0φ (t, φ 0 ))∗ , (gφ0 (t, φ 0 ))∗ and (B 0 )∗ are the adjoint of the variational operators A0φ (t, φ 0 (t)), gφ0 (t, φ 0 (t)) and B 0 , respectively. The functional is defined as f B0 (t, φ 0 (t), B 0 , E 0 ; B − f (t,φ 0 (t),B ε ,E 0 )− f (t,φ 0 (t),B 0 ,E 0 ) ε 0 0 ε 0 0 0 limε→0 f (t,φ (t),B ,E )−ε f (t,φ (t),B ,E ) .

B 0 , E − E 0 ) = w − limε→0

and f E0 (t, φ 0 (t), B 0 , E 0 ; B − B 0 , E − E 0 ) = w −

Proof. Since the map (B, E) → φ(B, E) has a (weak) Gˆateaux derivative on Pa,b × M, it follows that J (·, ·) as defined above also has a Gˆateaux derivative. Denote φ 0 ≡ φ(B 0 , E 0 ). Then in order that J (·, ·) attains its minimum at (B 0 , E 0 ) ∈ Pa,b × M, it is necessary that   J (B ε , E ε ) − J (B 0 , E 0 ) J 0 (B 0 , E 0 ; B − B 0 , E − E 0 ) ≡ lim ≥0 (4.8) ε→0 ε for all (B, E) ∈ Pa,b × M. Using the result of Lemma 4.1, it follows from the above that Z 0 0 0 0 0 0 ≤ J (B , E ; B − B , E − E ) = h f φ0 (t, φ 0 (t), B 0 , E 0 ), ψ(t)idt I Z Z 0 0 0 0 f E0 (t, φ 0 (t), B 0 , E 0 )dt f B (t, φ (t), B , E )dt + + I

(4.9)

I

for all (B, E) ∈ Pa,b × M, where ψ(t) is the Gˆateaux derivative as given by Lemma 4.1. Using (4.1) and (4.8), we obtain the adjoint system (4.6). Reversing the time flow, t → T − t, it follows from Theorem 2.1 that the system (4.6) has a unique weak solution p 0 ∈ L 2 (I, V ) ∩ C(I, H ) with dtd p 0 ∈ L 2 (I, H ) and p 0 (T ) = 0 with dtd p 0 ∈ L 2 (I, H ) and p 0 (T ) = 0. Using (4.6) and (4.9) and integrating by parts, we obtain  Z  d 0 0 0 0 0 0 0 0≤ (E ψ(t)) + Aφ (t, φ (t))ψ(t) + B ψ(t) − gφ (t, φ (t))ψ(t), p (t) dt I dt V ∗ ,V Z Z f E0 (t, φ 0 (t), B 0 , E 0 )dt, (4.10) f B0 (t, φ 0 (t), B 0 , E 0 )dt + + I

I

for all (B, E) ∈ Pa,b × M. From the sensitivity system (4.1) and (4.10), we obtain Z Z Z 0 0 0 0 0 0 0 0 ∗ h f φ (t, φ (t), B , E ), ψ(t)iV ,V dt + f B (t, φ (t), B , E )dt + f E0 (t, φ 0 (t), B 0 , E 0 )dt I I I  Z  d 0 0 0 0 0 0 (E ψ(t)) + Aφ (t, φ (t))ψ(t) + B ψ(t) − gφ (t, φ (t))ψ(t), p 0 (t) = dt I dt V ∗ ,V Z Z + f B0 (t, φ 0 (t), B 0 , E 0 )dt + f E0 (t, φ 0 (t), B 0 , E 0 )dt I  ZI d 0 0 =− (B − B )φ (t) + ((E − E 0 )φ 0 (t)), p 0 (t) dt dt I V ∗ ,V Z Z + f B0 (t, φ 0 (t), B 0 , E 0 )dt + f E0 (t, φ 0 (t), B 0 , E 0 )dt ≥ 0 I

I

for all (B, E) ∈ Pa,b × M. Hence characterization (4.7) is verified. This completes the proof.



Remark 4.1. We can get similar results to Lemma 4.1 and Theorem 4.1 for the general case 1 < q ≤ 2 ≤ p < ∞. Remark 4.2. Depending on the specific two operators in (1.1) and on the objective functional, one can sometimes solve explicitly for the optimal pair {B 0 , E 0 } in terms of the corresponding state and adjoint operators. Thus the

Y.H. Kang / Nonlinear Analysis 70 (2009) 1443–1458

1455

state and adjoint systems together with the two optimal operator characterization form the optimality systems. Under certain conditions, uniqueness of the optimality systems can also be obtained; this implies, in turn, uniqueness of the optimal pair (see Refs. [1,10–13,17–19,25–30] for various examples yielding an explicit characterization). In the first and second examples of the next section, we are also able obtain such explicit characterizations. 5. Illustrations In this section, we present two examples on which we illustrate the general theory derived in Sections 2–4. Example 1. We present here a simpler example dealing with a diffusive population model of a harvest [7]. The population has logistic growth with carrying capacity k (fixed). The dynamics of this system are governed by the following initial-boundary-value problem:  ∂  e(x) φ(x, t) − κ∆φ(x, t) + Bφ(x, t) = φ(x, t)(k − φ(x, t)) in Ω × (0, T ), ∂t (5.1) φ(x, t) = 0 on ∂Ω × [0, T ],   φ(x, 0) = φ0 (x) ∈ H in Ω . Here Ω is an open bounded subset of R n with smooth boundary. Define the operator E by (Eφ)(x) = e(x)φ(x), x ∈ Ω , φ ∈ H. E is a non-negative function on Ω with E ∈ L ∞ (Ω ) and there is present a climate change rate in any areas. Clearly, for E ∈ L + ∞ (Ω ), the operator E is a bounded linear positive self-adjoint operator in H . The constant parameter κ can be adjusted by environmental or nutrient changes. For the mathematical setting, we take p = q = 2, H = L 2 (Ω , R n ), V = H01 (Ω , R n ), with V ∗ being the dual of V . Let ∂t∂ φ = φt . For the operator A, we take, the L 2 (Ω , R n ) realization of the Laplacian −κ∆ with the Dirichlet boundary condition. It is obvious that the operator A ∈ L(V, V ∗ ) is coercive and hence monotone. Bφ = −βφ, β 6= 0 is an unknown constant. The nonlinear map g is realized by g(t, φ) = φ(k − φ). Let the admissible set M be a closed, bounded, and convex subset of L+ (H ) given by M = {E ∈ L + ∞ (H )|0 ≤ E(x) ≤ δ, ∃δ > 0}. Let the admissible set Pa,b be a relatively compact and convex subset of L(V, V ∗ ) given by Pa,b = {B ∈ L(V, V ∗ ) : kBkL(V,V ∗ ) ≤ b, hBξ, ξ i + akξ k2H ≥ 0 ∀ξ ∈ V }. and the solution set of the problem (5.1) is: W2,2 = {φ : φ ∈ L 2 (0, T ; V ), φt ∈ L 2 (0, T ; H ), Eφt ∈ L 2 (0, T ; V ∗ )}. Define the cost functional J (·, ·) by Z Z 1 ν µ ¯ J1 (B, E) = (φ(x, t) − φ(x, t))2 dxdt + (E(x))2 dx + (β)2 , 2 PT 2 Ω 2

(5.2)

where φ = φ(B, E), φ¯ is the target function, ν, µ are positive constants, and PT = Ω × (0, T ). We seek to keep the population near the desired level φ¯ while obtaining profit from harvest. The problem is to find (B 0 , E 0 ) such that J1 (B 0 , E 0 ) = inf{J1 (B, E) : (B, E) ∈ Pa,b × M}.

(5.3)

Using the results obtained in the previous sections, we get: Lemma 5.1. Consider the system (5.1) and suppose that the assumptions above hold. Let φ(B, E) denote the (weak) solution of the problem (5.1) corresponding to (B, E) ∈ Pa,b × M. Then at each point (B, E) ∈ Pa,b × M the function (B, E) → φ(B, E) has a weak Gˆateaux derivative in the direction (B − B 0 , E − E 0 ), denoted ψ(B 0 , E 0 ; B − B 0 , E − E 0 ), and it is the solution of the Cauchy problem  0  E ψt − κ∆ψ − (k − 2φ 0 )ψ + B 0 ψ = −(E − E 0 )φt 0 − (B − B 0 )φ 0 , ψ(x, t) = 0 on ∂Ω × [0, T ],  ψ(x, 0) = 0 in Ω satisfying ψ ∈ C(I, V ) ∩ C(I, H ), where φ 0 = φ(B 0 , E 0 ) is the solution of (5.1) corresponding to (B 0 , E 0 ).



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Using Lemma 5.1, we derive the following necessary conditions for optimality. Note that for this simple problem, we are able to solve for the necessary conditions of the optimal B 0 and E 0 explicitly in terms of the optimal state and the corresponding adjoint. Corollary 5.1. Suppose that the assumptions above hold. Consider the system (5.1) and the problem (5.3) with (5.2). Then in order that (B 0 , E 0 ) ∈ Pa,b × M is an optimal pair for the operator/operator, it is necessary that there exists a pair {φ 0 , p 0 } satisfying the state system (5.1), the adjoint equation  0 0 0 0 ∗ 0 0 0 0 ¯ −E pt − κ∆ p + (B ) p − (k − 2φ ) p = φ − φ, 0 (5.4) p (x, t) = 0 on ∂Ω × [0, T ],  0 p (x, T ) = 0 in Ω , and the optimality inequality Z TZ   p 0 −(E − E 0 )φt 0 − (B − B 0 )φ 0 dxdt 0≤ 0 Z Ω +ν E 0 (E − E 0 )dx + µβ 0 (β − β 0 ), ∀(B, E) ∈ Pa,b × M. Ω

Furthermore, we can take optimal pair (B 0 , E 0 ) ∈ Pa,b × M as follows:    Z 1 T 0 0 0 E (x) = min δ, max 0, p φt dt ν 0    Z Z 1 T 0 0 0 β = min b, max −b, p φ dxdt , µ 0 Ω where p 0 is a solution of the adjoint equation (5.4).



Example 2. For further illustration, consider the system of reaction–diffusion:  ∂  e(x) φ(x, t) − D(x, t)∆φ(x, t) + Bφ(x, t) = g(x, t, φ(x, t)) in PT = Ω × (0, T ), ∂t φ(x, t) = 0 on ∂Ω × [0, T ],   φ(x, 0) = φ0 (x) ∈ H in Ω ,

(5.5)

where φ is an k-vector-valued function on PT , φ0 (x) is given k-vector-valued function, φ ∈ L 2 (Ω , R k ). D = diag(d1 , . . . , dk ) is the diffusion matrix and di > 0, i = 1, 2, . . . , k. Let e(x) be an unknown matrix of density rate. e := diag(e1 , . . . , ek ) and ei (i = 1, 2, . . . k) be a non-negative function on Ω with ei ∈ L ∞ (Ω ). For the mathematical setting, we take p = q = 2, H = L 2 (Ω , R k ), V = W01,2 (Ω , R k ) with V ∗ = W −1,2 (Ω , R k ) being the dual V . For the nonlinear term, we assume that g(x, t, φ(x, t)) is a k-vector-valued function defined on PT × R k → R k which satisfies the following properties: (1) The function g is continuous in all the variables, (2) There exist a function h(t) ∈ L 2 (I, R + ) and constant α ≥ 0 such that kg(x, t, φ)k R k ≤ h(t) + αkφkk−1 , Rk

2 ≤ k ≤ p < ∞, a.e. in PT uniformly.

Then g satisfies the assumption H(g) of Section 2. Define the operator E by (Eϕ)(x) = e(x)ϕ(x),

x ∈ Ω , ϕ ∈ H.

k Clearly, for e ∈ L + ∞ (Ω , R ), the operator E is a bounded linear positive self-adjoint operator in H . Let A = −D∆, it is obvious that the operator A ∈ L(V, V ∗ ) and it is coercive and monotone [30]. A satisfies the assumption H(A) of Section 2. Define the operator B by Bϕ = β ∂ϕ ∂ x , constant β ≥ 0, β := diag(β1 , . . . , βk ). For the cost integrand f , one may choose the quadratic function f : I × H × Pa,b × M 7→ R as follows

Y.H. Kang / Nonlinear Analysis 70 (2009) 1443–1458

f (t, φ, B, E) =

1457

Z


1 ¯ ¯ C(x, t)(φ(x, t) − φ(x, t)), (φ(x, t) − φ(x, t)) R k dx 2 Ω Z k ν µ X + (E(x))2 dx + β2 2T Ω 2T i=1 i

¯ with C(x, t) is a symmetric positive semidefinite matrix-valued function on PT , φ(x, t) ∈ R k is target states. Then f (t, φ, B, E) satisfies assumption H ( f ) of Section 2. Let the admissible set M be a closed, bounded, and convex subset of L+ (H ). Let the admissible set Pa,b be a relatively compact and convex subset of L(V, V ∗ ) given by Pa,b = {B ∈ L(V, V ∗ ) : kBkL(V,V ∗ ) ≤ b, hBξ, ξ i + akξ k2H ≥ 0 ∀ξ ∈ V }. The solution set of the problem (5.5) is: W2,2 = {φ : φ ∈ L 2 (0, T ; V ), φt ∈ L 2 (0, T ; H ), Eφt ∈ L 2 (0, T ; V ∗ )}. Define the cost functional J (·, ·) by 1 J2 (B, E) = 2

Z

¯ ¯ (C(x, t)φ(x, t) − φ(x, t), φ(x, t) − φ(x, t))dxdt + PT

Z k ν µX β 2, (E(x))2 dx + 2 Ω 2 i=1 i

(5.6)

where φ = φ(B, E), φ¯ is the target function, ν, µ are positive constants, and PT = Ω × (0, T ). We seek to keep the population near the desired level φ¯ while obtaining profit. The problem is to find (B 0 , E 0 ) such that J (B 0 , E 0 ) = inf{J (B, E) : (B, E) ∈ Pa,b × M}.

(5.7)

Using the results obtained in the previous sections, we get Lemma 5.2. Consider the system (5.5) and suppose that the assumptions above hold. Let φ(B, E) denote the (weak) solution of the problem (5.5) corresponding to (B, E) ∈ Pa,b × M. Then at each point (B, E) ∈ Pa,b × M the function (B, E) → φ(B, E) has a weak Gˆateaux derivative in the direction (B − B 0 , E − E 0 ), denoted ψ(B 0 , E 0 ; B − B 0 , E − E 0 ), and it is the solution of the Cauchy problem  ∂   (E 0 ψ) − D∆ψ + B 0 ψ − gφ0 (t, φ 0 )ψ = −(E − E 0 )φt 0 − (B − B 0 )φ 0 in PT = Ω × (0, T ), ∂t (5.8) ψ(x, t) = 0 on ∂Ω × [0, T ],   ψ(x, 0) = 0 in Ω , satisfying ψ ∈ C(I, V ) ∩ C(I, H ), where φ 0 = φ(B 0 , E 0 ) is the solution of (5.5) corresponding to (B 0 , E 0 ).



Using Lemma 5.2, we derive the following necessary conditions for optimality. Note that for this simple problem, we are able to solve to the necessary conditions for the optimal B 0 and E 0 explicitly in terms of the optimal state and corresponding adjoint. Corollary 5.2. Suppose that the assumptions above hold. Consider the system (5.5) and the problem (5.7) with (5.6). Then in order that (B 0 , E 0 ) ∈ Pa,b × M is an optimal pair for the operator–operator, it is necessary that there exists a pair {φ 0 , p 0 } satisfying the state system (5.5), the adjoint equation  0 0 0 0 ∗ 0 0 0 ∗ 0 0 ¯ −E pt − D∆ p + (B ) p − (gφ (t, φ )) p = C(φ − φ), 0 p (x, t) = 0 on ∂Ω × [0, T ],  0 p (x, T ) = 0 in Ω and the optimality inequality

(5.9)

1458

Y.H. Kang / Nonlinear Analysis 70 (2009) 1443–1458

Z

T

0≤

Z   (E 0 − E)φt 0 (x, t) + (B 0 − B)φ 0 (x, t) p 0 (x, t)dxdt Ω

0

+ν

Z Ω

E 0 (x)(E − E 0 )(x)dx + µ

k X

βi0 (βi − βi0 )

∀(B, E) ∈ Pa,b × M,

i=1

where p 0 is a solution of the adjoint equation (5.9).



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