CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 14, Number 3, Fall 2006 NECESSARY CONDITIONS FOR OPTIMAL CONTROL SYSTEMS GOVERNED BY SEMI-LINEAR ELLIPTIC EQUATIONS HANG GAO AND MICHAEL Y. LI ABSTRACT. We study an optimal control problem for systems governed by a class of elliptic equations that may possess multiple solutions. Necessary conditions for optimal control of the elliptic problem are obtained by constructing a suitable sequence of approximating parabolic control problems. 1 Introduction and main results We consider the following control problem for systems governed by a class of semi-linear elliptic equations ( −A y = f (x, y(x), u(x)), x ∈ Ω (1.1) y ∂Ω = 0, where the operator A has the form Ay = n X aij (x) yxj i,j=1 xi . The region Ω ⊂ Rn is bounded with smooth boundary ∂Ω. The control function u(x) satisfies u(x) ∈ U for all x ∈ Ω, where U ⊂ Rn is a bounded closed convex set. A y(x) is a solution to (1.1) if y ∈ W 2,p (Ω), p > 1, −Ay = f (x, y(x), u(x)), a.e. x ∈ Ω, and y|∂Ω = 0. We make the following assumptions. (P1) aij ∈ C 1 (Ω), aij (x) = aji (x), and n X i,j=1 aij (x) ηi ηj ≥ λ |η|2 , ∀η ∈ Rn . c Copyright Applied Mathematics Institute, University of Alberta. 239 240 HANG GAO AND MICHAEL Y. LI (P2) f : Ω × R × U → R satisfies that f (x, y, k1 u1 + k2 u2 ) = k1 f (x, y, u1 ) + k2 f (x, y, u2 ), fy (x, ·, u) ∈ C(R), f (·, y, u) is measurable on Ω, and there exists a monotonically increasing function c : R+ → R+ such that |f (x, y, u)| ≤ c(|y|). In [3], the following problem ( −div |∇y|p−2 ∇y = a(x) |y|p y + b(x), (1.2) y ∂Ω = 0, x∈Ω was considered in with p > 1, where y(x) represents distribution of temperature. When p = 2, (1.2) is the Dirichlet problem of a semilinear elliptic equation. The right hand side of the equation is linear with respect to both a(x) and b(x). When the control function is chosen to be either the heat coefficient a(x) or the heat source b(x), system (1.2) satisfies our assumptions (P1) and (P2). A function y ∗ ∈ W 2,p (Ω) is said to be a upper solution to (1.1) if (1.3) ( −A y ≥ f (x, y(x), u(x)), y ∂Ω ≥ 0. a.e. x ∈ Ω Similarly, a lower solution y∗ ∈ W 2,p (Ω) satisfies (1.3) with both inequalities reversed. Define (1.4) U = { u(x) : u(x) ∈ U, u(·) is measurable on Ω } as the set of admissible controls. We make the following additional assumption for (1.1). (P3) The problem (1.1) has an upper solution y ∗ (x) and a lower solution y∗ (x) for all u ∈ U. Both y ∗ and y∗ are independent of u. The following lemma is similar to Theorem 1 in [1]. Lemma 1.1. Assume that the assumptions (P1)–(P3) are satisfied. Then, for each u ∈ U, problem (1.1) has at least two solutions y1 (x) and y2 (x) such that y∗ (x) ≤ y1 (x) ≤ y2 (x) ≤ y ∗ (x). NECESSARY CONDITIONS FOR OPTIMAL CONTROL 241 Set P = {(y, u) : u ∈ U, (y, u) satisfies (1.1)}. Any (y, u) ∈ P will be called an admissible pair. We define the following functional on P Z 2 [y(x, u) − B(x)] dx. (1.5) J(y, u) = Ω If there exists (y, u) ∈ P such that J(y, u) = inf{J(y, u) : (y, u) ∈ P}, then u is called an optimal control with y as an optimal state. We also call (y, u) an optimal pair. Problem E. Assume that (y, u) is an optimal pair. What conditions do u and y satisfy? This problem has been studied for systems governed by semi-linear elliptic equations (see [4, 5, 8, 12]) under the following assumption on f (1.6) fy (x, y, u) ≤ 0 for all (x, y, u). Condition (1.6) implies that the state variable is continuously dependent on the control variable. It plays an essential role in previous studies of Problem E; if (1.6) is not satisfied, then the problem (1.1) may have multiple solutions for each u. In the present paper, the restriction (1.6) is relaxed. To deal with the non-uniqueness, we approximate the elliptic control problem (1.1) with a suitable sequence of parabolic control systems. Necessary conditions for optimal control are first obtained for the approximating parabolic systems, which, in the limit, lead to necessary conditions for optimal control of the elliptic system. The main results of the present paper are stated in the following. Theorem 1.1. Assume that (P1)–(P3) are satisfied. Suppose that (¯ y (x), u ¯(x)) is an optimal pair and that fy (x, y(x), u(x)) < λ, where λ is given in (P1). Then there exist ξ(x) and ψ(x) such that kξk2,Ω ≤ C and ψ(x) satisfies the equation (1.7) ( −A ψ(x) = fy (x, y(x), u(x)) ψ(x) − 2 [y(x) − A(x)] , ψ ∂Ω = 0. 242 HANG GAO AND MICHAEL Y. LI Furthermore, the following maximum principle holds (1.8) H(x, u ¯(x)) = max H(x, u), u∈U a.e. x ∈ Ω, in which, H(x, u) := ψ(x) f (x, y(x), u) + ξ(x)u. The existence of optimal control is proved under a more relaxed assumption: (P2’) f (·, y, u) is measurable on Ω, f (x, · , ·) and fy (x, · , ·) are continuous on R × U, and there exists a continuous function c : [0, +∞) → [0, +∞), such that | f (x, y, u) | ≤ c(|y|). Theorem 1.2. Suppose that (P1), (P2’) and (P3) are satisfied. Then the problems (1.1) and (1.5) have at least one optimal pair. 2 Approximation by parabolic control systems (0, 1), Γ = ∂Ω × (0, 1), and Let Q = Ω × V = {v(x, t) ∈ [0, 1] : v(·) is measurable on Q}. Let n be an integer and v ∈ V, consider the following parabolic equation (2.1) wt (x, t) − Aw(x, t) = f (x, w(x, t), u(x)) 1 + v(x, t), (x, t) ∈ Q, n w = 0, w(x, 0) = y¯(x), Γ where y¯(x) is an optimal state in Problem E. From the standard parabolic theory (see [10]), we know that, for each (u, v) ∈ U × V, there exists a unique solution w = w(x, t; u, v) to problem (2.1) such that w ∈ Wp2,1 (Q) (p > 1). We define the following cost functional (2.2) Fn (u, v) = Z 2 Q [w(x, t) − B(x)] dx dt − √ + C0 n Z Ω Z Ω ψ(x)[w(x, 1) − y¯(x)] dx [u(x) − u ¯(x)]2 dx. NECESSARY CONDITIONS FOR OPTIMAL CONTROL 243 Here, w(x, t) is the solution of problem (2.1) corresponding to u(x) and v(x, t), ψ(x) is the solution of problem (1.7), (¯ y (x), u¯(x)) is an optimal pair in Problem E, and constant C0 > 0 is to be determined later. Set 1 . (2.3) Un = u ∈ U : ku − u ¯k2,Ω ≤ n An optimal control to the problem (2.1)–(2.2) is a pair (u, v) that minimizes the functional Fn . Theorem 2.1. For each integer n, there exists an optimal control to problem (2.1)–(2.2) in Un × V. Proof. It is easy to see that Fn (u, v) is bounded below for any (u, v) ∈ Un × V. There thus exists a minimizing sequence {(uk , vk )} ⊂ Un × V, such that lim Fn (uk , vk ) = inf{Fn (u, v) : (u, v) ∈ Un × V}. k→∞ For any k, equation (2.1) may be written as where wk t (x, t) − Awk (x, t) − ck (x, t) wk (x, t) 1 = f (x, 0, u(x)) + vk (x, t), (x, t) ∈ Q, n wk Γ = 0, wk (x, 0) = y¯(x), ck (x, t) = Z 1 fy (x, τ wk (x, t), uk (x)) dτ. 0 From the assumption (P2), we know that there exists constant C > 0, such that |ck (x, t)| ≤ C for all k. It follows from the classical theory that kwk kWp2,1 (Q) ≤ C, p > 1. There thus exists a subsequence {wki } ⊂ {wk }, and wn ∈ Wp2,1 (Q), such that wki → w ¯n weakly in Wp2,1 (Q), and thus wki → w n in Lp (Q) by the embedding theorem. Since kuki −uk2,Ω ≤ 1/n and kvki kp,Q ≤(measQ)1/p , there exist un , v n , and subsequences of {uki } and {vki }, denoted by 244 HANG GAO AND MICHAEL Y. LI themselves, such that uki → un weakly in L2 (Ω), and vki → v n weakly in Lp (Q). Let q be such that 1/p + 1/q = 1. For any ϕ ∈ Lq (Q), we have Z Q ϕ(x, t)[wki t (x, t) − Awki (x, t)] dx dt 1 ϕ(x, t) f (x, wki (x, t), uki (x)) + vki (x, t) dx dt. = n Q Z Since f (x, y, u) is linear in u and continuous in y, we can take a limit in the above equality as ki → ∞ to arrive at Z Q ϕ(x, t)[w nt (x, t) − Aw n (x, t)] dx dt 1 ϕ(x, t) f (x, w n (x, t), un (x)) + v n (x, t) dx dt. = n Q Z We also have w n |Γ = 0 and w n (x, 0) = y¯(x), by the H¨ older continuity of wki and w n . Therefore, w n (x, t) is a solution to the problem (2.1) corresponding to un (x) and v n (x, t). Next, we prove that (un , v n ) is an optimal control for the problem (2.1)–(2.2). Let Fn1 (u, v) = and Z Q [w(x, t) − B(x)]2 dx dt − √ Fn2 (u, v) = C0 n Z Ω Z Ω ψ(x)[w(x, t) − y¯(x)] dx [u(x) − u(x)]2 dx. Fn1 (u, v) It is easy to see that is weakly continuous. We will show that Fn2 (u, v) is weakly lower semi-continuous. In fact, for any ` > 0, let e` = u∈U : Z 2 Ω [u(x) − u(x)] dx ≤ ` . Suppose there exists a sequence {um } ⊂ e` such that um → u ˜ weakly in L2 (Ω). Then, k˜ u − uk22,Ω ≤ lim inf kum − uk22,Ω ≤ `, m→∞ NECESSARY CONDITIONS FOR OPTIMAL CONTROL 245 namely u ˜ ∈ e` . This implies that the set e` is sequentially weakly closed. From [2], we know that Fn2 (u, v) is weakly lower semi-continuous. Therefore, Fn (un , v n ) = Z 2 Q [w n (x, t) − B(x)] dx dt − √ + C0 n ≤ lim ki →∞ Z − Q Z Z Ω Z Ω ψ(x)[w n (x, 1) − y¯(x)] dx (un (x) − u(x))2 dx [wki (x, t) − B(x)]2 dx dt Ω ψ(x)[w ki (x, 1) − y¯(x)] dx √ + C0 n Z Ω (uki (x) − u(x))2 dx = inf{F (u, v) : (u, v) ∈ Un × V }. This implies Fn (un , v n ) = inf{F (u, v) : (u, v) ∈ Un × V }, completing the proof of Theorem 2.1. Theorem 2.2. For some C0 > 0 in (2.2), un satisfies kun − uk2,Ω < √ 1/ n. √ Proof. Suppose the contrary, namely, kun − uk2,Ω = 1/ n. Since y¯(x, t) = y¯(x) satisfies ( y¯t (x, t) − A¯ y (x, t) = f (x, y¯(x, t), u(x)), y¯|Γ = 0, y¯(x, 0) = y¯(x), we have (w n (x, t) − y¯)t − A(w n (x, t) − y¯) − cn (x, t)(w n (x, t) − y¯) 1 = f (x, y¯(x), un (x)) − f (x, y¯(x, t), u(x)) + v n (x, t), n (w n (x, t) − y¯)|Γ = 0, (w n (x, t) − y¯)(x, 0) = 0, 246 HANG GAO AND MICHAEL Y. LI where cn (x, t) = Z 1 0 fy (x, y¯(x) + τ (w n (x, t) − y¯(x)), un (x)) dτ, and for each n, |cn (x, t)| ≤ C1 . Choose a constant L, such that L + C1 < −1. Let hn (x, t) = eLt [wn (x, t) − y¯(x)]. Then (2.4) hn t (x, t) − Ahn (x, t) − (L + cn (x, t))hn (x, t) = eLt [f (x, y¯(x), un (x)) − f (x, y¯(x, t), u(x)) + hn Γ = 0, 1 v n (x, t)], n hn (x, 0) = 0. Multiplying (2.4) by hn (x, t), and by integration by parts over Q, we obtain Z Z Z 1 h2n (x, 1)dx + λ |∇hn (x, t)|2 dx dt + h2n (x, t) dx dt 2 Ω Q Q Z ≤ |fu (x, y¯(x), u(x)| |un (x) − u(x)| |hn (x, t)| dx dt Q + 1 n Z v n (x, t)|hn (x, t)| dx dt Q Z 1 ≤ C (un (x) − u(x)) dx + h2 (x, t) dx dt 4 Q n Ω Z 1 1 + C+ h2 (x, t) dx dt. n 4 Q n Z 2 It follows from this inequality that (2.5) Z Ω [w n (x, 1) − y¯(x)]2 dx ≤ C∗ n NECESSARY CONDITIONS FOR OPTIMAL CONTROL 247 and Z (2.6) Q [wn (x, t) − y¯(x)]2 dx dt ≤ C∗ . n Furthermore, Z C∗ ≤ √ (x, 1) − y ¯ (x)] dx ψ(x)[w , n n Ω (2.7) and (2.8) Z C∗ 2 [w n (x, t) − y¯(x)][w n (x, t) − B(x)] dx dt ≤ √ . n Q From the relation Z Z [¯ y (x) − B(x)]2 dx dt [¯ y (x) − B(x)]2 dx = Q Ω = Z Q y (x) − w n (x, t))(w n (x, t) − B(x)) {[¯ y(x) − wn (x, t)]2 + 2(¯ + [w n (x, t) − B(x)]2 } dx dt ≤ Z 2C ∗ [w n (x, t) − B(x)]2 dx dt + √ , n Q we obtain Fn (un , v n ) Z = [w n (x, t) − B(x)]2 dx dt Q − Z Ω √ ψ(x)[w n (x, 1) − y¯(x)] dx + C0 n 1 ≥ [¯ y (x) − B(x)]2 dx + (C0 − 3C ∗ ) √ . n Ω Z Z Ω [un (x) − u(x)]2 dx Choose C0 such that C0 − 3C ∗ > 0, we obtain Z y (x) − B(x)]2 dx = Fn (u, 0). (2.9) Fn (un , v n ) > [¯ Ω Since (u, 0) ∈ Un × V, inequality (2.9) indicates that (un , v n ) is not an optimal control of the problem √ (2.1)–(2.2) in Un × V. The contradiction shows that kun − uk2,Ω < 1/ n, completing the proof. 248 HANG GAO AND MICHAEL Y. LI Remark 2.1. From the proofs of the Theorems 2.1 and 2.2, we have that (1) lim kun − uk2,Ω = 0, n→∞ (2) lim (kwn − y¯k2,Ω + k∇(wn − y¯)k2,Ω ) = 0, n→∞ and (3) lim Fn (un , v n ) = J(y, u). n→∞ 3 Optimality conditions for the approximating parabolic systems In this section, we derive necessary conditions for optimal control for the parabolic system (2.1)–(2.2). From Theorem 2.2 we know that the optimal control function un is an interior point of Un , that is, there exists ρ0 > 0, such that for any ρ ∈ (0, ρ0 ) and u ∈ U, uρ = un + ρ(u − un ) ∈ Un . At the same time, for any v ∈ V, vρ = v n + ρ(v − v n ) ∈ V. Lemma 3.1. Suppose that wnρ (x, t) and wn (x, t) are the solutions of problem (2.1) corresponding to (uρ , vρ ) and (un , v n ), respectively. Then, (3.1) kwnρ (x, t) − w n (x, t)k2,Q + k∇(wnρ (x, t) − wn (x, t))k2,Q ≤ ρ, where C is a constant independent of ρ. Proof. Let hn (x, t) = eLt [wnρ (x, t) − w n (x, t)]. By the assumption of Lemma 3.1, we know that hn (x, t) satisfies hn t (x, t) − Ahn (x, t) − (L + cn (x, t))hn (x, t) Lt = e f (x, wn (x, t), uρ (x)) − f (x, w n (x, t), un (x)) (3.2) 1 + ρ(v − v n (x, t)) , n hn Γ = 0, hn (x, 0) = 0. NECESSARY CONDITIONS FOR OPTIMAL CONTROL 249 Multiplying (3.2) by hn (x, t), and by integration by parts over Q, we obtain Z Z Z 1 2 2 h2n (x, t) dx dt |∇hn (x, t)| dx dt + h (x, 1) dx + λ 2 Ω n Q Q Z Z 1 ≤ C (uρ (x) − un (x))2 dx + h2n (x, t) dx dt 4 Ω Q Z 1 1 h2 (x, t) dx dt, + Cρ2 + n 4 Q n which implies inequality (3.1). Lemma 3.2. Suppose the function zn (x, t) satisfies the equation (3.3) Then, (3.4) zn t (x, t) − Azn (x, t) − fy (x, w n (x, t), un (x))zn (x, t) = f (x, w n (x, t), u(x)) − f (x, w n (x, t), un (x)) + zn Γ = 0, 1 (v − v n (x, t)), n zn (x, 0) = 0. wnρ (x, t) = wn (x, t) + ρzn (x, t) + rρ (x, t), and (3.5) krρ k2,Q + k∇rρ k2,Q = o(ρ). Proof. Let (1/ρ)rρ (x, t) = (1/ρ)(wnρ (x, t) − w n (x, t)) − zn (x, t). By the linearity of f on u and the definition of uρ , we have 1 [f (x, w n , uρ ) − f (x, wn , un )] = f (x, w n , u) − f (x, wn , un ). ρ Let rρ (x, t) = (wnρ (x, t) − w n (x, t)) − ρzn (x, t). From (3.2) and (3.3) we obtain rρ t (x, t) − Arρ (x, t) − cn (x, t)rρ (x, t) = ρ [cn (x, t) − fy (x, w n (x, t), un (x))]zn (x, t), rρ (x, t)Γ = 0, rρ (x, t)(x, 0) = 0, 250 HANG GAO AND MICHAEL Y. LI where cn (x, t) = Z 1 0 fy (x, w n (x, t) + τ (wnρ (x, t) − wn (x, t)), uρ (x))dτ. Similar to the proof of Lemma 3.1, we can show krρ k2,Q + k∇rρ k2,Q ≤ C ρ k[cn − fy ( · , w n , un )]zn ]k2,Q . From Lemma 3.1 we know that lim [cn (x, t) − fy (x, w n (x, t), un (x))] = 0. ρ→0 Thus (3.5) holds and Lemma 3.2 is proved. Theorem 3.1. Suppose (un , v n ) is the optimal control for (2.1)–(2.2) and w n (x, t) is the corresponding optimal state. Then, there exists a function ψn (x, t) that satisfies ψn (x, t) − Aψn (x, t) − fy (x, w n (x, t), un (x))ψn (x, t) t (3.6) = −2[wn (x, t) − B(x)], ψn |Γ = 0, ψn (x, 1) = ψ(x), and a function ξn (x), kξn k2,Ω ≤ 2C0 , such that, for any (u, v) ∈ U × V, the following inequality holds (3.7) Z Q ψ(x, t)[f (x, w n (x, t), un (x)) − f (x, w n (x, t), u(x))] dx dt + Z Ω ξn (x)[un (x) − u(x)] dx 1 + n Z Q ψ(x, t)[v n (x, t) − v(x, t)] dx dt ≥ 0. Here ψ(x) is given by (1.7) and C0 , as in (2.2), is determined in Theorem 2.2. Proof. From Theorem 2.2 we know that un is an interior point of Un . For any u ∈ U and v ∈ V, there exists ρ0 > 0, such that for ρ ∈ (0, ρ0 ) and uρ = un + ρ(u − un ), vρ = v n + ρ(v − v n ), NECESSARY CONDITIONS FOR OPTIMAL CONTROL 251 we have (uρ , vρ ) ∈ Un × V. Therefore 0 ≤ Fn (uρ , vρ ) − Fn (un , v n ). (3.8) Direct calculation leads to (3.9) Fn (uρ , vρ ) − Fn (un , v n ) Z = {[wnρ (x, t) − B(x)]2 − [wn (x, t) − B(x)]2 } dx dt Q − Z Ω ψ(x)[wnρ (x, t) − w n (x, t)] dx √ + C0 n Z Ω [(uρ (x) − u(x))2 − (un (x) − u(x))2 ] dx = I1 − I2 + I3 , where (3.10) I1 = Z Q [wnρ (x, t) − w n (x, t)] × [wnρ (x, t) + wn (x, t) − 2B(x)] dx dt = Z Q [wnρ (x, t) − w n (x, t)][2(w n (x, t) − B(x)) + ρzn (x, t) + rρ (x, t)] dx dt = Z Q =− [wnρ (x, t) − w n (x, t)][2(w n (x, t) − B(x))] dx dt + o(ρ) Z Q [wnρ (x, t) − wn (x, t)] [ψn t (x, t) − Aψn (x, t) − fy (x, w n (x, t), un (x))ψn (x, t)] dx dt + o(ρ) = Z Ω ψ(x)[wnρ (x, t)(x, 1) − wn (x, t)(x, 1)] dx −ρ Z Q ψn (x, t)[zn t (x, t) − Azn (x, t) − fy (x, w n (x, t), un (x))zn (x, t)] dx dt + o(ρ) 252 HANG GAO AND MICHAEL Y. LI Z = I2 + ρ ψn (x, t)[f (x, w n (x, t), un (x)) Q − f (x, wn (x, t), u(x)) 1 (v n (x, t) − v(x, t))] dx dt + o(ρ), n + and (3.11) √ I3 = C 0 n Z √ Ω = ρ2C0 n =ρ Z where Ω [uρ (x) − un (x)][uρ (x) + un (x) − 2u(x)]dx Z Ω [u(x) − un (x)] [un (x) − u(x)]dx + o(ρ) ξn (x)[un (x) − u(x)]dx + o(ρ), √ ξn (x) = 2C0 n (u(x) − un (x)), and thus √ kξn k2,Ω = 2C0 n ku − un k2,Ω ≤ 2C0 . From (3.7)–(3.11) we obtain (3.12) 0≤ρ Z ψn (x, t)[f (x, w n (x, t), un (x)) Q − f (x, wn (x, t), u(x))] dx dt Z +ρ ξn (x)[un (x) − u(x)] dx Ω 1 + ρ n Z Q ψ(x, t)[v n (x, t) − v(x, t)] dx dt + o(ρ). Dividing inequality (3.12) by ρ and then taking a limit as ρ → 0, we obtain inequality (3.7). 4 Proofs of the main results proof of Theorems 1.1 and 1.2. In this section, we complete the Proof of Theorem 1.1. Let ψn (x) and ψ(x) be the solutions of the problems (3.6) and (1.7), respectively. We prove that ψn → ψ as n → ∞. NECESSARY CONDITIONS FOR OPTIMAL CONTROL 253 From equations (3.6) and (1.7), we can obtain − (ψn (x, t) − ψ(x))t − A(ψn (x, t) − ψ(x)) = fy (x, w n (x, t), un (x))ψn (x, t) (4.1) − fy (x, y¯(x), u(x))ψ(x) − 2[w n (x, t) − B(x)], (ψn − ψ)|Γ = 0, (ψn − ψ)(x, 1) = 0. Let hn (x, t) = e−Lt [ψn (x, t) − ψ(x)]. From (4.1), we have − hn t − Ahn + (L − fy (x, w n , un ))hn = e−Lt [fy (x, w n (x, t), un (x)) (4.2) − fy (x, y¯(x), u(x))]ψ(x) − 2e−Lt[w n (x, t) − B(x)], hn |Γ = 0, hn (x, 1) = 0. According to the assumption (P2), we can choose L such that L − fy (x, w n (x, t), un (x)) ≥ 2 for all n. Multiplying (4.2) by hn (x, t), and by integration by parts over Q, we have (4.3) 1 2 Z Ω h2n (x, 1) dx + λ 1 ≤ 2 Z Ω Z Q |∇hn (x, t)|2 dx dt + 2 Z Q Q h2n (x, t) dx dt {e−Lt [fy (x, w n (x, t), un (x)) − fy (x, y¯(x), u(x))]ψ(x)}2 dx + + Z 1 2 Z Q h2n (x, t) dx dt [e−Lt (w n (x, t) − y¯n (x))]2 dx dt + 1 2 Z Q h2n (x, t) dx dt. √ This √ relation and the fact that kun − uk2,Ω ≤ 1/ n and kwn − y¯k2,Q ≤ C/ n lead to lim {khn k2,Q + k∇hn k2,Q } = 0. n→∞ This implies limn→∞ ψn (x, t) = ψ(x). 254 HANG GAO AND MICHAEL Y. LI Since kξn k2,Ω ≤ C, there exists a subsequence {ξni } and a ξ ∈ L2 (Ω), such that kξk2,Ω ≤ C, and ξni → ξ weakly in L2 (Ω). Setting n = ni and letting ni → ∞ in (3.7), we obtain (4.4) Z Ω [H(x, u(x)) − H(x, u(x))] dx ≥ 0, where H(x, u) = ψ(x)f (x, y¯(x), u) + ξ(x)u. For any Lebesgue point x0 of the function H(x, u(x)) − H(x, u), we choose the function u, x ∈ Bρ (x0 ) ⊂ Ω, u(x) = . u(x), x ∈ Ω \ B . ρ Here, Bρ (x0 ) = {x : kx − x0 k < ρ}. From (4.4) we have (4.5) Z Bρ [H(x, u(x)) − H(x, u)] dx ≥ 0. Dividing (4.5) by ρ and then letting ρ → 0, we obtain H(x, u(x0 )) − H(x, u) ≥ 0. Since the set of Lebesgue point of the function H(x, u) is dense in Ω, we have H(x, u(x0 )) = max H(x, u) a.e. x ∈ Ω, u∈U completing the proof of Theorem 1.1. Now, we turn to the proof of Theorem 1.2. Proof of Theorem 1.2. For any (y, u) ∈ P, we have J(y, u) ≥ 0. There exists a minimizing sequence {(yk , uk )} ⊂ P. From [10], we know that, kyk k2,p,Ω ≤ C for all k. There thus exist y¯ ∈ W02,p (Ω) and a subsequence {yki } of {yk }, such that yki → y¯ weakly in W 2,p (Ω) and yki → y¯ in Lp (Q). Let hki (x) = f (x, yki (x), uki (x)). NECESSARY CONDITIONS FOR OPTIMAL CONTROL 255 It is clear that khki kp,Ω ≤ C. Thus, there exist a subsequence of {hki }, denoted by itself, and a function h ∈ W 2,p (Ω), such that hki → h weakly in Lp (Ω). Then, for any ϕ ∈ Lq (Ω), where 1/p + 1/q = 1, we have Z Z ϕ(x)A¯ y (x) dx = − lim ϕ(x)Ayki (x) dx − (4.6) ki →∞ Ω = lim ki →∞ = Z Z Ω ϕ(x)hki (x) dx Ω ϕ(x)h(x) dx. Ω Let Ek (x) = {f (x, yk (x), u) : u ∈ U } and E(x) = {f (x, y¯(x), u) : u ∈ U }. Then, hki (x) ∈ Eki (x), a.e. x ∈ Ω and as ki → ∞, we have Eki (x) → E(x), a.e. x ∈ Ω. Next, we prove that h(x) ∈ E(x), a.e. x ∈ Ω. In fact, since f (x, y, ·) is continuous on U and U is a closed set, E(x) is a closed set. Suppose it is not ture that h(x) ∈ E(x), a.e. x ∈ Ω, then, there exists a function ϕ(x), such that Z (4.7) [h(x) − f (x, y¯(x), u(x))]ϕ(x)dx ≥ C > 0, ∀u ∈ U. Ω Since hki → h weakly in Lp (Ω), and from (4.7), we have Z (4.8) [hki (x) − f (x, y¯(x), u(x))]ϕ(x)dx ≥ C/2 > 0, Ω ∀u ∈ U, for sufficiently large ki . On the other hand, taking u(x) = uki (x) in (4.8), we have Z [hki (x) − f (x, y¯(x), uki (x))]ϕ(x) dx Ω Z ≤ |fy (x, y¯(x) + τ (yki (x) − y¯(x)), uki (x))| Ω × |yki (x) − y¯(x)| |ϕ(x)| dx ≤ Ckyki − y¯kp,Ω kϕkq,Ω . 256 HANG GAO AND MICHAEL Y. LI Since yki → y¯ in Lp (Ω), we have Z [hki (x) − f (x, y¯(x), uki (x))]ϕ(x)dx ≤ C 2 Ω for sufficiently large ki . This contradicts (4.8). Therefore h(x) ∈ E(x) a.e. x ∈ Ω From the Selection Theorem [6, 11], we know that there exists a control function u ∈ U, such that h(x) = f (x, y¯(x), u(x)). It follows from (4.6) that (¯ y (x), u(x)) satisfies the equation ( −A¯ y(x) = f (x, y¯(x), u(x)), y¯|∂Ω = 0 and J(¯ y , u) = Z Ω [¯ y (x) − B(x)]2 dx = lim ki →∞ Z Ω [yki (x) − B(x)]2 dx ≤ J(y, u), ∀(y, u) ∈ P, namely, J(¯ y , u) = inf{J(y, u) : (y, u) ∈ P}, completing the proof of Theorem 1.2. Acknowledgments The research of HG is partially supported by the National Natural Sciences Foundation of China. 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Li Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 E-mail address: [email protected]
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