The existence of solution and optimal control problems for the klein - Gordon hemivariational inequality with strongly elliptic operator

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2008, Vol. 53, N ◦ . 5, pp. 20-30 THE EXISTENCE OF SOLUTION AND OPTIMAL CONTROL PROBLEMS FOR THE KLEIN-GORDON HEMIVARIATIONAL INEQUALITY WITH STRONGLY ELLIPTIC OPERATOR Pham Trieu Duong and Nguyen Thanh Nam Hanoi National University of Education Abstract. In this paper, we study the optimal control problems of systems governed by Klein-Gordon hemivariational inequalities. We establish the existence of solutions and the optimal control to these problems. Keywords and phrases: Optimal control problem, hemivariational inequality, monotone operator, hyperbolic equation. 1. Introduction The background of variational problems is in physics, especially in solid me- chanics, where non-monotone and multi-valued constitutive laws lead to hemivari- ational inequalities. In works [4,5] authors studied some applications of hemivaria- tional inequalities, but there is not much literature dealing with the optimal control problems for hemivariational inequalities. The optimal control problems for operator ∆ have been established in [7] where J.Y Park and J.U Jeong received the results on existence of solutions and optimal control. We study the optimal control problem for general strongly elliptic operator. Let Ω be a bounded domain in Rn (n ≥ 2) with the boundary ∂Ω. We first introduce the following abbreviations Q = Ω × (0, T ), Σ = ∂Ω × (0, T ), ‖ · ‖k,p = ‖ · ‖W k,p(Ω), ‖ · ‖p = ‖ · ‖Lp(Ω). For simplicity, we denote ‖ · ‖ = ‖ · ‖L2(Ω). We use the following symbol: (·, ·) is scalar product in L2(Ω) and will also be used for the notation of duality pairing between dual spaces. For each multi-index p = (p1, · · · , pn) ∈ Nn, |p| = p1 + . . . + pn and Dpu = ∂|p|u ∂p1x1 · · ·∂ pn xn = uxp11 ...x pn n is the generalized derivative up to order p with respect to x = (x1, ..., xn). 20 The existence of solution and optimal control problems... Let L(x, t,D) be the following differential operator: L(x, t,D) ≡ m∑ |p|,|q|=1 Dp(apqD q) (m ≥ 1) where apq ≡ apq(x, t), |p|, |q| = 1, . . . , m are continuous, real-valued functions on Q, apq = (−1) |p|+|q|aqp and ∣∣∂apq ∂t ∣∣ < a positive constant for all (x, t) ∈ Q; |p|, |q| = 1, . . . , m. Suppose that m∑ |p|,|q|=1 apq(x, t)ξ pξq ≥ γ0|ξ| 2m for all ξ ∈ Rn and (x, t) ∈ Q where ξ = (ξ1, . . . , ξn), ξp = ξ p1 1 ξ p2 2 . . . ξ pn n and γ0 = constant > 0. We denote a(t, φ(t), ψ(t)) = ∫ Ω m∑ |p|,|q|=1 (−1)m+|p|apq(x, t)D qφ(t)Dpψ(t)dx It is easy to show that a is a bi-linear form onWm,20 (Ω)×W m,2 0 (Ω) and a(t, φ(t), ψ(t)) = (L(t)φ, ψ). For all φ(t) ∈Wm,20 (Ω) we have the Garding inequality a(t, φ(t), φ(t)) + λ2‖φ(t)‖ 2 ≥ λ1‖φ(t)‖ 2 m,2 where λ1, λ2 are constants, not independent of φ, x, t. We consider in Q the problem: Minimize J(y, u) with all pair (y, u) satisfy Klein-Gordon hemivariational inequality (denoted by (∗)) y′′ + αy′ + (−1)mβL(x, t,D)y + δ|y|γy + Ξ = f +Bu, (1.1) ∂jy ∂νj = 0, j = 0, . . . , m− 1, (1.2) y(0, x) = y0(x); y ′(0, x) = y1(x) ∀x ∈ Ω, (1.3) Ξ(x, t) ∈ φ(y(x, t)) for a.e (x, t) ∈ Q (1.4) where ν is outer unit normal to S; α, β, δ are positive constants (independent of t); δ ≥ 1. Function φ is discontinuous and nonlinear multi-valued mapping by filling in jumps of locally bounded b. u denotes the control variable in the real space Hilbert 21 Pham Trieu Duong and Nguyen Thanh Nam U , B is a bounded linear operator and f is a given function. The cost function J(y, u) is given by J(y, u) = ∫ T 0 [g(y(t)) + h(u(t))] dt where g, h are convex functions. In this paper we consider the existence of solution and optimal control of problem (1.1)− (1.4). 2. Main results Definition 2.1. Function y is said to be a weak solution of (∗) if y ∈ L∞(0, T ;Wm,20 (Ω)∩ Lγ+2(Ω)), y′ ∈ L∞(0, T ;L2(Ω)) there exists Ξ ∈ L∞(0, T ;L2(Ω)) and the following relations hold:∫ T 0 (y′′(t), η(t)w) dt+α ∫ T 0 (y′(t), η(t)w) dt+β ∫ T 0 a(t, y(t), η(t)w) dt+ ∫ T 0 (Ξ, η(t)w) dt + δ ∫ T 0 (|y(t)|γy(t), η(t)w) dt = ∫ T 0 (f(t), η(t)w) dt+ ∫ T 0 (Bu, η(t)w) dt for all w ∈Wm,20 (Ω) ∩ L γ+2(Ω) and η ∈ D(0, T ) y(0, x) = y0; y ′(0, x) = y1, on Ω Ξ(x, t) ∈ φ(y(x, t)) for a.e (x, t) ∈ Q. We will use the evolutional triple Wm,20 (Ω) ∩ L γ+2(Ω) ⊂ L2(Ω) ⊂ W−m,20 (Ω) + L (γ+2)′(Ω) which means that embeddingWm,20 (Ω)∩L γ+2(Ω) ↪→ L2(Ω) and L2(Ω) ↪→ W−m,20 (Ω)+ L(γ+2) ′ (Ω) are continuous, dense and compact. (Wm,20 (Ω) ∩ L γ+2(Ω) is defined with norm ‖ · ‖m,2 + ‖ · ‖γ+2 ). Scalar function k, given by k(s) = |s|γs. It is easy to show that |k′(s)| ≤ (γ + 1)|s|γ. We recall the Sobolev embedding: ifm < N 2 we have continuous embeddingWm,20 (Ω) ↪→ L 2N N − 2m (Ω). We suppose that N 4 ≤ m < N 2 and 1 ≤ γ ≤ N 2(N − 2m) (or m = N 2 and γ ≥ 1) then the nonlinear operator k :Wm,20 (Ω)→ L 2(Ω), y 7−→ k ◦ y 22 The existence of solution and optimal control problems... is well defined, especially it is locally Lipschitz. We have the following well-known lemma which is proved in [5]: Lemma 2.1. The operator k : Wm,20 (Ω) → L 2(Ω); y 7−→ k ◦ y is locally Lipschitz. That is, there exists a constant l > 0 such that: ‖k(ψ)− k(ρ)‖ ≤ l(‖ψ‖m,2 + ‖ρ‖m,2) γ‖ψ − ρ‖m,2. We assume that the following conditions hold (Hb) b : Q×R is a locally bounded function satisfying the following conditions: (i) (x, t) −→ b(x, t, ξ) is continuous on Q for all ξ ∈ R, (ii) (x, t, ξ) −→ b(x, t, ξ) is measurable in Q× R, (iii) |b(x, t, ξ)| ≤ µ(1 + |ξ|) ∀(x, t, ξ) ∈ Q× R (µ = constant > 0), (iv) b(x, t, ξ)ξ ≥ θξ2 ∀ξ ∈ R (θ = constant > 0). (HB) B : L 2(0, T ;U) −→ L2(0, T ;L2(Ω)) is bounded linear operator, where U is a real Hilbert space. (HU) Uad is a closed convex, bounded subset of L 2(0, T ;U). (Hg) g : L 2(Ω) −→ (−∞,+∞] is convex and lower semi-continuous. Moreover there exists r1 > 0 and r2 ∈ R such that g(y) ≥ r1‖y‖2 + r2 ∀y ∈ L2(Ω). (Hh) h : U −→ (−∞,+∞] is convex lower semi-continuous. Moreover, there exists r3 > 0 and r4 ∈ R such that h(u) ≥ r3‖u‖U + r4, ∀u ∈ U . The multi-valued function φ : Q × R −→ 2R is obtained by filling in jumps of a function b(x, t, ·) : R −→ R by means of the functions b, b, b, b : R −→ R as follows: b(x, t, ξ) = ess inf |s−ξ|≤ b(x, t, s); b(x, t, ξ) = ess sup |s−ξ|≤ b(x, t, s), b(x, t, ξ) = lim →0+ b(x, t, ξ); b(x, t, ξ) = lim →0+ b(x, t, ξ), φ(x, t, ξ) = [b(x, t, ξ), b(x, t, ξ)]. We shall need a regularization of b defined by: bn(x, t, s) = n ∫ ∞ −∞ b(x, t, s− τ)ρ(nτ)dτ, 23 Pham Trieu Duong and Nguyen Thanh Nam where ρ ∈ C∞0 ((−1, 1)), ρ ≥ 0 and 1∫ −1 ρ(τ)dτ = 1. It is easy to show that |bn(x, t, s)| ≤ µ(2 + |s|)n ∫ 1 n − 1 n ρ(nτ)dτ = µ(2 + |s|). Theorem 2.1. Under hypotheses (Hb), (HB), (HL), and (y0, y1, f) ∈ (W m,2 0 (Ω) ∩ Lγ+2(Ω)) × L2(Ω) × L2(0, T ;L2(Ω)). Then (∗) has at least one weak solution for every u ∈ L2(0, T ;U). Theorem 2.2. Assume that the conditions of Theorem 2.1; (HU), (Hg), (Hh) hold. Then the optimal control problem (P ) has at least one solution. 3. Existence of solution for the Klein - Gordon hemivari- ational inequality Proof of Theorem 2.1. We represent by {wj}j≥0 a basis in W m,2 0 (Ω) ∩ L γ+2(Ω) which is orthogonal in L2(Ω). Let Vn = Span{w1, . . . , wn}. We denote yn(t) = n∑ j=1 gjn(t)wj (3.1) to be solution to the approximate equation: (y′′n(t), wj) + α(y ′ n(t), wj) + βa(t, yn(t), wj) + δ(|yn(t)| γyn(t), wj)+ + (bn(yn(t)), wj) = (f(t), wj) + (Bu,wj) ∀j = 1, . . . , n (3.2) yn(0) = y0n → y0 in W m,2 0 (Ω) ∩ L γ+2(Ω) y′n(t) = y1n → y1 in L 2(Ω) as n→∞. Because the function k(yn) = |yn| γyn is locally Lipschitz, by standard methods of ODE we can prove the existence of a solution to the above equation on some interval [0, tn). Step 1: We will prove ‖yn(t)‖m,2 + ‖yn(t)‖γ+2 ≤ constant (independent of n, t). From that, we get tn = T . Indeed: By (3.1), in equation (3.2) we can replace wj by y ′ n(t). Putting En(t) = 1 2 ‖y′n(t)‖ 2+ δ γ + 2 ‖yn(t)‖ γ+2 γ+2 we obtain d dt En(t) + α‖y ′ n(t)‖ 2 + βa(t, yn(t), y ′ n(t)) = (f(t), y′n(t)) + (Bu, y ′ n(t))− (b n(yn(t)), y ′ n(t)). (3.3) 24 The existence of solution and optimal control problems... By apq = (−1) |p|+|q|aqp, we have 2 m∑ |p|,|q|=1 (−1)m+|p|apqD qyn(t)D py′n(t) = d dt [ m∑ |p|,|q|=1 (−1)m+|p|apqD qyn(t)D pyn(t)] − m∑ |p|,|q|=1 (−1)m+|p| ∂apq ∂t Dqyn(t)D pyn(t). Integrating (3.3) over (0, t) with t ≤ tn, then adding side by side with βλ2‖yn(t)‖ 2 = β(2λ2 ∫ t 0 (yn(s), y ′ n(s))ds+ λ2‖y0n‖ 2), (where λ2 is coefficient in the Garding inequality), we come to 2En(t)+2α ∫ t 0 ‖y′n(s)‖ 2 ds+ β[a(t, yn(t), yn(t)) + λ2‖yn(t)‖ 2] = 2 ∫ t 0 (f(s), y′n(s)) ds+ 2 ∫ t 0 (Bu, y′n(s)) ds− 2 ∫ t 0 (bn(yn(s)), y ′ n(s)) ds + 2En(0) + βa(0, y0n, y0n) + βλ2‖y0n‖ 2 + 2βλ2 ∫ t 0 (yn(s), y ′ n(s)) ds + ∫ t 0 m∑ |p|,|q|=1 (−1)m+|p| ∂apq ∂s Dqyn(s)D pyn(s) dxds. (3.4) We will provide some estimates of terms of (3.4) by using the Gronwall inequality. It is easy to show that 2En(0) + βa(0, y0n, y0n) ≤ C (3.5) where C is denoted a general constant which is positive and independent of n, t. By continuous embedding Wm,20 (Ω) ↪→ L 2(Ω) we have∫ t 0 ‖bn(yn(s))‖ 2 ds ≤ ∫ t 0 ∫ Ω 2µ2(4 + |yn(x, s)| 2) dxds ≤ C + C ∫ t 0 ‖yn(s)‖ 2 m,2 ds. (3.6) Combining Schwartz inequality and Cauchy inequality we have: | ∫ t 0 (bn(yn(s)), y ′ n(s)) ds| ≤ 1 2 (∫ t 0 ‖bn(yn(s))‖ 2 ds+ ∫ t 0 ‖y′n(s)‖ 2 ds ) ≤ C + C ∫ t 0 ‖yn(s)‖ 2 m,2ds+ C ∫ t 0 ‖y′n(s)‖ 2ds. (3.7) 25 Pham Trieu Duong and Nguyen Thanh Nam By a similar way we have similar inequalities for ∫ t 0 (f(s), y′n(s)) ds and∫ t 0 (Bu(s), y′n(s)) ds. Using inequality m∑ i,j=1 aibj ≤ m 2 m∑ i,j=1 (a2i + b 2 j ), and hypotheses ∣∣∂apq ∂s ∣∣ ≤ C for all (x, s) ∈ Q we get∣∣ ∫ t 0 m∑ |p|,|q|=1 (−1)m+|p| ∂apq ∂s Dqyn(s)D pyn(s) ds ∣∣ ≤ C ∫ t 0 ‖yn(s)‖ 2 m,2 ds. (3.8) Moreover we have |(yn(s), y ′ n(s))| ≤ ‖yn(s)‖.‖y ′ n(s)‖ ≤ ≤ 1 2 (‖yn(s)‖ 2 + ‖y′n(s)‖ 2) ≤ C‖y′n(s)‖ 2 + C‖yn(s)‖ 2 m,2, thus | ∫ t 0 (yn(s), y ′ n(s)) ds| ≤ C ∫ t 0 ‖y′n(s)‖ 2 ds+ C ∫ t 0 ‖yn(s)‖ 2 m,2 ds. (3.9) Using the Garding inequality we have a(t, yn(t), yn(t)) + λ2‖yn(t)‖ 2 ≥ λ1‖yn(t)‖ 2 m,2. (3.10) By estimates (3.4), (3.7)− (3.10) we get ‖yn(t)‖ 2 m,2 + ‖y ′ n(t)‖ 2 ≤ C + C ∫ t 0 ‖y′n(t)‖ 2ds+ C ∫ t 0 ‖yn(s)‖ 2 m,2ds. From that, using the Gronwall inequality we deduce ‖y′n(t)‖+ ‖yn(t)‖m,2 + ‖yn(t)‖γ+2 ≤ constant (independent of n, t) (3.11) (the inequality holds for almost every t ∈ (0, tn]), hence tn = T . Combining (3.6) and (3.11) we get ‖bn(yn(t))‖ ≤ constant (independent of n, t). (3.12) Step 2: Passage to the limit Combining (3.6) and (3.12) we have that relabelling if necessary yn ∗ ⇀ y in L∞(0, T ;Wm,20 (Ω) ∩ L γ+2(Ω)), y′n ∗ ⇀ y′ in L∞(0, T ;L2(Ω)), bn(yn) ∗ ⇀ Ξ in L∞(0, T ;L2(Ω)), Dpyn ∗ ⇀ Dpy in L∞(0, T ;L2(Ω)). (3.13) 26 The existence of solution and optimal control problems... From (3.13), using the Aubin-Lion compactness theorem we get yn → y in L 2(0, T ;L2(Ω)). Hence, there exists a subsequence of yn (which is denoted again by yn) such that yn a.e −→ y in Q. Moreover from ‖|yn(t)| γyn(t)‖(γ+2)′ = ‖yn(t)‖γ+2 we deduce that |yn| γyn ∗ ⇀ |y|γy in L∞(0, T ;L(γ+2) ′ (Ω)). (3.14) Using the estimates (3.13) and (3.14), in (3.2) letting n→∞ we obtain (y′′(t), w) + α(y′(t), w) + βa(t, y(t), w) + δ(|y(t)|γy(t), w)+ + (Ξ, w) = (f(t), w) + (Bu,w) in D′(0, T ) for all w ∈Wm,20 (Ω) ∩ L γ+2(Ω). (3.15) Step 3: We will prove that (y,Ξ) is a weak solution of (∗). By y(t) ∈ Wm,20 (Ω) the boundary conditions are satisfied. By a similar way in Section 1.4 of Chapter 1 in [3] we have y′(0) = y1 and y(0) = y0, hence the initial conditions are also satisfied. Now we will check the conditions of Ξ. Because yn a.e −→ y in Q then according to Lusin theorem, for each η > 0 we can choose W ⊂ Q such that mease(W ) < η and yn → y uniformly in Q\W . For each  > 0 ∃n0 > 2  such that |yn(x, t)− y(x, t)| <  2 ∀(x, t) ∈ Q\W,n > n0. Then, if |yn(x, t)− s| < 1 n we have |y(x, t)− s| n0 and (x, t) ∈ Q\W It follows that b(y(x, t)) ≤ b n(yn(x, t)) ≤ b(y(x, t)). For φ ∈ L 2(Q) and φ ≥ 0 then∫ Q\W b(y(x, t))φ(x, t) dxdt ≤ ∫ Q\W bn(yn(x, t))φ(x, t) dxdt ≤ ∫ Q\W b(y(x, t))φ(x, t) dxdt. Letting n → ∞, then letting  → 0 by using Lebesgue monotonous convergence theorem we obtain∫ Q\W b(y(x, t))φ(x, t) dxdt ≤ ∫ Q\W Ξ(x, t)φ(x, t) dxdt ≤ ∫ Q\W b(y(x, t))φ(x, t) dxdt. 27 Pham Trieu Duong and Nguyen Thanh Nam The above inequalities hold for all φ ≥ 0, φ ∈ L2(Q), it means Ξ(x, t) ∈ φ(y(x, t)) for a.e (x, t) ∈ Q\W . Let η → 0 then we get Ξ(x, t) ∈ φ(y(x, t)) for a.e (x, t) ∈ Q We completed the proof of the Theorem 2.1. 4. Existence of the solutions of the optimal control problem We denote by S(u) the set of all weak solutions of the problem (∗) for a given u ∈ Uad. Theorem 2.1 implies that S(u) 6= φ for all u ∈ Uad. Let us consider the following optimal control problem (P) Minimize{J(y, u) : u ∈ Uad, y ∈ S(u)}. We have the following lemma which is proved in the similar way as in [7]. Lemma 4.1. For a given u ∈ Uad, the following estimate holds sup y∈S(u) {‖y‖L∞(0,T ;Wm,20 (Ω)∩Lγ+2(Ω)) + ‖y ′ ‖L∞(0,T ;L2(Ω))} ≤ C (independent of u). Proof of Theorem 2.2. Let d = inf{J(y, u)|y ∈ S(u), u ∈ S(u), u ∈ Uad}. By hypotheses of Hg, Hh, it is clear that J(y, u) ≥ T∫ 0 [r2 + r4]dt, hence d > −∞. Because of definition of the maximal lower bound, there exists (yn, un) ∈ S(un)× Uad such that lim n→∞ J(yn, un) = d, where (yn, un) satisfies: (y′′n(t), w) + α(y ′ n(t), w)+βa(t, yn(t), w) + δ(|yn(t)| γyn(t), w) + (Ξn(t), w) = (f(t), w) + (Bun(t), w) in D ′(0, T ) ∀w ∈Wm,20 (Ω) ∩ L γ+2(Ω), ∂jyn ∂νj = 0, j = 0, . . . , m− 1, ν is outer unit normal to S, yn(0, x) = y0(x); y ′ n(0, x) = y1(x) on Ω, Ξn(x, t) ∈ φ(yn(x, t)) a.e (x, t) ∈ Q = Ω× (0, T ), d ≤ J(yn, un) ≤ d+ 1 n , n = 1, 2, . . . By hypothesis (Hh) : (un) is a bounded subset of L 2(0, T ;U). Accordingly, there exists a subsequence (denoted again by un) such that as n→∞, we have un ⇀ u ∗ in L2(0, T ;U). 28 The existence of solution and optimal control problems... According to the result of Lemma 4.1 and by a similar way in the proof of Theorem 2.1, we imply that there exists a subsequence of yn (denoted again by yn) such that yn ∗ ⇀ y∗ in L∞(0, T ;Wm,20 (Ω) ∩ L γ+2(Ω)), y′n ∗ ⇀ y∗ ′ in L∞(0, T ;L2(Ω)), Ξn ∗ ⇀ Ξ∗ in L∞(0, T ;L2(Ω)), Dpyn ∗ ⇀ Dpy in L∞(0, T ;L2(Ω)), |yn| γyn ∗ ⇀ |y∗|γy∗ in L∞(0, T ;L(γ+2) ′ (Ω)). It follows that (y∗ ′′ (t), w) + α(y∗ ′ (t), w) + βa(t, y∗(t), w) + δ(|y∗(t)|γy∗(t), w)+ + (Ξ∗n(t), w) = (f(t), w) + (Bu ∗(t), w) in D′(0, T ), ∀w ∈Wm,20 (Ω) ∩ L γ+2(Ω). Checking the boundary and initial conditions is similar to the proof of Theorem 2.1. We will verify the condition: Ξ∗(x, t) ∈ φ(x, t, y∗(x, t)) for a.e (x, t) ∈ Q. By a similar way of the proof of Theorem 2.1, we get yn → y in L 2(0, T ;L2(Ω)). Accordingly, there exists a subsequence of yn (denoted again by yn) such that yn(x, t) a.e −−→ y∗(x, t) in Q. Because of Lusin theorem, we can choose W ⊂ Q such that mease(W ) 0) and yn → y ∗ uniformly on Q\W . Thus, for each  > 0 there exists n0 such that |yn(x, t)− y ∗(x, t)| <  2 ∀ (x, t) ∈ Q\W,n > n0. It follows that∫ Q\W b  2 (yn(x, t))φ(x, t) dxdt ≤ ∫ Q\W Ξn(x, t)φ(x, t) dxdt ≤ ∫ Q\W b  2 (yn(x, t))φ(x, t) dxdt. On the other hand, for all φ ∈ L2(Q) satisfy φ ≥ 0 and n > n0 we have b  2 (yn(x, t)) = ess inf |s−yn|≤  2 b(x, t, s) ≥ ess inf |s−y∗|≤ b(x, t, s) = b(y ∗(x, t)) and b  2 (yn(x, t)) = ess sup |s−yn|≤  2 b(x, t, s) ≤ ess sup |s−y∗|≤ b(x, t, s) = b(y ∗(x, t)). 29 Pham Trieu Duong and Nguyen Thanh Nam Hence∫ Q\W b(y ∗(x, t))φ(x, t) dxdt ≤ ∫ Q\W Ξn(x, t)φ(x, t) dxdt ≤ ∫ Q\W b(y ∗(x, t))φ(x, t) dxdt By the similar way of the proof of Theorem 2.1, letting n→∞, → 0, η → 0 we obtain Ξ∗(x, t) ∈ φ(x, t, y∗(x, t)) for almost every (x, t) ∈ Q. 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