Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

and Applied Analysis 3 We are interested in the following optimal control problem: min u∈Uad⊂X J ( u, y u ) 1 2 {∫T 0 ∥ ∥y − zd ∥ ∥2 0,Ωdt ∫T 0 ‖u‖0,ΩUdt } , 2.1 subject to yt − div ( A∇yt D∇y ∫ t 0 C t, τ ∇y x, τ dτ ) f Bu, in Ω × 0, T , y 0, on ∂Ω × 0, T , y|t 0 y0, in Ω, 2.2 where u is control, y is state, zd is the observation, Uad is a closed convex subset, f x, t ∈ L2 0, T ;L2 Ω , and zd and y0 ∈ H1 Ω are some suitable functions to be specified later. B is a linear bounded operator from L2 ΩU to L2 Ω independent of t. And A A x ( ai,j · ) n×n ∈ ( C∞ ( Ω ))n×n , D D x ( di,j · ) n×n ∈ ( C∞ ( Ω ))n×n , 2.3 such that there is a constant c > 0 satisfying that for any vector X ∈ R as follows: XAX ≥ c‖X‖Rn , XDX ≥ c‖X‖Rn , 2.4 C C x, t, τ ci,j x, t, τ n×n ∈ C∞ 0, T ;L2 Ω n×n . Let ( f1, f2 ) ∫ Ω f1f2, ∀ ( f1, f2 ) ∈ H ×H, u, v U ∫ ΩU uv, ∀ u, v ∈ U ×U, a z,ω A∇z,∇ω , d z,ω D∇z,∇ω , c t, τ ; z,ω C t, τ ∇z,∇ω , ∀z,w ∈ V × V. 2.5 In the case that f1 ∈ V and f2 ∈ V ∗, the dual pair f1, f2 is understood as 〈f1, f2〉V×V ∗ . Assume that there are constants c and C, such that for all t and τ in 0, T as follows: a a z, z c‖z‖1,Ω, b |a z,w | C‖z‖1,Ω‖w‖1,Ω, |d z,w | C‖z‖1,Ω‖w‖1,Ω, c |c t, τ ; z,w | C‖z‖1,Ω‖w‖1,Ω. 2.6 for any z and w in V . 4 Abstract and Applied Analysis Then the weak form of the state equation reads as ( yt,w ) a ( yt,w ) d ( y,w ) ∫ t 0 c ( t, τ ;y τ , w ) dτ ( f Bu,w ) ∀w ∈ V, t ∈ 0, T , y |t 0 y0. 2.7 It is well known see, e.g., 1 that the above weak formulation has at least one solution in y ∈ W 0, T {w ∈ L∞ 0, T ;H1 Ω , w′ t ∈ L2 0, T ;H1 Ω }. Therefore, the weak form of the control problem 2.1 and 2.2 reads as OCP min u∈Uad J ( u, y u ) , ( yt,w ) a ( yt,w ) d ( y,w ) ∫ t 0 c ( t, τ ;y τ , w ) dτ ( f Bu,w ) ∀w ∈ V, t ∈ 0, T , y |t 0 y0. 2.8 In the following, we first give the existence and uniqueness of the solution of the system 2.8 . Theorem 2.1. Assume that the condition 2.6 (a)–(c) holds. There exists the unique solution u, y for the minimization problem 2.8 such that u ∈ L2 0, T ;L2 ΩU , y ∈ L∞ 0, T ;H1 Ω , and yt ∈ L2 0, T ;H1 Ω . Proof. Let { u, y }n 1 be a minimization sequence for the system 2.8 , then the sequence {un}n 1 is bounded in L2 0, T ;L2 ΩU . Thus there is a subsequence of {un}n 1 still denote by {un}n 1 such that u converges to u∗ weakly in L2 0, T ;L2 ΩU . For the subsequence {un}n 1, we have ( y t ,w ) a ( y t ,w ) d ( y,w ) ∫ t 0 c ( t, τ ;y τ , w t ) dτ ( f Bu,w ) ∀w ∈ V, t ∈ 0, T . 2.9 By setting w y and integrating from 0 to t in 2.9 , we give ∥yn t ∥2 1,Ω ∫ t 0 ∥yn ∥∥2 1,Ωdτ C {∥∥y0 ∥∥∥ 1,Ω C ∫ t 0 ∥f ∥2 −1,Ω ‖u‖0,ΩU ) dt ∫ t 0 ∫ τ 0 ∥y s ∥2 1,Ωdsdτ } . 2.10 Applying Gronwall’s inequality to 2.10 yields ∥yn ∥∥2 L∞ 0,T ;H1 Ω ∥yn ∥∥2 L2 0,T ;H1 Ω C {∥∥y0 ∥∥ 2 1,Ω ∫T 0 ∥f ∥2 −1,Ω ‖u‖0,ΩU )} . 2.11 Abstract and Applied Analysis 5 So {un}n 1 is a bounded set in L2 0, T ;L2 ΩU and {yn}n 1 is a bounded set in L∞ 0, T ; H1 Ω . Thus u −→ u weakly in L2 ( 0, T ;L2 ΩU ) , y −→ y weakly in L∞ ( 0, T ;H1 Ω ) , y T −→ y T weakly in H1 Ω . 2.12and Applied Analysis 5 So {un}n 1 is a bounded set in L2 0, T ;L2 ΩU and {yn}n 1 is a bounded set in L∞ 0, T ; H1 Ω . Thus u −→ u weakly in L2 ( 0, T ;L2 ΩU ) , y −→ y weakly in L∞ ( 0, T ;H1 Ω ) , y T −→ y T weakly in H1 Ω . 2.12 Let W {w; w ∈ L∞ 0, T ;H1 Ω , w′ t ∈ L2 0, T ;H1 Ω }. By integrating time from 0 to T in 2.9 and taking limit as n → ∞, we obtain ( y T , w T ) a ( y T , w T ) − ∫T 0 [( y,w′ t ) a ( y,w′ t ) d ( y,w )] ∫T 0 ∫ t 0 c ( t, τ ;y τ , w t ) dτ dt ( y0, w 0 ) a ( y0, w 0 ) ∫T 0 ( f Bu,w ) , ∀w ∈ W. 2.13