OPTIMAL CONTROL OF PDE PROBLEM SET 3 Submission deadline and submission point: 14:00 on Wednesday 29 Mar 2017, School office Problem 1. Determining the adjoint operator - Let aΩ; u 2 L2(Ω), aΓ; v 2 L2(@Ω), bΩ; cΩ 2 L1(Ω), bΓ; cΓ 2 L1(@Ω) be given. Assume cΓ ≥ 0, [30 marks] cΩ ≥ 0, and at least one of the two functions is strictly positive. Define y 2 H1(Ω) and p 2 H1(Ω) as the solutions of the elliptic problems −∆y + cΩy = bΩu; −∆p + cΩp = aΩ in Ω; @νy + cΓy = bΓv; @νp + cΓp = aΓ on @Ω: Show that the following duality relation holds: ZΩ aΩy dx + Z@Ω aΓy ds(x) = ZΩ bΩup dx + Z@Ω bΓvp ds(x): Problem 2. Existence and uniqueness of an optimal control - Let Ω ⊂ R3 with smooth boundary Γ = @Ω, λ ≥ 0 and yΩ, β, ua, ub be given functions. Consider [30 marks] the following optimal control problem min J(y; u) = 1 2 ZΩ jy(x) − yΩ(x)j2 dx + λ 2 ZΩ ju(x)j2 dx subject to − ∆y + y = βu in Ω; @νy = 0 on Γ: and ua(x) ≤ u(x) ≤ ub(x) for almost every x 2 Ω: 1. Making suitable assumptions, show that this optimal control problem admits a least one optimal control with associated optimal state. 2. Is this optimal control unique? Problem 3. Ga ^teaux and Fr´echet derivative - Let fU; jj · jjUg; fV; jj · jjV g be real Banach spaces, U ⊂ U a non-empty and open set, and [40 marks] f : U ! V . 1. When is f called Ga ^teaux differentiable at u 2 U? 2. When is f called Fr´echet differentiable at u 2 U? 3. Show that the functional g(u) = sin(u(1)) is Fr´echet differentiable at every u 2 C([0; 1]). 1