Optimal Control of Partial Differential Equations

5.5. EXERCISES 59
Theorem 5.4.6 Assume that f ∈ L2 (Q), z0 ∈ L2 (Ω), and zd ∈ L2 (0, T ; L2 (Ω)). Let (¯z, ū) be
the unique solution to problem (P4). The optimal control u is defined by u = 1 ∂p
, where p is
β ∂n
the solution to the equation
− ∂p
∂t − ∆p = 0 in Q, p = 0 on Σ, p(x, T ) = (−∆)−1 (¯z(T ) − zT ) in Ω. (5.4.27)
Proof. We set F4(u) = J4(z(f, z0, u), u). If wu is the solution to equation 5.4.21, and p the
solution to equation 5.4.27, with the formula stated in Theorem 5.4.5(ii), we have
ūu.
The proof is complete.
5.5 Exercises
Exercise 5.5.1
F4(ū)u = 〈wu(T ), (−∆) −1 (¯z(T ) − zT )〉 H −1 (Ω)×H 1 0 (Ω) + β
=
Σ
− ∂p
+ βū u.
∂n
The notation are the ones of section 5.2. Let (un)n be a sequence in L 2 (0, T ; L 2 (ω)), converging
to u for the weak topology of L 2 (0, T ; L 2 (ω)). Let zn be the solution to equation (5.2.1)
corresponding to un, and zu be the solution to equation (5.2.1) corresponding to u. Prove that
(zn(T ))n converges to zu(T ) for the weak topology of L 2 (Ω). Prove that the control problem
(P1) admits a unique solution.
Exercise 5.5.2
Prove that the control problem (P2) of section 5.3 admits a unique solution.
Exercise 5.5.3
The notation are the ones of section 5.4. Let (un)n be a sequence in L 2 (Σ), converging to
u for the weak topology of L 2 (Σ). Let zn be the solution to equation (5.4.20) corresponding
to un, and zu be the solution to equation (5.4.20) corresponding to u. Prove that (zn(T ))n
converges to zu(T ) for the weak topology of H −1 (Ω). Prove that the control problem (P4)
admits a unique solution.
Exercise 5.5.4
Let Ω be a bounded domain in R N , with a boundary Γ of class C 2 . Let T > 0, set Q = Ω×(0, T )
and Σ = Γ × (0, T ). We consider a convection-diffusion equation with a distributed control
∂z
∂t − ∆z + V · ∇z = f + χωu in Q, z = 0 on Σ, z(x, 0) = z0 in Ω. (5.5.28)
Σ