Optimal Control of Partial Differential Equations

70 CHAPTER 6. CONTROL OF THE WAVE EQUATION
Theorem 6.4.5 Assume that f ∈ L 2 (0, T ; L 2 (Ω)), z0 ∈ L 2 (Ω), z1 ∈ (H 1 (Ω)) ′ , and zd ∈
C([0, T ]; L 2 (Ω)). Problem (P2) admits a unique solution (¯z, ū). Moreover the optimal control
ū is defined by ū = − 1
β p|Σ, where p is the solution to the equation
∂ 2 p
∂t 2 − ∆p = ¯z − zd in Q,
p(T ) = 0,
∂p
∂n
= 0 on Σ = Γ×]0, T [,
∂p
∂t (T ) = ¯z(T ) − zd(T ) in Ω.
Proof. We leave the reader adapt the proof of Theorem 6.3.1.
6.5 Trace regularity
(6.4.12)
To study the wave equation with a control in a Dirichlet boundary condition, we have to
establish a sharp regularity result stated below.
Theorem 6.5.1 Let y be the solution to the equation
∂2y ∂t2 − ∆y = f in Q, y = 0 on Σ, y(x, 0) = y0 and ∂y
∂t (x, 0) = y1 in Ω. (6.5.13)
We have
∂y
∂n L2
(Σ) ≤ C fL2 (Q) + y0H 1
0 (Ω) + y1L2 (Ω) . (6.5.14)
The proof can be found in [14, Theorem 2.2].
6.6 Dirichlet boundary control
We consider the wave equation with a control in a Dirichlet boundary condition
∂2z ∂t2 − ∆z = f in Q, z = u on Σ, z(x, 0) = z0 and ∂z
∂t (x, 0) = z1 in Ω. (6.6.15)
As for the heat equation with a Dirichlet boundary control, the solution to equation (6.6.15)
is defined by the transposition method.
Definition 6.6.1 A function z ∈ C([0, T ]; L2 (Ω)) ∩ C1 ([0, T ]; H−1 (Ω)) is called a weak solution
to equation (6.6.15) if, and only if,
fy dxdt =
−〈 ∂z
∂t (0), y(0)〉 H −1 (Ω)×H 1 0 (Ω) −
Q
Q
Ω
zϕ dxdt + 〈 ∂z
∂t (T ), yT 〉 H −1 (Ω)×H 1 0 (Ω)
z(T )νT dx +
Ω
z(0) ∂y
∂y
(0) dx + u dsdt (6.6.16)
∂t Σ ∂n
for all (ϕ, yT , νT ) ∈ L 1 (0, T ; L 2 (Ω)) × H 1 0(Ω) × L 2 (Ω), where y is the solution to
∂ 2 y
∂t 2 − ∆y = ϕ in Q, y = 0 on Σ, y(x, T ) = yT and ∂y
∂t (x, T ) = νT in Ω. (6.6.17)