Optimal Control of Partial Differential Equations

Chapter 6
Control of the wave equation
6.1 Introduction
We first begin by problems with a distributed control. We study the wave equation via the
semigroup theory with initial data in H 1 0(Ω) × L 2 (Ω) (section 6.2.1), and in L 2 (Ω) × H −1 (Ω)
(section 6.2.2). These results are next used to derive optimality conditions in the case of
functionals involving observations of the derivative of the state (Theorem 6.3.1). The case
of Neumann boundary controls is briefly presented in section 6.4. To obtain fine regularity
results in the case of Dirichlet boundary controls, we need a trace regularity result for solutions
to the wave equation with homogeneous boundary conditions (Theorem 6.5.1). Equations
with nonhomogeneous Dirichlet boundary conditions is studied by the transposition method
(Theorem 6.6.1). We derive optimality conditions for functionals involving observations in
C([0, T ]; H −1 (Ω)) (Theorem 6.6.2).
The notation Ω, Γ, T , Q, Σ, as well as the assumptions on Ω and Γ, are the ones of the
previous chapter.
6.2 Existence and regularity results
6.2.1 The wave equation in H 1 0(Ω) × L 2 (Ω)
To study equation
∂2z ∂t2 − ∆z = f in Q, z = 0 on Σ, z(x, 0) = z0 and ∂z
∂t (x, 0) = z1 in Ω, (6.2.1)
with (z0, z1) ∈ H1 0(Ω) × L2 (Ω) and f ∈ L2 (Q), we transform the equation in a first order
), equation (6.2.1) may be written in the form
evolution equation. Setting y = (z, dz
dt
with
y1
Ay = A
y2
=
dy
dt
y2
∆y1
= Ay + F, y(0) = y0,
0
, F =
f
z0
, and y0 =
z1
Set Y = H 1 0(Ω) × L 2 (Ω). The domain of A in Y is D(A) = (H 2 (Ω) ∩ H 1 0(Ω)) × H 1 0(Ω).
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