Optimal Control of Partial Differential Equations

6.4. NEUMANN BOUNDARY CONTROL 69
Theorem 6.4.2 For every f ∈ L 2 (Q), every z0 ∈ H 1 (Ω), every z1 ∈ L 2 (Ω), equation (6.4.10)
admits a unique weak solution z(f, z0, z1), moreover the operator
(f, z0, z1) ↦→ z(f, z0, z1)
is linear and continuous from L 2 (Q)×H 1 (Ω)×L 2 (Ω) into C([0, T ]; H 1 (Ω))∩C 1 ([0, T ]; L 2 (Ω)).
To study the wave equation with nonhomogeneous boundary conditions, we set D( A) =
H1 (Ω) × L2 (Ω), Y = L2 (Ω) × (H1 (Ω)) ′ , and
Ay =
y1
y2
A =
,
y2 Ãy1 − y1
where
Ãy1, ζ
(H1 (Ω)) ′
= − ∇y1 · ∇(−∆ + I)
Ω
−1 ζ.
Theorem 6.4.3 The operator ( A, D( A)) is the infinitesimal generator of a semigroup of contractions
on Y .
Proof. The proof is similar to the one of Theorem 6.2.3.
Now, we consider the wave equation with a control in a Neumann boundary condition:
∂2z ∂z
− ∆z = f in Q,
∂t2 ∂n = u on Σ, z(x, 0) = z0 and ∂z
∂t (x, 0) = z1 in Ω. (6.4.11)
For any u ∈ L2 (Γ), the mapping ζ ↦→
Γ uζ is a continuous linear on H1 (Ω). Thus it can be
identified with an element of (H1 (Ω)) ′ . Thus for u ∈ L2 (Σ), the mapping ζ ↦→
u(·)ζ is an
element of L2 (0, T ; (H1 (Ω)) ′ ). Let us denote this mapping by û. We set
y1
0
V =
0
û
, F =
0
f
, L
Equation (6.4.11) may be written in the form
y2
=
y1
dy
dt = ( A + L)y + F + V, y(0) = y0,
, and y0 =
z0
with F and V belong to L 2 (0, T ; L 2 (Ω)) × L 2 (0, T ; (H 1 (Ω)) ′ ), y0 ∈ L 2 (Ω) × (H 1 (Ω)) ′ .
Theorem 6.4.4 For every (f, u, z0, z1) ∈ L 2 (Q)×L 2 (Σ)×L 2 (Ω)×(H 1 (Ω)) ′ , equation (6.4.11)
admits a unique weak solution z(f, u, z0, z1) in C([0, T ]; L 2 (Ω))∩C 1 ([0, T ]; (H 1 (Ω)) ′ ). Moreover
the mapping (f, u, z0, z1) ↦→ z(f, u, z0, z1) is continuous from L 2 (Q)×L 2 (Σ)×L 2 (Ω)×(H 1 (Ω)) ′
into C([0, T ]; L 2 (Ω)) ∩ C 1 ([0, T ]; (H 1 (Ω)) ′ ).
Proof. The result is a direct consequence of Theorem 6.4.3.
We consider the control problem
(P4) inf{J4(z, u) | (z, u) ∈ C([0, T ]; L 2 (Ω)) × L 2 (0, T ; L 2 (Ω)), (z, u) satisfies (6.4.11)},
with
J4(z, u) = 1
(z − zd)
2 Q
2 + 1
(z(T ) − zd(T ))
2 Ω
2 + β
u
2 Σ
2 .
z1
Γ
.