Optimal Control of Partial Differential Equations

68 CHAPTER 6. CONTROL OF THE WAVE EQUATION
Let us establish the optimality conditions for (P2). As usual we set F2(u) = J2(z(u), u),
where z(u) is the solution to (6.3.6). We have F ′ 2(ū2)u =
Ω (∇¯z2(T ) − ∇zd(T ))∇wu(T ) +
β
Q χωū2u, where wu is the solution to
Since
∂2w ∂t2 − ∆w = χωu in Q, w = 0 on Σ, w(x, 0) = 0 and ∂w
(x, 0) = 0 in Ω.
∂t
(∇¯z2(T ) − ∇zd(T ))∇wu(T ) = 〈−∆(¯z2(T ) − zd(T )), wu(T )〉 H−1 (Ω)×H1 0 (Ω),
Ω
applying formula (6.2.3) to p2 and wu, we obtain
F ′
2(ū2)u = (χω(βū2 + p2)u = 0
Q
for every u ∈ L 2 (0, T ; L 2 (ω)). Thus the optimality condition for (P2) is proved. The proof of
the other results is left to the reader.
Comments. As for the heat equation with distributed controls, equation (6.3.6) is of the
form
y ′ = Ay + F + Bu, y(0) = y0,
with
Ay = A
y1
y2
=
y2
∆y1
0
, F =
f
z0
, y0 =
z1
0
, and Bu =
χωu
Thus problem (P1) is a particular case of control problems studied in Chapter 7.
6.4 Neumann boundary control
We first study the equation
∂2z − ∆z = f in Q,
∂t2 ∂z
∂n = 0 on Σ, z(x, 0) = z0 and ∂z
∂t (x, 0) = z1 in Ω. (6.4.10)
We set D(A) = {y1 ∈ H 2 (Ω) | ∂y1
∂n = 0} × H1 (Ω), Y = H 1 (Ω) × L 2 (Ω), and
Ay = A
y1
y2
y2
=
∆y1 − y1
, Ly =
Equation (6.4.10) may be written in the form
dy
dt
0
y1
, F =
= (A + L)y + F, y(0) = y0.
0
f
, and y0 =
Theorem 6.4.1 The operator (A, D(A)) is the infinitesimal generator of a strongly continuous
semigroup of contractions on Y .
Proof. We leave the reader adapt the proof of Theorem 6.2.1.
z0
z1
.
.