Optimal Control of Partial Differential Equations

72 CHAPTER 6. CONTROL OF THE WAVE EQUATION
and
∂zn
∂t
− ∂zn
∂t C([0,T ];H −1 (Ω)) = sup τ∈]0,T ]sup yτ H 1 0 (Ω) =1
≤ Cun − um L 2 (Σ).
Σ
∂y(yτ, 0)
∂n
(un − um) dsdt
Thus (zn)n is a Cauchy sequence in C([0, T ]; L 2 (Ω)), and ( ∂zn
∂t )n is a Cauchy sequence in
C([0, T ]; H−1 (Ω)). It is clear that the limit of the sequence (zn)n is z(0, u, 0, 0), and the limit
of the sequence ( ∂zn
. The proof is complete.
∂t )n is ∂z(0,u,0,0)
∂t
We consider the control problem
(P5) inf{J5(z, u) | (z, u) ∈ C([0, T ]; L 2 (Ω)) × L 2 (0, T ; L 2 (Γ)), (z, u) satisfies (6.6.15)},
with
J5(z, u) = c1
2
Q
(z − zd) 2 + c2
2 z(T ) − zd(T ) 2
L 2 (Ω)
where c1, c2, and c3 are nonnegative constants, and β > 0.
c3
+
2 |∂z
∂zd
(T ) − (T )|2H
∂t ∂t −1
β
(Ω) + u
2 Σ
2 ,
Theorem 6.6.2 Assume that f ∈ L2 (Q), z0 ∈ L2 (Ω), z1 ∈ H−1 (Ω), and zd ∈ C([0, T ]; L2 (Ω))∩
C1 ([0, T ]; H−1 (Ω)). Problem (P5) admits a unique solution (¯z, ū). Moreover the optimal control
ū is defined by ū = 1 ∂p
, where p is the solution to the equation
β ∂n
∂2p ∂t2 − ∆p = c1(z − zd) in Q,
p(T ) = c3(−∆)
p = 0 on Σ = Γ×]0, T [,
−1 ( ∂z ∂zd
(T ) − (T )),
∂t ∂t
∂p
∂t (T ) = c2(z(T ) − zd(T )) in Ω.
(6.6.19)
Proof. Set F5(u) = J5(z(f, z0, z1, u), u), where z(f, z0, z1, u) is the solution to equation
(6.6.15). We have
F5(ū)u = c1(¯z − zd)wu + c2(¯z(T ) − zd(T ))wu(T )
Q
+c3〈 ∂wu
∂t (T ), (−∆)−1 ( ∂¯z ∂zd
(T ) −
∂t ∂t (T ))〉 H−1 (Ω)×H1 0 (Ω)
+ β ūu,
Σ
where wu = z(0, 0, 0, u). The functions wu and p satisfy the Green formula
c1(¯z − zd)wu + c2(¯z(T ) − zd(T ))wu(T )
Observe that (−∆) −1 ( ∂z
∂t
Q
+c3〈 ∂wu
∂t (T ), (−∆)−1 ( ∂¯z
∂t
Ω
Ω
(T ) − ∂zd
∂t (T ))〉 H −1 (Ω)×H 1 0 (Ω) = −
Σ
∂p
∂n u.
(T )− ∂zd
∂t (T )) belongs to H1 0(Ω), and (z(T )−zd(T )) belongs to L 2 (Ω).
Therefore, due to Theorem 6.5.1, ∂p
∂n belongs to L2 (Σ). Since all the terms in the above formula
are well defined, this formula can be proved for regular data, and next proved by a passage to
the limit. Due to this formula, we have
F5(ū)u = −
This completes the proof.
Σ
∂p
u + β ūu,
∂n Σ