Optimal Control of Partial Differential Equations

106 CHAPTER 9. CONTROL OF A SEMILINEAR PARABOLIC EQUATION
Theorem 9.6.4 The imbedding from W (0, T ; H 1 0(Ω), H −1 (Ω)) into L 2 (Q) is compact.
Theorem 9.6.5 Let z be in L 4 (Q). For all z0 ∈ L 2 (Ω), all f ∈ L 2 (Q), equation
∂w
∂t − ∆w + 2∂x1(zw) = f in Q, w = 0 on Σ, w(0) = z0 in Ω,
admits a unique solution in W (0, T ; H 1 0(Ω), H −1 (Ω)). Moreover the mapping (z0, f) ↦→ w is
continuous from L 2 (Ω) × L 2 (Q) into W (0, T ; H 1 0(Ω), H −1 (Ω)).
Proof. This theorem can be proved by using a fixed point method as in exercise 5.5.4 (see
exercise 9.7.1).
9.7 Exercises
Exercise 9.7.1
Adapt the fixed point method of exercise 5.5.4 to prove Theorem 9.6.5.
Exercise 9.7.2 (Variational method)
We want to give another proof of Theorem 9.4.1. Assumptions and notation are the ones of
Theorem 9.4.1. Let (ψn)n be a Hilbertian basis in H 1 0(Ω), and let (φn)n be the basis obtained by
applying the Gram-Schmidt process to (ψn)n for the scalar product of L 2 (Ω). Thus (φn)n is a
Hilbertian basis in L 2 (Ω) whose elements belong to H 1 0(Ω). Denote by Hm = vect(ψ0, . . . , ψm)
the vector space generated by (ψ0, . . . , ψm). We have
∩∞ H
m=0Hm
1 0 1
= H0(Ω), and
Ω φiφj = δij. We also assume that the family (ψn)n is orthogonal in H1 0(Ω) (which is
satisfied if we choose a family of eigenfunctions of the Laplace operator). Denote by Pm the
orthogonal projection in L2 (Ω) on Hm. Observe that a function z belongs to H1 (0, T ; Hm) if
and only if z is of the form z = Σm j=0gjφj, with gj ∈ H1 (0, T ).
1 - Prove that the variational equation
find z = Σ m j=0gjφj ∈ H 1 (0, T ; Hm) such that
d
dt 〈z(t), ζ〉 =〈∇z(t), ∇ζ〉 + 〈f, ζ〉 − 〈φ(z), ζ〉 and 〈z(0), ζ〉 = 〈z0, ζ〉,
(9.7.12)
for all ζ ∈ Hm, is equivalent to a system of ordinary differential equations in R N satisfied by
g = (g0, . . . , gm) T . Prove that this system admits a unique solution g m = (g m 0 , . . . , g m m) T , and
that the corresponding function zm = Σ m j=0g m j φj obeys
1
|zm(T )|
2 Ω
2 T
+ |∇zm|
0 Ω
2 = 1
|z0m|
2 Ω
2 T
+ fmzm,
0 Ω
where z0m = Pm(z0) and fm(t) = Pm(f(t)). Prove that 〈zm(·), φj〉 H 1 (0,T ) ≤ C, where C is
independent of m and j.