Optimal Control of Partial Differential Equations

32 CHAPTER 3. CONTROL OF SEMILINEAR ELLIPTIC EQUATIONS
Proof. The proof is completely analogous to the previous one.
We give a definition of solution to equation (3.1.1) very similar to the one for equation (3.3.6).
Definition 3.3.1 A function z ∈ H1 (Ω) is a solution to equation (3.1.1) if ψ(z) ∈ L 2N
N+2 (Γ)
when N > 2, ψ(z) ∈ Lr (Γ) with r > 1 when N = 2, and
a(z, φ) + ψ(z)φ = fφ + uφ for all φ ∈ H 1 (Ω).
Γ
Ω
Theorem 3.3.7 For all f ∈ L r (Ω), with r > N
2 , and u ∈ Ls (Γ), with s > N − 1, equation
(3.1.1) admits a unique solution in H 1 (Ω) ∩ L ∞ (Ω), this solution belongs to C(Ω) and
z C(Ω) + z H 1 (Ω) ≤ C(fL r (Ω) + uL s (Γ)).
It also belongs to C 0,ν (Ω) for some 0 < ν ≤ 1, and
z C 0,ν (Ω) ≤ C
Γ
fL r (Ω) + uL s (Γ) + ψ(z C(Ω) )
Proof. Let k > C(fL r (Ω) + uL s (Γ)), where C is the constant in the estimate of Theorem
3.3.5. Let zk be the solution to equation (3.3.6). Due to Theorem 3.3.5 zk C(Ω) < k. Thus
ψk(zk) = ψ(zk). This means that zk is also a solution to equation (3.1.1). To prove the
uniqueness, we argue by contradiction. Suppose that z1 and z2 are two solutions to equation
(3.1.1) in H 1 (Ω) ∩ L ∞ (Ω). Then w = z1 − z2 is the solution to
Aw = 0 in Ω,
∂w
+ bw = 0 on Γ,
∂nA
with b(x) = 1
0 ψ′ (z2(x) + θ(z1(x) − z2(x)))dθ. The functions z1 and z2 being bounded, b ∈
L ∞ (Γ). From the comparison principle, it follows that w = 0.
The estimate in C(Ω) follows from the estimate of zk. The estimate in H1 (Ω) follows from
the variational formulation. The estimate in C0,ν (Ω) can be deduced from Theorem 3.3.4 by
= −ψ(z) + u. The proof is complete.
writing the boundary condition in the form ∂z
∂nA
3.4 Control problems
From now on we suppose that the exponent N − 1 < s < ∞ is given fixed, and f ∈ Lr (Ω)
with r > N is also given. In this case the nonlinear equation (3.1.1) admits a unique solution
2
for all u ∈ Ls (Γ). We are going to study the control problem
(P1) inf{J1(z, u) | (z, u) ∈ H 1 (Ω) × Uad, (z, u) satisfies (3.1.1)},
where Uad is a closed convex subset in Ls (Γ), and
J1(z, u) = 1
(z − zd)
2
2
+ β
Ω
We suppose that either β > 0 or Uad is bounded in L s (Γ). The function zd belongs to L 2 (Ω).
Γ
|u| s .
.