Optimal Control of Partial Differential Equations

Chapter 9
Control of a semilinear parabolic
equation
9.1 Introduction
In this chapter we study control problems for a semilinear parabolic equation of Burgers’ type
in dimension 2. For a L 2 -distributed control, we prove the existence of a unique solution to the
state equation in C([0, T ]; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0(Ω)). Following the approach of the previous
chapters we use the semigroup theory. We first prove the existence of a local solution for
initial data in L 2p (Ω) for p > 2, and next the existence of a global solution by establishing
an energy estimate. The existence of a (uinque) solution for L 2 -initial data is obtained by
approximation. The classical method to study this kind of equation is the variational method
(also called the Faedo-Galerkin method). This approach is treated in exercise 9.7.2. This
chapter can be considered as an introduction to the optimal control of the Navier-Stokes
equations [27]. Indeed the proof of optimality conditions is very similar in both cases.
To study the state equation and the control problem, we need additional regularity results
on parabolic equations. These results are stated in Appendix (section 9.6).
9.2 Distributed control
Let Ω be a bounded domain in R 2 , with a regular boundary Γ. Let T > 0, set Q = Ω × (0, T )
and Σ = Γ × (0, T ). We consider the equation
∂z
∂t − ∆z + Φ(z) = f + χωu in Q, z = 0 on Σ, z(x, 0) = z0 in Ω, (9.2.1)
with f ∈ L2 (Q), u ∈ L2 (0, T ; L2 (ω)), z0 ∈ L2 (Ω). The function f is a given source term,
χω is the characteristic function of ω, ω is an open subset of Ω, and the function u is a
control variable. The nonlinear term φ is defined by φ(z) = 2z∂x1z = ∂x1(z 2 ). Any other
combination of first order partial derivatives may be considered, for example we can as well
consider φ(z) = Σ 2 i=1z∂xi
z. We first want to prove the existence of a unique weak solution (in a
sense to be precised) to equation (9.2.1). Recall that, if z0 ∈ L 2 (Ω) and g ∈ L 2 (0, T ; H −1 (Ω)),
equation
∂z
∂t − ∆z = g in Q, z = 0 on Σ, z(x, 0) = z0 in Ω, (9.2.2)
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