Optimal Control of Partial Differential Equations

Chapter 5
Control of the heat equation
5.1 Introduction
We begin with distributed controls (section 5.2). Solutions of the heat equation are defined
via the semigroup theory, but we explain how we can recover regularity results in
W (0, T ; H 1 0(Ω), H −1 (Ω)) (Theorem 5.2.3). Since we study optimal control problems of evolution
equations for the first time, we carefully explain how we can calculate the gradient, with
respect to the control variable, of a functional depending of the state variable via the adjoint
state method. The case of Neumann boundary controls is studied in section 5.3. Estimates in
W (0, T ; H 1 (Ω), (H 1 (Ω)) ′ ) are obtained by an approximation process, using the Neumann operator
(see the proof of Theorem 5.3.6). Section 5.4 deals with Dirichlet boundary controls. In
that case the solutions do not belong to C([0, T ]; L 2 (Ω)), but only to C([0, T ]; H −1 (Ω)). We
carefully study control problems for functionals involving observations in C([0, T ]; H −1 (Ω))
(see section 5.4.2).
We only study problems without control constraints. But the extension of existence results
and optimality conditions to problems with control constraints is straightforward.
5.2 Distributed control
Let Ω be a bounded domain in R N , with a boundary Γ of class C 2 . Let T > 0, set Q = Ω×(0, T )
and Σ = Γ × (0, T ). We consider the heat equation with a distributed control
∂z
∂t − ∆z = f + χωu in Q, z = 0 on Σ, z(x, 0) = z0 in Ω. (5.2.1)
The function f is a given source of temperature, χω is the characteristic function of ω, ω is an
open subset of Ω, and the function u is a control variable. We consider the control problem
(P1) inf{J1(z, u) | (z, u) ∈ C([0, T ]; L 2 (Ω)) × L 2 (0, T ; L 2 (ω)), (z, u) satisfies (5.2.1)},
where
J1(z, u) = 1
(z − zd)
2 Q
2 + 1
(z(T ) − zd(T ))
2 Ω
2 + β
χωu
2 Q
2 ,
and β > 0. In this section, we assume that f ∈ L 2 (Q) and that zd ∈ C([0, T ]; L 2 (Ω)).
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