Optimal Control of Partial Differential Equations

Chapter 2
Control of elliptic equations
2.1 Introduction
In this chapter we study optimal control problems for elliptic equations. We first consider
problems with controls acting in a Neumann boundary condition. We derive optimality conditions
for problems without control constraints. In section 2.3 we explain how these optimality
conditions may be useful to compute the optimal solution. The case of a Dirichlet boundary
control is studied in section 2.4. For a control in a Dirichlet boundary condition, we define
solutions by the transposition method. Extensions to problems with control constraints are
considered in section 2.5. The existence of optimal solutions is proved in section 2.6. In section
2.7, we extend the results of section 2.2 to problems defined by various functionals.
2.2 Neumann boundary control
Let Ω be a bounded domain in RN , with a boundary Γ of class C2 . We consider the elliptic
equation
∂z
−∆z + z = f in Ω, = u on Γ. (2.2.1)
∂n
The function f is a given source term and the function u is a control variable.
A classical control problem consists in finding a control function u ∈ L2 (Γ) which minimizes
the cost functional
J1(z, u) = 1
(z − zd)
2 Ω
2 + β
u
2 Γ
2 ,
where the pair (z, u) satisfies equation (2.2.1). The term β
2 Γ u2 (with β > 0) is proportional to
the consumed energy. Thus, minimizing J1 is a compromise between the energy consumption
and finding u so that the distribution z is close to the desired profile zd. In the case of equation
(2.2.1) we say that the control function is a boundary control because the control acts on Γ.
Before studying the above control problem, we first recall some results useful for equation
(2.2.1). The existence of a unique solution z ∈ H1 (Ω) to equation (2.2.1) may be proved with
the Lax-Milgram theorem. More precisely we have the following theorem.
Theorem 2.2.1 ([12]) For every f ∈ L 2 (Ω) and every u ∈ L 2 (Γ), equation (2.2.1) admits a
unique weak solution z(f, u) in H 1 (Ω), moreover the operator
(f, u) ↦→ z(f, u)
13