Optimal Control of Partial Differential Equations

Chapter 7
Control of evolution equations
with bounded control operators
7.1 Introduction
The purpose of this chapter is to extend results obtained for the control of the wave and heat
equations to other linear evolution equations. We consider equations of the form
z ′ = Az + Bu + f, z(0) = z0. (7.1.1)
We have already seen that the controlled equations of chapters 5 and 6 may be written in this
form. Other examples will also be considered.
We make the following assumptions.
Assumption (H1)
Z and U are two Hilbert spaces.
The unbounded operator A, with domain D(A) dense in Z, is the infinitesimal generator of a
strongly continuous semigroup on Z. This semigroup will be denoted by (e tA )t≥0.
The operator B belongs to L(U; Z).
We here suppose that B is a bounded operator from U into Z. The case of unbounded control
operators will be studied in the next chapter.
Associated with equation (7.1.1), we shall study the control problem
(P ) inf{J(z, u) | (z, u) ∈ C([0, T ]; Z) × L 2 (0, T ; U), (z, u) satisfies (7.1.1)}.
with
T
J(z, u) = 1
|Cz(t) − yd(t)|
2 0
2 Y + 1
2 |Dz(T ) − yT | 2 1
YT + |u(t)|
2 0
2 U. (7.1.2)
This problem is often referred as a ’Linear Quadratic Regulation’ problem (LQR problem in
short). We make the following assumption on the operators C and D.
Assumption (H2)
Y and YT are Hilbert spaces.
The operator C belongs to L(Z; Y ), and the operator D belongs to L(Z; YT ). The function
yd belongs to L 2 (0, T ; Y ) and yT ∈ YT .
In this chapter we identify Z ′ with Z, and U ′ with U.
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