Nonlinear Optimal Control Theory

Chapter 3

Relaxed Controls

3.1 Introduction

In this chapter we define relaxed controls and the re laxed control problem

and determine some of the properties of relaxed co ntrols. For problems with

well-behaved compact constraint sets, relaxed controls have a very use ful com-

pactness property. Also, at a given point in the subset R of (t, x) space, the set

of directions that the state of a relaxed system may take is convex. This prop-

erty is needed in existence theorems. We also shall prove an implicit function

theorem for measur able functions that permits a definition of relaxed controls

alternative to the one given in the next section. This theorem ...