Nonlinear Optimal Control Theory

Chapter 4

Existence Theorems; Compact

Constraints

4.1 Introduction

Examples 3.1.1 and 3.1.2 in Section 3.1 of Chapter 3 showed that if the set

of admissible directions is not convex, then existence may fail, even though

the dynamics and the constraints ex hibit regular behavior. Relaxed controls

were introduced to provide convexity of the set of admissible directions. If the

constraint sets are compact and exhibit a certain regular behavior, then the

relaxed controls were shown to have a compactness property that will be used

to prove the existence theorems in Section 4.3 and the necessary conditions of

later chapters. In the next section we shall present ...