ABSTRACT
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 exhibit 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 examples of non-existence that illustrate the need for conditions on the behavior of the constraint sets, terminal sets, and dynamics. In Section 4.4 we introduce a convexity condition implying that an optimal relaxed trajectory is an ordinary trajectory, and thus is a solution of the ordinary problem. In Section 4.5 we give examples from applied areas that are covered by the existence theorems of Section 4.4.
