Existence without Convexity

Abstract

The existence theorems in the preceding chapter required certain regularity in the behavior of the data of the problem and required that certain sets I ⁺(t,x) be convex. In the case of non- compact constraints it was also required that the trajectories be equi-absolutely continuous, and reasonable growth conditions to ensure this were formulated. All of the conditions placed on the problem can be justified except, perhaps, the requirement that the sets I ⁺(t,x) be convex. This requirement essentially restricts us to systems whose state equations are linear in the control, whose payoff function f⁰ is convex in z, and for which the constraint sets Ω(t,x) are convex. In this chapter we investigate systems in which the sets I ⁺(t,x) need not be convex.