Existence without Convexity
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.
KeywordsExtreme Point Original Problem Measurable Subset Compact Interval Admissible Pair
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