Nonlinear Images of Sets. II: Applications to Differential Games

This paper is a continuation of our previous study on weak convexity of nonlinear images of sets [see G. E. Ivanov, Nonlinear images of sets. I: Strong and weak convexity, J. Convex Analysis 27(1) (2020) 363--382]. We consider smoothness of a set in a special sense called "bodily smoothness" and split this smoothness property of a set S into two properties: weak convexity of S and weak convexity of the closure of the complementary to S. We obtain sufficient conditions for bodily smoothness of the image of the Cartesian product of two sets S × U, wherein S is bodily smooth and U is strongly convex. We apply this result for nonlinear differential games and obtain sufficient conditions for the game reachable sets to be smooth.