Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Grigorii E. Ivanov

Department of Higher Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Moscow Region, Russia

g.e.ivanov@mail.ru

Keywords

  • C1
  • 1 mapping
  • covering mapping
  • bi-Lipschitz mapping
  • strong convexity
  • weak convexity
  • Minkowski sum
  • Minkowski operator
  • differential game

Mathematics Subject Classification

52A05, 46T20, 49N70

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Suggested citation

G. E. Ivanov. “Nonlinear Images of Sets. II: Applications to Differential Games.” Journal of Convex Analysis 27 (2020), No. 4, 1123–1136. https://doi.org/10.68381/jca27059

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.