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Viscosity solutions to evolution problems of star-shaped reachable sets

Abstract

The article deals with Lipschitz continuous differential inclusions that yield star-shaped reachable sets. The purpose of the paper is to show that the radial function of such reachable sets is a viscosity solution to a certain partial differential equation. As a result, the existing theory of viscosity solutions to first-order partial differential equations was applied to resolve the existence, uniqueness, and some calculation aspects. Several relaxations concerning the forms of the inclusion and the initial set were also considered.

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Correspondence to S. S. Mazurenko.

Additional information

This research was supported by the National Sustainability Programme of the Czech Ministry of Education, Youth and Sports (LO1214) and the RECETOX research infrastructure (LM2015051 and CZ.02.1.01/0.0/0.0/16_013/0001761).

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Mazurenko, S.S. Viscosity solutions to evolution problems of star-shaped reachable sets. Nonlinear Differ. Equ. Appl. 25, 29 (2018). https://doi.org/10.1007/s00030-018-0516-8

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Mathematics Subject Classification

  • 35D40
  • 93B03
  • 34A60

Keywords

  • Differential inclusion
  • Generalized solutions
  • Radial function
  • Minkowski function
  • Gauge function