
Minimax Theory and its Applications 05 (2020), No. 2, 151180 Copyright Heldermann Verlag 2020 A Hybrid Differential Game with Switching ThermostaticType Dynamics and Costs Fabio Bagagiolo Dept. of Mathematics, University of Trento, Italy fabio.bagagiolo@unitn.it Rosario Maggistro Dept. of Management, Ca' Foscari University, Venice, Italy rosario.maggistro@unive.it Marta Zoppello Dept. of Mathematical Sciences, Politecnico di Torino, Italy marta.zoppello@polito.it We consider an infinite horizon zerosum differential game where the dynamics of each player and the running costs depend on the evolution of some discrete (switching) variables. In particular, such switching variables evolve according to the switching law of a socalled thermostatic delayed relay, applied to the players' states. We first address the problem of the continuity of both lower and upper value function. Then, by a suitable representation of the problem as a coupling of several exittime differential games, we characterize those value functions as, respectively, the unique solution of a coupling of several Dirichlet problems for HamiltonJacobiIsaacs equations. The concept of viscosity solutions and a suitable definition of boundary conditions in the viscosity sense are used in the paper. Keywords: Differential games, hybrid systems, switching, exit costs, HamiltonJacobiIsaacs equations, viscosity solutions, nonanticipating strategies, delayed thermostat. MSC: 47J40, 49N70, 49L25. [ Fulltextpdf (267 KB)] for subscribers only. 