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Minimax Theory and its Applications 05 (2020), No. 2, 151--180
Copyright Heldermann Verlag 2020

A Hybrid Differential Game with Switching Thermostatic-Type Dynamics and Costs

Fabio Bagagiolo
Dept. of Mathematics, University of Trento, Italy

Rosario Maggistro
Dept. of Management, Ca' Foscari University, Venice, Italy

Marta Zoppello
Dept. of Mathematical Sciences, Politecnico di Torino, Italy

We consider an infinite horizon zero-sum 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 so-called 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 exit-time differential games, we characterize those value functions as, respectively, the unique solution of a coupling of several Dirichlet problems for Hamilton-Jacobi-Isaacs 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, Hamilton-Jacobi-Isaacs equations, viscosity solutions, non-anticipating strategies, delayed thermostat.

MSC: 47J40, 49N70, 49L25.

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