A Differential Game with Exit Costs

Abstract

We study a differential game where two players separately control their own dynamics, pay a running cost, and moreover pay an exit cost (quitting the game) when they leave a fixed domain. In particular, each player has its own domain and the exit cost consists of three different exit costs, depending whether either the first player only leaves its domain, or the second player only leaves its domain, or they both simultaneously leave their own domain. We prove that, under suitable hypotheses, the lower and upper value are continuous and are, respectively, the unique viscosity solution of a suitable Dirichlet problem for a Hamilton-Jacobi-Isaacs equation. The continuity of the values relies on the existence of suitable non-anticipating strategies respecting the domain-constraint. This problem is also treated in this work.

Relationship Between the Maximum Principle and Dynamic Programming for Minimax Problems

“…Several works have been devoted to analyze of the regularity of the value function for state-constrained optimal control problems; we refer for instance to [22,14,13,18,17,16] and the references therein. See also [3,5,7] for similar results for the case of differential games.…”