Uniform Tauberian theorem in differential games

Abstract

This paper establishes the uniform Tauberian theorem for differential zero-sum games. Under rather mild conditions imposed on the dynamics and running cost, two parameterized families of games are considered, i.e., the ones with the payoff functions defined as the Cesaro mean and Abel mean of the running cost. The asymptotic behavior of value in these games is investigated as the game horizon tends to infinity and the discounting parameter tends to zero, respectively. It is demonstrated that the uniform convergence of value on an invariant subset of the phase space in one family implies the uniform convergence of value in the other family and that the limit values in the both families coincide. The dynamic programming principle acts as the cornerstone of proof.