Value Function of Differential Games without Isaacs Conditions. An Approach with Non-Anticipative Mixed Strategies

TL;DR

This paper establishes the existence of a value in differential games without Isaacs condition by introducing non-anticipative mixed strategies and characterizing the value as a viscosity solution of the Hamilton-Jacobi-Isaacs equation.

Contribution

It introduces a new concept of mixed strategies for differential games without Isaacs condition and proves the existence of the game value as a limit of lower and upper values.

Findings

Existence of the value in mixed strategies for differential games without Isaacs condition.

Characterization of the value as a viscosity solution of the Hamilton-Jacobi-Isaacs equation.

Convergence of lower and upper value functions along refining partitions.

Abstract

In the present paper we investigate the problem of the existence of a value for differential games without Isaacs condition. For this we introduce a suitable concept of mixed strategies along a partition of the time interval, which are associated with classical nonanticipative strategies (with delay). Imposing on the underlying controls for both players a conditional independence property, we obtain the existence of the value in mixed strategies as the limit of the lower as well as of the upper value functions along a sequence of partitions which mesh tends to zero. Moreover, we characterize this value in mixed strategies as the unique viscosity solution of the corresponding Hamilton-Jacobi-Isaacs equation.