A strategy-based proof of the existence of the value in zero-sum differential games

TL;DR

This paper provides a self-contained proof of the existence of the value in zero-sum differential games using a strategy-based approach inspired by extremal aiming, under Isaacs' condition.

Contribution

It introduces a new proof method for the value's existence in differential games based on constructing epsilon-optimal strategies.

Findings

Existence of the value proven under Isaacs' condition

Strategy construction based on extremal aiming

Self-contained proof approach

Abstract

The value of a zero-sum differential games is known to exist, under Isaacs' condition, as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. In this note we provide a self-contained proof based on the construction of $\ep$-optimal strategies, which is inspired by the "extremal aiming" method from Krasovskii and Subbotin.