A zero-sum differential game for two opponent masses

TL;DR

This paper studies a zero-sum differential game involving two masses governed by a controlled transport equation, establishing the uniqueness of the value function using viscosity solutions in an infinite-dimensional Hilbert space.

Contribution

It introduces a novel framework for analyzing differential games with infinite-dimensional state spaces and proves the uniqueness of the value function in this setting.

Findings

Proved the uniqueness of the value function as a viscosity solution.

Formulated the game as an infinite-dimensional Isaacs equation.

Applied viscosity solutions theory in Hilbert spaces.

Abstract

We investigate an infinite dimensional partial differential equation of Isaacs' type, which arises from a zero-sum differential game between two masses. The evolution of the two masses is described by a controlled transport/continuity equation, where the control is given by the vector velocity field. Our study is set in the framework of the viscosity solutions theory in Hilbert spaces, and we prove the uniqueness of the value functions as solutions of the Isaacs equation.