A single player and a mass of agents: a pursuit evasion-like game

TL;DR

This paper models a pursuit-evasion game involving one player and a mass of agents, using differential game theory and PDEs, and proves the uniqueness of the value function as a viscosity solution.

Contribution

It introduces a novel approach to pursuit-evasion games with a mass of agents using infinite-dimensional PDEs and adapted strategies, extending existing differential game frameworks.

Findings

Derivation of an infinite-dimensional Isaacs equation

Proof of the value function as a unique viscosity solution

Application of dynamic programming to a PDE-based pursuit-evasion game

Abstract

We study a finite-horizon differential game of pursuit-evasion like, between a single player and a mass of agents. The player and the mass directly control their own evolution, which for the mass is given by a first order PDE of transport equation type. Using also an adapted concept of non-anticipating strategies, we derive an infinite dimensional Isaacs equation, and by dynamic programming techniques we prove that the value function is the unique viscosity solution on a suitable invariant subset of a Hilbert space.