Differential Games for a Mixed ODE-PDE System

TL;DR

This paper studies a zero-sum differential game involving a coupled hyperbolic PDE and ODE system, establishing well-posedness, stability, and characterizing the value functions via Hamilton-Jacobi-Isaacs equations.

Contribution

It introduces a framework for analyzing differential games with mixed PDE-ODE systems, proving well-posedness and stability, and defining value functions through viscosity solutions.

Findings

Proved well-posedness of the coupled PDE-ODE system.

Established stability results for fixed controls.

Derived Hamilton-Jacobi-Isaacs equations for the value functions.

Abstract

Motivated by a vaccination coverage problem, we consider here a zero-sum differential game governed by a differential system consisting of a hyperbolic partial differential equation (PDE) and an ordinary differential equation (ODE). Two players act through their respective controls to influence the evolution of the system with the aim of minimizing their objective functionals $\mathcal F_1$ and $\mathcal F_2$, under the assumption that $\mathcal F_1 +\mathcal F_2 = 0$. First we prove a well posedness and a stability result for the differential system, once the control functions are fixed. Then we introduce the concept of non-anticipating strategies for both players and we consider the associated value functions, which solve two infinite-dimensional Hamilton-Jacobi-Isaacs equations in the viscosity sense.