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A zero-sum differential game for two opponent masses

Fabio Bagagiolo, Rossana Capuani, Luciano Marzufero

TL;DR

The paper tackles a zero-sum differential game between two mass populations, each governed by a transport/continuity equation in a Hilbert-space setting, and derives infinite-dimensional Isaacs equations for the lower and upper value functions via dynamic programming. It provides a rigorous compact invariant set framework, proves that the value functions are the unique continuous viscosity solutions of the corresponding HJI equations, and discusses conditions under which the game has a value. The work also offers two one-dimensional examples illustrating the qualitative behavior of optimal strategies and highlights a potential parabolic (diffusion) extension with a vanishing-viscosity perspective for future study. Together, these results lay a foundation for feedback controls in infinite-dimensional mass-transport games with possible applications to multi-agent systems and mean-field-type interactions.

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.

A zero-sum differential game for two opponent masses

TL;DR

The paper tackles a zero-sum differential game between two mass populations, each governed by a transport/continuity equation in a Hilbert-space setting, and derives infinite-dimensional Isaacs equations for the lower and upper value functions via dynamic programming. It provides a rigorous compact invariant set framework, proves that the value functions are the unique continuous viscosity solutions of the corresponding HJI equations, and discusses conditions under which the game has a value. The work also offers two one-dimensional examples illustrating the qualitative behavior of optimal strategies and highlights a potential parabolic (diffusion) extension with a vanishing-viscosity perspective for future study. Together, these results lay a foundation for feedback controls in infinite-dimensional mass-transport games with possible applications to multi-agent systems and mean-field-type interactions.

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.
Paper Structure (9 sections, 7 theorems, 105 equations)

This paper contains 9 sections, 7 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. On the mass distribution: estimates and technicalities
  3. The continuity equation for the mass
  4. A suitable invariant set for the evolution of the mass
  5. The differential game model
  6. Dynamic Programming Principle and regularity of the value function
  7. The Hamilton-Jacobi-Isaacs equation for $\underbar V$
  8. Two one-dimensional examples
  9. Something on a possible parabolic case

Key Result

Proposition 1

Suppose that $(H)$ holds and $\bar{m}\in H^1(\mathbb R^d)\cap W^{1, \infty}({\mathbb{R}}^d)$. Then Moreover, it holds the following for every $t_1,t_2\in[0, T]$, $s\in[\max\{t_1,t_2\}, T]$ and for some $\tilde{L}$ independent of $t_1,t_2$ and $\beta$.

Theorems & Definitions (14)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 1
  • Theorem 3.1
  • proof
  • ...and 4 more