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

Fabio Bagagiolo, Rossana Capuani, Luciano Marzufero

TL;DR

We address a zero-sum pursuit-evasion-like differential game where a single agent competes against a mass whose density evolves under a transport-type continuity equation. The authors derive an infinite-dimensional Hamilton-Jacobi-Isaacs equation on a viable Hilbert-space subset and prove that the lower value function is the unique viscosity solution via a Dynamic Programming Principle based on non-anticipating strategies. They establish regularity and compactness properties for the density $m$, define a viable domain $\tilde{X}$ to avoid boundary complications, and formulate the HJI equation with a corresponding viscosity framework, including a comparison principle for uniqueness. The paper also illustrates the framework with two one-dimensional examples to shed light on the structure of optimal strategies in simplified settings and to connect with mean-field-type considerations. Overall, the results advance the theory of infinite-dimensional viscosity solutions for differential games with distributed mass dynamics and open avenues for mean-field game interpretations and applications.

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.

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

TL;DR

We address a zero-sum pursuit-evasion-like differential game where a single agent competes against a mass whose density evolves under a transport-type continuity equation. The authors derive an infinite-dimensional Hamilton-Jacobi-Isaacs equation on a viable Hilbert-space subset and prove that the lower value function is the unique viscosity solution via a Dynamic Programming Principle based on non-anticipating strategies. They establish regularity and compactness properties for the density , define a viable domain to avoid boundary complications, and formulate the HJI equation with a corresponding viscosity framework, including a comparison principle for uniqueness. The paper also illustrates the framework with two one-dimensional examples to shed light on the structure of optimal strategies in simplified settings and to connect with mean-field-type considerations. Overall, the results advance the theory of infinite-dimensional viscosity solutions for differential games with distributed mass dynamics and open avenues for mean-field game interpretations and applications.

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.
Paper Structure (9 sections, 8 theorems, 139 equations)

This paper contains 9 sections, 8 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. On the continuity equation for the mass
  3. Spatial and time estimates on the density $m$
  4. The differential game model: first results
  5. Dynamic Programming Principle and regularity of the value function
  6. The Hamilton-Jacobi-Isaacs equation for $\underbar V$
  7. Comments on two one-dimensional examples
  8. First example
  9. Second example

Key Result

Proposition 1

Under hypothesis vectorfields, for any Borel measure $\bar{\mu}$ the solution of the continuity equation conteq in the sense of distribution is given by the push-forward where, for any Borel measurable set $A\subset{\mathbb{R}}^d$, $\mu(\cdot, s)(A)=\bar{\mu}(\Phi^{-1}(\cdot, t, s)(A))$.

Theorems & Definitions (18)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 1
  • Theorem 4.1
  • ...and 8 more