Compatibility of state constraints and dynamics for multiagent control systems

Giulia Cavagnari,Antonio Marigonda,Marc Quincampoix

doi:10.1007/s00028-021-00724-z

Abstract

This study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on {mathbb {R}}^d representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton–Jacobi–Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton–Jacobi equations in the Wasserstein space.

Highlights

In classical control theory, a single agent controls a dynamics x(t) ∈ F(x(t)), x(0) = x0, (1.1)where F : Rd ⇒ Rd is a set valued map, associating with each x ∈ Rd the subsetF(x) of Rd of the admissible velocities from x
The evolution of the controlled multi-agent system can be represented by the following two-scale dynamics
In the framework of mean field approximation of multi agent system, starting from a control problem for a large number of the agents, the problem is recasted in the framework of probability measures

Summary

Introduction

In the framework of mean field approximation of multi agent system, starting from a control problem for a large number of the (discrete) agents, the problem is recasted in the framework of probability measures (see the recent [15] or the preprint [12] for -convergence results for optimal control problems with nonlocal dynamics). This procedure reduces the dimensionality and the number of equations, possibly leading to a simpler and treatable problem from the point of view of numerics.

Notations

Admissible trajectories

Viability problem and the value function

Hamilton Jacobi Bellman equation

Viscosity characterization of viability and invariance

An example