Mean-Field Sparse Jurdjevic--Quinn Control

TL;DR

This paper develops a control method for nonlinear mean-field equations with non-local velocity, using a sparse, Lyapunov-based approach inspired by the Jurdjevic--Quinn procedure, applicable to multi-agent kinetic models.

Contribution

It extends the classical Jurdjevic--Quinn stabilization method to infinite-dimensional mean-field equations with sparsity constraints on control support.

Findings

Control support can be made arbitrarily small while stabilizing the system.

The method applies to a broad class of kinetic equations modeling collective dynamics.

The approach ensures dissipativity and stabilization using Lyapunov functions.

Abstract

We consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, we address the problem of controlling such equations by means of a time-varying bounded control action localized on a time-varying control subset with bounded Lebesgue measure (sparsity space constraint). Finite-dimensional versions are given by control-affine systems, which can be stabilized by the well known Jurdjevic--Quinn procedure. In this paper, assuming that the uncontrolled dynamics are dissipative, we develop an approach in the spirit of the classical Jurdjevic--Quinn theorem, showing how to steer the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function, and enjoy sparsity properties in the sense that the control support is small. Finally, we show that our result applies to a large class of kinetic equations modelling multi-agent dynamics.