Computation and Control of Unstable Steady States for Mean Field Multiagent Systems

TL;DR

This paper introduces a numerical framework to identify and control all steady states, including unstable ones, in mean field multiagent systems, enabling better understanding and manipulation of complex collective behaviors.

Contribution

The authors develop an efficient spectral Galerkin-based numerical scheme combined with deflated Newton's method to find all steady states of McKean-Vlasov PDEs, including unstable equilibria, and propose an optimal control strategy to steer systems toward desired states.

Findings

Successfully identified all steady states, stable and unstable, in example models.

Demonstrated stabilization of unstable equilibria using receding horizon control.

Validated approach on opinion dynamics and biophysical models.

Abstract

We study interacting particle systems driven by noise, modeling phenomena such as opinion dynamics. We are interested in systems that exhibit phase transitions i.e. non-uniqueness of stationary states for the corresponding McKean-Vlasov PDE, in the mean field limit. We develop an efficient numerical scheme for identifying all steady states (both stable and unstable) of the mean field McKean-Vlasov PDE, based on a spectral Galerkin approximation combined with a deflated Newton's method to handle the multiplicity of solutions. Having found all possible equilibra, we formulate an optimal control strategy for steering the dynamics towards a chosen unstable steady state. The control is computed using iterated open-loop solvers in a receding horizon fashion. We demonstrate the effectiveness of the proposed steady state computation and stabilization methodology on several examples, including the noisy Hegselmann-Krause model for opinion dynamics and the Haken-Kelso-Bunz model from biophysics. The numerical experiments validate the ability of the approach to capture the rich self-organization landscape of these systems and to stabilize unstable configurations of interest. The proposed computational framework opens up new possibilities for understanding and controlling the collective behavior of noise-driven interacting particle systems, with potential applications in various fields such as social dynamics, biological synchronization, and collective behavior in physical and social systems.