Stabilizing particles across scales
Presenter
May 8, 2024
Abstract
This talk explores the control of interacting particle systems to desired stationary configurations across scales, connecting microscopic particle dynamics with macroscopic mean-field descriptions. We discuss two approaches: stabilizing the McKean-Vlasov PDE around unstable steady states and optimally controlling consensus-based optimization (CBO) dynamics.
For interacting particle systems and their mean-field limit governed by the McKean-Vlasov PDE, we propose a numerical method combining spectral Galerkin approximation with deflated Newton's method to identify multiple steady states. The deflation technique systematically eliminates known solutions, enabling the discovery of distinct stationary configurations. To stabilize the particle ensemble around desired unstable steady states, we formulate an optimal control problem, where the control enters as an additional drift term. We derive optimality conditions and propose a gradient-based algorithm, employing model predictive control.
We also introduce a controlled CBO framework that incorporates a feedback control term derived from the numerical solution of an auxiliary Hamilton-Jacobi-Bellman equation. This control guides particles towards the global minimizer of the objective function. We establish the well-posedness of the controlled CBO system and demonstrate its improved performance over standard CBO methods.