Steering Large Agent Populations Using Mean-Field Schrödinger Bridges With Gaussian Mixture Models

We address the problem of controlling a swarm of identical, interacting cooperative agents, as captured by the time-varying probability measure of their state. Available methods for solving this problem for distributions with continuous support rely either on spatial discretizations of the problem’s domain or on approximating optimal solutions using neural networks trained through stochastic optimization schemes. For agents following Linear Time Varying dynamics, and for Gaussian Mixture Model boundary distributions, we propose a highly efficient parameterization to approximate the optimal solutions of the corresponding Mean-Field Schrödinger Bridge (MFSB) in closed form, without any learning step. Our proposed approach consists of a mixture of elementary policies, each solving a Gaussian-to-Gaussian Covariance Steering problem from the components of the initial mixture to the components of the terminal mixture. Leveraging the semidefinite formulation of the Covariance Steering problem, the proposed solver can handle probabilistic constraints on the system’s state while maintaining numerical tractability. We illustrate our approach on a variety of numerical examples.