Exponential integrators for mean-field selective optimal control problems

TL;DR

This paper develops exponential integrator-based algorithms for solving mean-field optimal control problems with selective actions, involving non-local continuity equations and diffusion, demonstrated through applications in opinion formation and pedestrian dynamics.

Contribution

It introduces a novel exponential integrator approach for efficiently solving the forward-backward optimality system in mean-field control problems with non-local terms.

Findings

Effective numerical algorithms for mean-field control problems

Successful application to opinion formation models

Enhanced computational efficiency with exponential integrators

Abstract

In this paper we consider mean-field optimal control problems with selective action of the control, where the constraint is a continuity equation involving a non-local term and diffusion. First order optimality conditions are formally derived in a general framework, accounting for boundary conditions. Hence, the optimality system is used to construct a reduced gradient method, where we introduce a novel algorithm for the numerical realization of the forward and the backward equations, based on exponential integrators. We illustrate extensive numerical experiments on different control problems for collective motion in the context of opinion formation and pedestrian dynamics.