Optimal control of nonlocal continuity equations: numerical solution

TL;DR

This paper develops a numerical method for solving optimal control problems involving nonlocal continuity equations on probability measures, using a new Pontryagin maximum principle formulation and demonstrating its effectiveness on oscillator models.

Contribution

It introduces a novel PMP-based descent method with a decoupled Hamiltonian system for nonlocal continuity equations, including convergence proof and practical implementation.

Findings

Convergence of the proposed numerical algorithm is proven.

The method is demonstrated on a Kuramoto-type oscillator model.

A new form of PMP for nonlocal measure-based control problems is derived.

Abstract

The paper addresses an optimal ensemble control problem for nonlocal continuity equations on the space of probability measures. We admit the general nonlinear cost functional, and an option to directly control the nonlocal terms of the driving vector field. For this problem, we design a descent method based on Pontryagin's maximum principle (PMP). To this end, we derive a new form of PMP with a decoupled Hamiltonian system. Specifically, we extract the adjoint system of linear nonlocal balance laws on the space of signed measures and prove its well-posedness. As an implementation of the designed descent method, we propose an indirect deterministic numeric algorithm with backtracking. We prove the convergence of the algorithm and illustrate its modus operandi by treating a simple case involving a Kuramoto-type model of a population of interacting oscillators.