# Optimal control of nonlocal continuity equations: numerical solution

arXiv: https://arxiv.org/abs/2212.06608 · HTML: https://mlsanity.com/paper/2212.06608/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2022.

## Built on by (most cited first, 1 of 1)

- [A convergence result of a continuous model of deep learning via a Lojasiewicz--Simon inequality](https://mlsanity.com/paper/2311.15365/) (2023)

## Abstract (arXiv)

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.