Indirect methods in optimal control on Banach spaces

TL;DR

This paper develops a new indirect descent method for optimal control problems in Banach spaces, addressing limitations of classical schemes by ensuring stable monotone convergence, demonstrated through neural field control applications.

Contribution

It introduces an alternative approach based on exact cost-increment formulas and finite differences, improving convergence stability over traditional methods.

Findings

Method exhibits stable monotone convergence in neural field control

Revisits classical Pontryagin-based schemes highlighting their limitations

Demonstrates effectiveness through numerical analysis

Abstract

This work focuses on indirect descent methods for optimal control problems governed by nonlinear ordinary differential equations in Banach spaces, viewed as abstract models of distributed dynamics. As a reference line, we revisit the classical schemes, rooted in Pontryagin's maximum principle, and highlight their sensitivity to local convexity and lack of monotone convergence. We then develop an alternative method based on exact cost-increment formulas and finite-difference probes of the terminal cost. We show that our method exhibits stable monotone convergence in numerical analysis of an Amari-type neural field control problem.