Indirect methods in optimal control on Banach spaces

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.