Pontryagin's Principle Based Algorithms for Optimal Control Problems of Parabolic Equation

TL;DR

This paper develops Pontryagin's principle-based algorithms, including an augmented MSA with convergence proof, to solve constrained optimal control problems for semilinear parabolic equations, validated by numerical experiments.

Contribution

It introduces an augmented MSA algorithm with convergence analysis for state-constrained parabolic control problems, advancing existing methods.

Findings

Error estimates for derivatives under bounded conditions

Convergence of the augmented MSA algorithm proven

Numerical experiments demonstrate effectiveness

Abstract

This paper applies the Method of Successive Approximations (MSA) based on Pontryagin's principle to solve optimal control problems with state constraints for semilinear parabolic equations. Error estimates for the first and second derivatives of the function are derived under \( L^{\infty} \)-bounded conditions. An augmented MSA is developed using the augmented Lagrangian method, and its convergence is proven. The effectiveness of the proposed method is demonstrated through numerical experiments.