Augmented Lagrange method for optimal control problems of parabolic equation with state constraints

TL;DR

This paper develops an augmented Lagrange method combined with the Method of Successive Approximations to solve optimal control problems with state constraints in parabolic equations, demonstrating convergence and validating through numerical experiments.

Contribution

It introduces a novel augmented Lagrange algorithm for constrained parabolic control problems with proven convergence properties and practical validation.

Findings

Strong convergence of primal variables established

Weak convergence of dual variables proven

Numerical experiments confirm algorithm effectiveness

Abstract

The augmented Lagrange method is employed to address the optimal control problem involving pointwise state constraints in parabolic equations. The strong convergence of the primal variables and the weak convergence of the dual variables are rigorously established. The sub-problems arising in the algorithm are solved using the Method of Successive Approximations (MSA), derived from Pontryagin's principle. Numerical experiments are provided to validate the convergence of the proposed algorithm.