A Lagrange Multiplier Method for Semilinear Elliptic State Constrained Optimal Control Problems

TL;DR

This paper develops an augmented Lagrange method for solving semilinear elliptic optimal control problems with pointwise state constraints, demonstrating convergence properties and providing numerical results.

Contribution

It introduces a novel Lagrange multiplier approach with proven convergence for a class of constrained elliptic control problems.

Findings

Strong convergence of primal variables to local solutions

Weak convergence of adjoint states and multipliers

Numerical experiments validating the method

Abstract

In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.