Efficient Solution of State-Constrained Distributed Parabolic Optimal Control Problems

TL;DR

This paper presents an efficient space-time finite element approach for solving heat equation-based optimal control problems with state constraints, utilizing anisotropic Sobolev norms and semi-smooth Newton methods.

Contribution

It introduces a novel finite element discretization and iterative solution technique for state-constrained parabolic optimal control problems.

Findings

Efficient realization of anisotropic Sobolev norms in finite element meshes

Development of a semi-smooth Newton active set method for variational inequalities

Demonstration of the method's effectiveness in numerical experiments

Abstract

We consider a space-time finite element method for the numerical solution of a distributed tracking-type optimal control problem subject to the heat equation with state constraints. The cost or regularization term is formulated in an anisotropic Sobolev norm for the state, and the optimal state is then characterized as the unique solution of a first kind variational inequality. We discuss an efficient realization of the anisotropic Sobolev norm in the case of a space-time tensor-product finite element mesh, and the iterative solution of the resulting discrete variational inequality by means of a semi-smooth Newton method, i.e., using an active set strategy.