Variational discretization of a control-constrained parabolic bang-bang optimal control problem

TL;DR

This paper develops a variational discretization approach for a control-constrained parabolic optimal control problem, providing error estimates and numerical validation, especially for bang-bang solutions.

Contribution

It introduces a novel variational discretization method for the problem without Tikhonov regularization, with improved error estimates for bang-bang solutions.

Findings

Established a-priori error estimates for the discretized problem.

Derived robust error estimates that remain valid as regularization parameter tends to zero.

Numerical example confirms the theoretical error bounds and effectiveness of the approach.

Abstract

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional. For the numerical treatment, we use variational discretization of its Tikhonov regularization: For the state and the adjoint equation, we apply Petrov-Galerkin schemes from [Daniels et al 2015] in time and usual conforming finite elements in space. We prove a-priori estimates for the error between the discretized regularized problem and the limit problem. Since these estimates are not robust if the regularization parameter tends to zero, we establish robust estimates, which --- depending on the problem's regularity --- enhance the previous ones. In the special case of bang-bang solutions, these estimates are further improved. A numerical example confirms our analytical findings.