Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation

TL;DR

This paper introduces a variational discretization method for one-dimensional elliptic optimal control problems involving BV functions, utilizing mixed formulations and Raviart-Thomas finite elements, with proven error estimates and numerical validation.

Contribution

It develops a novel variational discretization approach combining mixed formulations and Raviart-Thomas elements for BV control problems, providing error analysis and numerical confirmation.

Findings

Error estimates for the discretization are established.

Numerical experiments confirm the theoretical error bounds.

The method effectively handles BV control functions in elliptic problems.

Abstract

We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments.