Maximal discrete sparsity in parabolic optimal control with measures

TL;DR

This paper develops a new discretization method for parabolic optimal control problems with measure controls, proving convergence and demonstrating advantages through numerical experiments.

Contribution

It introduces a Petrov-Galerkin discretization approach for measure controls in parabolic problems, ensuring strong convergence of states and weak-* convergence of controls.

Findings

Strong convergence of discrete states in $L^q$

Weak-* convergence of controls in measures

Numerical experiments validate the approach

Abstract

We consider variational discretization of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in $L^q$ and weakly-$*$ in $\mathcal{M}$, respectively, to their smooth counterparts, where $q \in (1,\min\{2,1+2/d\}]$ is the spatial dimension. Furthermore, we compare our approach to a approach by Casas, E. and Kunisch, K., where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.