Iterative hard-thresholding applied to optimal control problems with $L^0(\Omega)$ control cost

TL;DR

This paper explores the application of the hard-thresholding method to optimal control problems with an $L^0$ control cost, proving convergence properties and demonstrating robustness and superiority over $L^1$ regularization through numerical experiments.

Contribution

It introduces a novel convergence analysis for the hard-thresholding method in $L^0$ control problems and compares its effectiveness to $L^1$ regularization.

Findings

Objective functional values are lower semicontinuous along iterates.

Weak limit points are also strong limit points under certain conditions.

The method shows robustness and superior solutions compared to $L^1$ regularization.

Abstract

We investigate the hard-thresholding method applied to optimal control problems with $L^0(\Omega)$ control cost, which penalizes the measure of the support of the control. As the underlying measure space is non-atomic, arguments of convergence proofs in $l^2$ or $\mathbb R^n$ cannot be applied. Nevertheless, we prove the surprising property that the values of the objective functional are lower semicontinuous along the iterates. That is, the function value in a weak limit point is less or equal than the lim-inf of the function values along the iterates. Under a compactness assumption, we can prove that weak limit points are strong limit points, which enables us to prove certain stationarity conditions for the limit points. Numerical experiments are carried out, which show the performance of the method. These indicates that the method is robust with respect to discretization. In addition, we show that solutions obtained by the thresholding algorithm are superior to solutions of $L^1(\Omega)$-regularized problems.