A topological derivative-based algorithm to solve optimal control problems with $L^0(\Omega)$ control cost

TL;DR

This paper introduces a novel topological derivative-based algorithm for solving optimal control problems with $L^0$ control cost, focusing on the support of controls as an optimization variable, and demonstrates its convergence properties.

Contribution

The paper develops a new gradient descent algorithm utilizing topological derivatives for $L^0$ control cost problems, which is a novel approach in this context.

Findings

Algorithm produces a minimizing sequence under certain assumptions.

Topological derivatives effectively guide support modifications in control optimization.

The method offers a new perspective on sparse control problems with $L^0$ costs.

Abstract

In this paper, we consider optimization problems with $L^0$-cost of the controls. Here, we take the support of the control as independent optimization variable. Topological derivatives of the corresponding value function with respect to variations of the support are derived. These topological derivatives are used in a novel gradient descent algorithm with Armijo line-search. Under suitable assumptions, the algorithm produces a minimizing sequence.