Optimal Control of the Linear Wave Equation by Time-Depending BV-Controls: A Semi-Smooth Newton Approach

TL;DR

This paper develops a semi-smooth Newton method for solving an optimal control problem of the linear wave equation with BV semi-norm controls, promoting piecewise constant controls and analyzing their properties.

Contribution

It introduces a novel approach combining BV controls with a semi-smooth Newton algorithm and investigates the asymptotic behavior of regularized solutions.

Findings

Optimal controls can be a mix of different measure types.

Semi-smooth Newton method achieves super-linear convergence.

Numerical examples confirm theoretical analysis.

Abstract

An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by H1 functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the H1 regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the distributional derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.