Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach

Engel Sebastian; Kunisch Karl

doi:10.3934/mcrf.2020012

Article Contents

2020, Volume 10, Issue 3: 591-622. Doi: 10.3934/mcrf.2020012

This issue Previous Article A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices Next Article Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix

Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach

Engel Sebastian^1, , and
Kunisch Karl^2,

1.
Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Heinrichstr. 36, Graz, 8010, Austria
2.
Radon Institute, Austrian Academy of Sciences, and Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Heinrichstr. 36, Graz, 8010, Austria

^* Corresponding author: Sebastian Engel
^* Corresponding author: Sebastian Engel

Received: September 30, 2018

Revised: August 31, 2019

Early access: December 2019

Published: July 2020

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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 $ H^1 $ functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the $ H^1- $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.

Keywords:

Mathematics Subject Classification: Primary: 26A45, 35L05, 49J20, 49J52; Secondary: 35L10, 49K20.

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Figure 4. In this figure we see one possible shape for $ \overline{u}_i $

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