Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations

Frank Pörner; Daniel Wachsmuth

doi:10.3934/mcrf.2018013

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2018, Volume 8, Issue 1: 315-335. Doi: 10.3934/mcrf.2018013

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Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations

Frank Pörner and
Daniel Wachsmuth^,

Institut für Mathematik, Universität Würzburg, D-97974 Würzburg, Germany

* Corresponding author: Daniel Wachsmuth
* Corresponding author: Daniel Wachsmuth

Received: March 31, 2017

Revised: July 31, 2017

Published: January 2018

This work was supported by the German Research Foundation DFG under project grant Wa 3626/1-1.

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Abstract

In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under suitable conditions. These conditions comprise second-order sufficient optimality conditions as well as regularity conditions on the control, which consists of a source condition and a condition on the active sets. In addition, we show that these conditions are necessary for convergence rates under certain conditions. We also consider sparse optimal control problems and derive regularization error estimates for them. Numerical experiments underline the theoretical findings.

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Mathematics Subject Classification: Primary: 49M20; Secondary: 49K20, 49N45.

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Figure 1. Error $\|u_\alpha - \bar u\|_{L^2(\Omega)}$ for $f(y) = \text{exp}(y)$ in a double logarithmic plot for different values for $h$ and $\alpha$. For comparison we plotted a triangle with slope $\frac{1}{2}$.

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