Maximal discrete sparsity in parabolic optimal control with measures

Evelyn Herberg; Michael Hinze; Henrik Schumacher

doi:10.3934/mcrf.2020018

Article Contents

2020, Volume 10, Issue 4: 735-759. Doi: 10.3934/mcrf.2020018

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Maximal discrete sparsity in parabolic optimal control with measures

1.
Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz, Universitätsstraße 1, 56070 Koblenz, Germany
2.
Institut für Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

^* Corresponding author: Evelyn Herberg
^* Corresponding author: Evelyn Herberg

Received: May 31, 2018

Revised: October 31, 2019

Early access: March 2020

Published: October 2020

The second author acknowledges support of the priority programme 1962 (SPP 1962) funded by the Deutsche Forschungsgemeinschaft

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Abstract

We consider variational discretization [18] of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in $ L^q $ and weakly-$ * $ in $ \mathcal{M} $, respectively, to their smooth counterparts, where $ q \in (1,\min\{2,1+2/d\}] $ is the spatial dimension. Furthermore, we compare our approach to [8], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.

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Mathematics Subject Classification: Primary: 49J20, 49M25, 65K10; Secondary: 49M29.

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Figure 1. Numerical setup on $ 4 \times 12 $ space-time grid with $ q = \tfrac{4}{3} $. From left to right: control $ u = \delta _{(1/2,1/2)} $, associated state $ y(u) $ (sampled from the analytic solution with spacial Fourier modes), discrete desired state $ y_{\operatorname{d}} $ and calculated controls $ u_{\sigma,0} $ and $ u_{\operatorname{DG},0} $ for $ \alpha = 0 $. Here the controls are represented by their coefficients. In the case of piecewise constant controls in time, which are used in the discontinuous Galerkin setting, this leads to coefficient values, which are scaled with $ \tfrac{1}{\tau} = 8 $

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Figure 2. The dependence on the penalty parameter $ \alpha $ of the measure norm of $ u_{\sigma,\alpha} $ and $ u_{\operatorname{DG},\alpha} $ (left) and the errors $ y_{\sigma,\alpha} - y_{\operatorname{d}} $ and $ y_{\operatorname{DG},\alpha} - y_{\operatorname{d}} $ in the $ L^{4/3} $ norm (right)

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Figure 3. Top row: The measure control and the optimal controls $ u_{\sigma,0.456} $ and $ u_{\operatorname{DG},0.456} $. Bottom row: The associated state $ y(u) $ (sampled from the analytic solution with spacial Fourier modes) and the associated states $ y_{\sigma,0.456} $ and $ y_{\operatorname{DG},0.456} $

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Figure 4. Numerical setup on a $ 4 \times 48 $ space-time grid with $ q = \tfrac{4}{3} $. From left to right: true control $ u_{\operatorname{true}} = \delta _{(1/2,1/2)} $, interpolation of the associated state $ y(u_{\operatorname{true}}) $, adjoint state $ w_{\bar{\alpha}} $ multiplied by $ -1 $ for easier visualization and desired state $ y_{\operatorname{d}} $ calculated using Fenchel duality

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Figure 5. Top row: The difference of the measure norms of the true control $ u_{\operatorname{true}} $ and calculated optimal control $ u_i $, $ i \in \{\sigma, \operatorname{DG}\} $ for $ \tau = \tfrac{h}{2} $ (left) and $ \tau = \tfrac{h^2}{2} $ (right). Bottom row: The $ L^{4/3} $ norm of the difference of the associated states $ y_i - y_{\operatorname{true}} $, $ i \in \{\sigma, \operatorname{DG}\} $ for $ \tau = \tfrac{h}{2} $ (left) and $ \tau = \tfrac{h^2}{2} $ (right)

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