Error analysis for global minima of semilinear optimal control problems

Ahmad Ahmad Ali; Klaus Deckelnick; Michael Hinze

doi:10.3934/mcrf.2018009

Article Contents

2018, Volume 8, Issue 1: 195-215. Doi: 10.3934/mcrf.2018009

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Error analysis for global minima of semilinear optimal control problems

1.
Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55,20146 Hamburg, Germany
2.
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2,39106 Magdeburg, Germany

* Corresponding author: Michael Hinze
* Corresponding author: Michael Hinze

Received: March 31, 2017

Revised: August 31, 2017

Published: January 2018

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Abstract

In [2] we consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization, where we provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. With the present work we complement our investigations of [2] in that we prove an error estimate for those discrete global solutions. Numerical experiments confirm our analytical findings.

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Mathematics Subject Classification: Primary: 65K10, 90C26; Secondary: 35Q93.

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Figure 1. Example 1: $\|\bar p_h\|_{L^4}$ and $\eta(\alpha)$ vs. $\alpha$.

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Figure 2. Example 1: The unique global minimum together with its state and the associated multipliers.

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Figure 3. Example 1: Errors for the optimal control and its state versus the mesh size.

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Figure 5. Example 2: The unique global minimum together with its state and the associated multipliers.

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Figure 6. Example 2: Errors for the optimal control and its state versus the mesh size.

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Table 1. Example 1: EOC for the optimal control and its state.

Levels	EOC$_{u_{L2}}$	EOC$_{y_{H1}}$	EOC$_{y_{L2}}$	EOC$_{y_{L\infty}}$
2-3	1.187645	0.833842	1.464788	1.058334
3-4	1.078183	0.948273	1.708822	1.758387
4-5	1.027290	0.985352	1.794456	1.657899
5-6	1.016702	0.997996	1.831198	1.514376

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Table 2. Example 2: EOC for the optimal control and its state.

Levels	EOC$_{u_{L2}}$	EOC$_{y_{H1}}$	EOC$_{y_{L2}}$	EOC$_{y_{L\infty}}$
2-3	1.910131	1.015420	1.686197	1.399920
3-4	1.983722	1.017850	1.934978	1.876141
4-5	1.982064	1.005320	1.996849	1.993434
5-6	1.944839	1.003374	2.035042	2.027269

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References

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