Optimal control of a non-smooth quasilinear elliptic equation

Christian Clason; Vu Huu Nhu; Arnd Rösch

doi:10.3934/mcrf.2020052

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2021, Volume 11, Issue 3: 521-554. Doi: 10.3934/mcrf.2020052

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Optimal control of a non-smooth quasilinear elliptic equation

Faculty of Mathematics, University Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

^* Corresponding author: Christian Clason.
^* Corresponding author: Christian Clason.

Received: September 30, 2018

Revised: November 30, 2018

Early access: December 2020

Published: September 2021

This work was supported by the DFG under the grants CL 487/2-1 and RO 2462/6-1, both within the priority programme SPP 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

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Abstract

This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a $ PC^1 $ function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.

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Mathematics Subject Classification: 49K20, 49J52, 49M15.

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Figure 1. constructed exact solution for $ \alpha = 10^{-7} $, $ \beta = 0.85 $

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Table 1. numerical results. number of Newton iterations and relative errors for state $\bar y$ and adjoint $\bar w$ in dependence of $n_h$, $\alpha$, and $\beta$

$n_h$	$\alpha$	$\beta$	$\frac{\\| y_h - \bar y\\|_{H^1_0(\Omega)}}{\\|\bar y\\|_{H^1_0(\Omega)}}$	$\frac{\\| w_h - \bar w\\|_{H^1_0(\Omega)}}{\\|\bar w\\|_{H^1_0(\Omega)}}$	#SSN	$\\|y_d \\|_{L^\infty(\Omega)} $
100	1·10⁻⁶	0.8	3.27·10⁻³	2.92·10⁻²	2	2.07·10⁻⁴
200	1·10⁻⁶	0.8	1.66·10⁻³	1.54·10⁻²	4	2.07·10⁻⁴
400	1·10⁻⁶	0.8	8.36·10⁻⁴	7.92·10⁻³	3	2.07·10⁻⁴
800	1·10⁻⁶	0.8	4.19·10⁻⁴	4.03·10⁻³	3	2.07·10⁻⁴
1000	1·10⁻⁶	0.8	3.36·10⁻⁴	3.24·10⁻³	3	2.07·10⁻⁴
(A) dependence on $n_h$
$n_h$	$\alpha$	$\beta$	$\frac{\\| y_h - \bar y\\|_{H^1_0(\Omega)}}{\\|\bar y\\|_{H^1_0(\Omega)}}$	$\frac{\\| w_h - \bar w\\|_{H^1_0(\Omega)}}{\\|\bar w\\|_{H^1_0(\Omega)}}$	#SSN	$\\|y_d \\|_{L^\infty(\Omega)} $
800	1·10⁻²	0.8	6.36·10⁻²	1.36·10⁻²	4	9.83·10⁻²
800	1·10⁻⁴	0.8	8.76·10⁻³	7.32·10⁻³	3	9.83·10⁻²
800	1·10⁻⁶	0.8	4.19·10⁻⁴	4.03·10⁻³	3	2.07·10⁻⁴
800	1·10⁻⁸	0.8	2.32·10⁻⁵	2.19·10⁻³	25	2.03·10⁻⁴
(B) dependence on $\alpha$
$n_h$	$\alpha$	$\beta$	$\frac{\\| y_h - \bar y\\|_{H^1_0(\Omega)}}{\\|\bar y\\|_{H^1_0(\Omega)}}$	$\frac{\\| w_h - \bar w\\|_{H^1_0(\Omega)}}{\\|\bar w\\|_{H^1_0(\Omega)}}$	#SSN	$\\|y_d \\|_{L^\infty(\Omega)} $
800	1·10⁻⁵	0.5	7.03·10⁻³	1.20·10⁻²	3	1.50·10⁻⁵
800	1·10⁻⁵	0.7	2.68·10⁻³	7.11·10⁻³	3	1.07·10⁻⁴
800	1·10⁻⁵	0.9	1.41·10⁻³	4.27·10⁻³	4	5.07·10⁻⁴
800	1·10⁻⁵	1.0	8.65·10⁻⁵	3.39·10⁻³	6	9.50·10⁻⁴
(C) dependence on $\beta$

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