# American Institute of Mathematical Sciences

September  2021, 11(3): 521-554. doi: 10.3934/mcrf.2020052

## 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.

Received  October 2018 Revised  December 2018 Published  September 2021 Early access  December 2020

Fund Project: 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".

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.

Citation: Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2021, 11 (3) : 521-554. doi: 10.3934/mcrf.2020052
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##### References:
constructed exact solution for $\alpha = 10^{-7}$, $\beta = 0.85$
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 $\beta$
 $n_h$ $\alpha$ $\beta$ $\frac{\| y_h - \bar y\|_{H^1_0(\Omega)}}{\|\bar y\|_{H^1_0(\Omega)}}$ $\frac{\| w_h - \bar w\|_{H^1_0(\Omega)}}{\|\bar w\|_{H^1_0(\Omega)}}$ #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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