    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  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, doi: 10.3934/mcrf.2020052
##### References:
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Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar  C. Christof, C. Clason, C. Meyer and S. Walter, Optimal control of a non-smooth semilinear elliptic equation, Mathematical Control and Related Fields, 8 (2018), 247-276.  doi: 10.3934/mcrf.2018011.  Google Scholar  C. Clason and K. Kunisch, A convex analysis approach to multi-material topology optimization, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016), 1917-1936.  doi: 10.1051/m2an/2016012.  Google Scholar  P. Fusek, D. Klatte and B. Kummer, Examples and counterexamples in Lipschitz analysis, Control Cybernet., 31 (2002), 471-492. Google Scholar  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2011. Google Scholar  P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub, Program, 1985. Google Scholar  R. Herzog, C. Meyer and G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening, SIAM J. Optim., 23 (2013), 321-352.  doi: 10.1137/110821147.  Google Scholar  K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, SIAM, 2008. doi: 10.1137/1.9780898718614.  Google Scholar  A. Logg and G. N. Wells, DOLFIN: automated finite element computing, ACM Transactions on Mathematical Software, 37 (2010), Art. 20, 28pp. doi: 10.1145/1731022.1731030.  Google Scholar  A. Logg, G. N. Wells and J. Hake, DOLFIN: A C++/Python Finite Element Library, Springer, 2012. Google Scholar  C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM J. Control Optim., 55 (2017), 2206-2234.  doi: 10.1137/15M1040426.  Google Scholar  C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-69952-1.  Google Scholar  P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems, Theory, Algorithms and Applications, Marcel Dekker, 1994. Google Scholar  S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-4340-7.  Google Scholar  D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/BFb0085564.  Google Scholar  M. Ulbrich, Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, SIAM J. Optim., 13 (2002), 805-842.  doi: 10.1137/S1052623400371569.  Google Scholar  A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.  Google Scholar  E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar  Y. B. Zel'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press. Google Scholar

show all references

##### References:
  R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New YorkLondon, 1975. Google Scholar  V. Barbu, Optimal Control of Variational Inequalities, vol. 100 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984. Google Scholar  A. Bejan, Convection Heat Transfer, 4th edition, J. Wiley & Sons, 2013. doi: 10.1002/9781118671627. Google Scholar  A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/978-3-662-12905-0.  Google Scholar  L. M. Betz, Second-order sufficient optimality conditions for optimal control of non-smooth, semilinear parabolic equations, SIAM Journal on Control and Optimization, 57 (2019), 4033-4062.  doi: 10.1137/19M1239106.  Google Scholar  J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, Berlin, Heidelberg, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar  E. Casas and V. Dhamo, Error estimates for the numerical approximation of a quasilinear Neumann problem under minimal regularity of the data, Numer. Math., 117 (2011), 115-145.  doi: 10.1007/s00211-010-0344-1.  Google Scholar  E. Casas and V. Dhamo, Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations, Comput. Optim. Appl., 52 (2012), 719-756.  doi: 10.1007/s10589-011-9440-0.  Google Scholar  E. Casas and F. Tröltzsch, First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J. Control Optim., 48 (2009), 688-718.  doi: 10.1137/080720048.  Google Scholar  E. Casas and F. Tröltzsch, Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations, ESAIM Control Optim. Calc. Var., 17 (2011), 771-800.  doi: 10.1051/cocv/2010025.  Google Scholar  M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar  C. Christof, C. Clason, C. Meyer and S. Walter, Optimal control of a non-smooth semilinear elliptic equation, Mathematical Control and Related Fields, 8 (2018), 247-276.  doi: 10.3934/mcrf.2018011.  Google Scholar  C. Clason and K. Kunisch, A convex analysis approach to multi-material topology optimization, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016), 1917-1936.  doi: 10.1051/m2an/2016012.  Google Scholar  P. Fusek, D. Klatte and B. Kummer, Examples and counterexamples in Lipschitz analysis, Control Cybernet., 31 (2002), 471-492. Google Scholar  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2011. Google Scholar  P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub, Program, 1985. Google Scholar  R. Herzog, C. Meyer and G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening, SIAM J. Optim., 23 (2013), 321-352.  doi: 10.1137/110821147.  Google Scholar  K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, SIAM, 2008. doi: 10.1137/1.9780898718614.  Google Scholar  A. Logg and G. N. Wells, DOLFIN: automated finite element computing, ACM Transactions on Mathematical Software, 37 (2010), Art. 20, 28pp. doi: 10.1145/1731022.1731030.  Google Scholar  A. Logg, G. N. Wells and J. Hake, DOLFIN: A C++/Python Finite Element Library, Springer, 2012. Google Scholar  C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM J. Control Optim., 55 (2017), 2206-2234.  doi: 10.1137/15M1040426.  Google Scholar  C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-69952-1.  Google Scholar  P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems, Theory, Algorithms and Applications, Marcel Dekker, 1994. Google Scholar  S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-4340-7.  Google Scholar  D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/BFb0085564.  Google Scholar  M. Ulbrich, Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, SIAM J. Optim., 13 (2002), 805-842.  doi: 10.1137/S1052623400371569.  Google Scholar  A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.  Google Scholar  E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar  Y. B. Zel'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press. Google Scholar
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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