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

Optimal control of a non-smooth quasilinear elliptic equation

Christian Clason , , Vu Huu Nhu and  Arnd Rösch

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.

Keywords: Optimal control, non-smooth optimization, optimality system, quasilinear elliptic equation.
Mathematics Subject Classification: 49K20, 49J52, 49M15.
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
References:
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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

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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

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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

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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

[10]

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

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M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar

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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

[13]

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

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P. Fusek, D. Klatte and B. Kummer, Examples and counterexamples in Lipschitz analysis, Control Cybernet., 31 (2002), 471-492.   Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2011.  Google Scholar

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P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub, Program, 1985.  Google Scholar

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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

[18]

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

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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

[20]

A. Logg, G. N. Wells and J. Hake, DOLFIN: A C++/Python Finite Element Library, Springer, 2012. Google Scholar

[21]

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

[22]

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

[23]

P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems, Theory, Algorithms and Applications, Marcel Dekker, 1994.  Google Scholar

[24]

S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-4340-7.  Google Scholar

[25]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/BFb0085564.  Google Scholar

[26]

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

[27]

A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.  Google Scholar

[28]

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

[29]

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:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New YorkLondon, 1975.  Google Scholar

[2]

V. Barbu, Optimal Control of Variational Inequalities, vol. 100 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

[3]

A. Bejan, Convection Heat Transfer, 4th edition, J. Wiley & Sons, 2013. doi: 10.1002/9781118671627.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar

[12]

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

[13]

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

[14]

P. Fusek, D. Klatte and B. Kummer, Examples and counterexamples in Lipschitz analysis, Control Cybernet., 31 (2002), 471-492.   Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2011.  Google Scholar

[16]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub, Program, 1985.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

A. Logg, G. N. Wells and J. Hake, DOLFIN: A C++/Python Finite Element Library, Springer, 2012. Google Scholar

[21]

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

[22]

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

[23]

P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems, Theory, Algorithms and Applications, Marcel Dekker, 1994.  Google Scholar

[24]

S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-4340-7.  Google Scholar

[25]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/BFb0085564.  Google Scholar

[26]

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

[27]

A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.  Google Scholar

[28]

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

[29]

Y. B. Zel'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press. Google Scholar

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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$n_h$ $\alpha$ $\beta$ $\frac{\| y_h - \bar y\|_{H^1_0(\Omega)}}{\|\bar y\|_{H^1_0(\Omega)}}$ $\frac{\| w_h - \bar w\|_{H^1_0(\Omega)}}{\|\bar w\|_{H^1_0(\Omega)}}$ #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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