American Institute of Mathematical Sciences

Advanced
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:
[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

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $\beta$
Table Options

Download as excel

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

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011
[2]

Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control & Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289
[3]

Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004
[4]

Mikhail I. Belishev, Aleksei F. Vakulenko. Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$. Evolution Equations & Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247
[5]

Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078
[6]

Paul Glendinning. Non-smooth pitchfork bifurcations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 457-464. doi: 10.3934/dcdsb.2004.4.457
[7]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[8]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[9]

Roberto Triggiani. Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space. Evolution Equations & Control Theory, 2016, 5 (4) : 489-514. doi: 10.3934/eect.2016016
[10]

Xiaoshan Chen, Xun Li, Fahuai Yi. Optimal stopping investment with non-smooth utility over an infinite time horizon. Journal of Industrial & Management Optimization, 2019, 15 (1) : 81-96. doi: 10.3934/jimo.2018033
[11]

Luis Bayón, Jose Maria Grau, Maria del Mar Ruiz, Pedro Maria Suárez. A hydrothermal problem with non-smooth Lagrangian. Journal of Industrial & Management Optimization, 2014, 10 (3) : 761-776. doi: 10.3934/jimo.2014.10.761
[12]

Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559
[13]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073
[14]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[15]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[16]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011
[17]

Nicola Gigli, Sunra Mosconi. The Abresch-Gromoll inequality in a non-smooth setting. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1481-1509. doi: 10.3934/dcds.2014.34.1481
[18]

Yanni Xiao, Tingting Zhao, Sanyi Tang. Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Mathematical Biosciences & Engineering, 2013, 10 (2) : 445-461. doi: 10.3934/mbe.2013.10.445
[19]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063
[20]

Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (78)
  • HTML views (232)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences