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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 |
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 and Related Fields,
2021, 11
(3)
: 521-554.
doi: 10.3934/mcrf.2020052
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References:
| [1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New YorkLondon, 1975. |
| [2] |
V. Barbu, Optimal Control of Variational Inequalities, vol. 100 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
| [3] |
A. Bejan, Convection Heat Transfer, 4th edition, J. Wiley & Sons, 2013.
doi: 10.1002/9781118671627. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-9982-5. |
| [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. |
| [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. |
| [14] |
P. Fusek, D. Klatte and B. Kummer,
Examples and counterexamples in Lipschitz analysis, Control Cybernet., 31 (2002), 471-492.
|
| [15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2011. |
| [16] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub, Program, 1985. |
| [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. |
| [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. |
| [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. |
| [20] |
A. Logg, G. N. Wells and J. Hake, DOLFIN: A C++/Python Finite Element Library, Springer, 2012. |
| [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. |
| [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. |
| [23] |
P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems, Theory, Algorithms and Applications, Marcel Dekker, 1994. |
| [24] |
S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer Science & Business Media, 2012.
doi: 10.1007/978-1-4614-4340-7. |
| [25] |
D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer-Verlag, Berlin, Heidelberg, 1990.
doi: 10.1007/BFb0085564. |
| [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. |
| [27] |
A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4078-5. |
| [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. |
| [29] |
Y. B. Zel'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press. |
show all references
References:
| [1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New YorkLondon, 1975. |
| [2] |
V. Barbu, Optimal Control of Variational Inequalities, vol. 100 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
| [3] |
A. Bejan, Convection Heat Transfer, 4th edition, J. Wiley & Sons, 2013.
doi: 10.1002/9781118671627. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-9982-5. |
| [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. |
| [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. |
| [14] |
P. Fusek, D. Klatte and B. Kummer,
Examples and counterexamples in Lipschitz analysis, Control Cybernet., 31 (2002), 471-492.
|
| [15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2011. |
| [16] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub, Program, 1985. |
| [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. |
| [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. |
| [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. |
| [20] |
A. Logg, G. N. Wells and J. Hake, DOLFIN: A C++/Python Finite Element Library, Springer, 2012. |
| [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. |
| [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. |
| [23] |
P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems, Theory, Algorithms and Applications, Marcel Dekker, 1994. |
| [24] |
S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer Science & Business Media, 2012.
doi: 10.1007/978-1-4614-4340-7. |
| [25] |
D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer-Verlag, Berlin, Heidelberg, 1990.
doi: 10.1007/BFb0085564. |
| [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. |
| [27] |
A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4078-5. |
| [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. |
| [29] |
Y. B. Zel'dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press. |
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 $\beta$ | ||||||
| $n_h$ | $\alpha$ | $\beta$ | $\frac{\| y_h - \bar y\|_{H^1_0(\Omega)}}{\|\bar y\|_{H^1_0(\Omega)}}$ | $\frac{\| w_h - \bar w\|_{H^1_0(\Omega)}}{\|\bar w\|_{H^1_0(\Omega)}}$ | #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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