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Optimal control of a non-smooth quasilinear elliptic equation

  • * Corresponding author: Christian Clason.

    * Corresponding author: Christian Clason. 
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".
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  • 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.

    Mathematics Subject Classification: 49K20, 49J52, 49M15.

    Citation:

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  • Figure 1.  constructed exact solution for $ \alpha = 10^{-7} $, $ \beta = 0.85 $

    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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  • [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. ChristofC. ClasonC. 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. FusekD. 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. HerzogC. 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.
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