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December  2016, 6(4): 595-628. doi: 10.3934/mcrf.2016017

## Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

 1 heSam Université, Conservatoire National des Arts et Métiers, M2N, Case 2D 5000, 292 rue Saint-Martin, 75003 Paris, France 2 Department of Differential Equations, Dnipropetrovsk National University, Gagarin av., 72, 49010 Dnipropetrovsk 3 heSam Université Conservatoire National des Arts et Métiers, M2N, Case 2D 5000, 292 rue Saint-Martin, 75003 Paris, France

Received  October 2015 Revised  July 2016 Published  October 2016

In this paper we study we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control $A(x)$ belongs to $L^2$-space (rather than $L^\infty)$. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.
Citation: 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
##### References:
 [1] R. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar [2] M. Briane and J. Casado-Diaz, Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients,, J. of Diff. Equa., 245 (2008), 2038.  doi: 10.1016/j.jde.2008.07.027.  Google Scholar [3] G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems,, Revista Matematica Complutense, 24 (2011), 83.  doi: 10.1007/s13163-010-0030-y.  Google Scholar [4] D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford University Press, (1999).   Google Scholar [5] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications,, AMS, (2000).   Google Scholar [6] A. D. Ioffe and V. M. Tichomirov, Theory of Extremal Problems,, North-Holland, (1979).   Google Scholar [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2000).   Google Scholar [8] T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic dirichlet boundary value problems,, Asymptotic Analysis, 98 (2016), 155.  doi: 10.3233/ASY-161365.  Google Scholar [9] T. Horsin and P. I. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. i. existence results,, Math. Control Relat. Fields, 5 (2015), 73.  doi: 10.3934/mcrf.2015.5.73.  Google Scholar [10] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications,, Academic Press, (1980).   Google Scholar [11] P. I. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients,, Descrete and Continuous Dynamical System, 34 (2014), 2105.  doi: 10.3934/dcds.2014.34.2105.  Google Scholar [12] P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis,, Birkhäuser, (2011).  doi: 10.1007/978-0-8176-8149-4.  Google Scholar [13] P. I. Kogut and G. Leugering, Optimal $L^1$-control in coefficients for dirichlet elliptic problems: W-optimal solutions,, Journal of Optimization Theory and Applications, 150 (2011), 205.  doi: 10.1007/s10957-011-9840-4.  Google Scholar [14] P. I. Kogut and G. Leugering, Optimal $L^1$-control in coefficients for dirichlet elliptic problems: H-optimal solutions,, Zeitschrift für Analysis und ihre Anwendungen, 31 (2012), 31.  doi: 10.4171/ZAA/1447.  Google Scholar [15] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications,, Springer-Verlag, (1972).   Google Scholar [16] T. Jin, V. Mazya and J. van Schaftinger, Pathological solutions to elliptic problems in divergence form with continuous coefficients,, C. R. Math. Acad. Sci. Paris, 347 (2009), 773.  doi: 10.1016/j.crma.2009.05.008.  Google Scholar [17] J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 3 (1964), 385.   Google Scholar [18] J. L. Vazquez and N. B. Zographopoulos, Functional aspects of the Hardy inequlity. Appearance of a hidden energy,, Journal of Evolution Equations, 12 (2012), 713.  doi: 10.1007/s00028-012-0151-5.  Google Scholar [19] J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. of Functional Analysis, 173 (2000), 103.  doi: 10.1006/jfan.1999.3556.  Google Scholar [20] V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156.  doi: 10.1007/BF02465783.  Google Scholar [21] V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189 (1998), 27.  doi: 10.1070/SM1998v189n08ABEH000344.  Google Scholar [22] V. V. Zhikov, Remarks on the uniqueness of a solution of the Dirichlet problem for second-order elliptic equations with lower-order terms,, Functional Analysis and Its Applications, 38 (2004), 173.  doi: 10.1023/B:FAIA.0000042802.86050.5e.  Google Scholar [23] V. V. Zhikov, private, communication., ().   Google Scholar [24] Ph. Destuynder, Analyse, Traitement et Synthèse D'images Numériques,, Hermes, (2006).   Google Scholar [25] O. Faugeras, Three-Dimensional Computer Vision,, MIT Press, (1993).   Google Scholar [26] D. Cioranescu and F. Murat, A strange term coming from nowhere,, in Topic in the Math. Modelling of Composit Materials, 31 (1997), 45.   Google Scholar

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##### References:
 [1] R. Adams, Sobolev Spaces,, Academic Press, (1975).   Google Scholar [2] M. Briane and J. Casado-Diaz, Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients,, J. of Diff. Equa., 245 (2008), 2038.  doi: 10.1016/j.jde.2008.07.027.  Google Scholar [3] G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems,, Revista Matematica Complutense, 24 (2011), 83.  doi: 10.1007/s13163-010-0030-y.  Google Scholar [4] D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford University Press, (1999).   Google Scholar [5] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications,, AMS, (2000).   Google Scholar [6] A. D. Ioffe and V. M. Tichomirov, Theory of Extremal Problems,, North-Holland, (1979).   Google Scholar [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2000).   Google Scholar [8] T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic dirichlet boundary value problems,, Asymptotic Analysis, 98 (2016), 155.  doi: 10.3233/ASY-161365.  Google Scholar [9] T. Horsin and P. I. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. i. existence results,, Math. Control Relat. Fields, 5 (2015), 73.  doi: 10.3934/mcrf.2015.5.73.  Google Scholar [10] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications,, Academic Press, (1980).   Google Scholar [11] P. I. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients,, Descrete and Continuous Dynamical System, 34 (2014), 2105.  doi: 10.3934/dcds.2014.34.2105.  Google Scholar [12] P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis,, Birkhäuser, (2011).  doi: 10.1007/978-0-8176-8149-4.  Google Scholar [13] P. I. Kogut and G. Leugering, Optimal $L^1$-control in coefficients for dirichlet elliptic problems: W-optimal solutions,, Journal of Optimization Theory and Applications, 150 (2011), 205.  doi: 10.1007/s10957-011-9840-4.  Google Scholar [14] P. I. Kogut and G. Leugering, Optimal $L^1$-control in coefficients for dirichlet elliptic problems: H-optimal solutions,, Zeitschrift für Analysis und ihre Anwendungen, 31 (2012), 31.  doi: 10.4171/ZAA/1447.  Google Scholar [15] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications,, Springer-Verlag, (1972).   Google Scholar [16] T. Jin, V. Mazya and J. van Schaftinger, Pathological solutions to elliptic problems in divergence form with continuous coefficients,, C. R. Math. Acad. Sci. Paris, 347 (2009), 773.  doi: 10.1016/j.crma.2009.05.008.  Google Scholar [17] J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 3 (1964), 385.   Google Scholar [18] J. L. Vazquez and N. B. Zographopoulos, Functional aspects of the Hardy inequlity. Appearance of a hidden energy,, Journal of Evolution Equations, 12 (2012), 713.  doi: 10.1007/s00028-012-0151-5.  Google Scholar [19] J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. of Functional Analysis, 173 (2000), 103.  doi: 10.1006/jfan.1999.3556.  Google Scholar [20] V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156.  doi: 10.1007/BF02465783.  Google Scholar [21] V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189 (1998), 27.  doi: 10.1070/SM1998v189n08ABEH000344.  Google Scholar [22] V. V. Zhikov, Remarks on the uniqueness of a solution of the Dirichlet problem for second-order elliptic equations with lower-order terms,, Functional Analysis and Its Applications, 38 (2004), 173.  doi: 10.1023/B:FAIA.0000042802.86050.5e.  Google Scholar [23] V. V. Zhikov, private, communication., ().   Google Scholar [24] Ph. Destuynder, Analyse, Traitement et Synthèse D'images Numériques,, Hermes, (2006).   Google Scholar [25] O. Faugeras, Three-Dimensional Computer Vision,, MIT Press, (1993).   Google Scholar [26] D. Cioranescu and F. Murat, A strange term coming from nowhere,, in Topic in the Math. Modelling of Composit Materials, 31 (1997), 45.   Google Scholar
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