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

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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

Thierry Horsin ^1, , Peter I. Kogut ^2, and Olivier Wilk ^3,

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

Abstract
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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.

Keywords: non-variational solutions, Control in coefficients, fictitious control., variational convergence.

Mathematics Subject Classification: Primary: 35Q93, 49J20, 49N05, 65N99; Secondary: 49J30, 35J75, 65N2.

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

References:

[1]	R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[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), 2038-2054. doi: 10.1016/j.jde.2008.07.027.
[3]	G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems, Revista Matematica Complutense, 24 (2011), 83-94. doi: 10.1007/s13163-010-0030-y.
[4]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
[5]	A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, AMS, Providence, RI, 2000.
[6]	A. D. Ioffe and V. M. Tichomirov, Theory of Extremal Problems, North-Holland, Amsterdam-New York, 1979.
[7]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2000.
[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), 155-188. doi: 10.3233/ASY-161365.
[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), 73-96. doi: 10.3934/mcrf.2015.5.73.
[10]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.
[11]	P. I. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Descrete and Continuous Dynamical System, Series A, 34 (2014), 2105-2133. doi: 10.3934/dcds.2014.34.2105.
[12]	P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011. doi: 10.1007/978-0-8176-8149-4.
[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), 205-232. doi: 10.1007/s10957-011-9840-4.
[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), 31-53. doi: 10.4171/ZAA/1447.
[15]	J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.
[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), 773-778. doi: 10.1016/j.crma.2009.05.008.
[17]	J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 3 (1964), 385-387.
[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), 713-739. doi: 10.1007/s00028-012-0151-5.
[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), 103-153. doi: 10.1006/jfan.1999.3556.
[20]	V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783.
[21]	V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. doi: 10.1070/SM1998v189n08ABEH000344.
[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), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.
[23]	V. V. Zhikov, private communication.
[24]	Ph. Destuynder, Analyse, Traitement et Synthèse D'images Numériques, Hermes, Paris, 2006.
[25]	O. Faugeras, Three-Dimensional Computer Vision, MIT Press, Cambridge, 1993.
[26]	D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topic in the Math. Modelling of Composit Materials, Boston, Birkhäuser, Prog. Non-linear Diff. Equ. Appl., 31 (1997), 45-93.

show all references

References:

[1]	R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[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), 2038-2054. doi: 10.1016/j.jde.2008.07.027.
[3]	G. Buttazzo and P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems, Revista Matematica Complutense, 24 (2011), 83-94. doi: 10.1007/s13163-010-0030-y.
[4]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
[5]	A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, AMS, Providence, RI, 2000.
[6]	A. D. Ioffe and V. M. Tichomirov, Theory of Extremal Problems, North-Holland, Amsterdam-New York, 1979.
[7]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2000.
[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), 155-188. doi: 10.3233/ASY-161365.
[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), 73-96. doi: 10.3934/mcrf.2015.5.73.
[10]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.
[11]	P. I. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Descrete and Continuous Dynamical System, Series A, 34 (2014), 2105-2133. doi: 10.3934/dcds.2014.34.2105.
[12]	P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011. doi: 10.1007/978-0-8176-8149-4.
[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), 205-232. doi: 10.1007/s10957-011-9840-4.
[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), 31-53. doi: 10.4171/ZAA/1447.
[15]	J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.
[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), 773-778. doi: 10.1016/j.crma.2009.05.008.
[17]	J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 3 (1964), 385-387.
[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), 713-739. doi: 10.1007/s00028-012-0151-5.
[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), 103-153. doi: 10.1006/jfan.1999.3556.
[20]	V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783.
[21]	V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. doi: 10.1070/SM1998v189n08ABEH000344.
[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), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.
[23]	V. V. Zhikov, private communication.
[24]	Ph. Destuynder, Analyse, Traitement et Synthèse D'images Numériques, Hermes, Paris, 2006.
[25]	O. Faugeras, Three-Dimensional Computer Vision, MIT Press, Cambridge, 1993.
[26]	D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topic in the Math. Modelling of Composit Materials, Boston, Birkhäuser, Prog. Non-linear Diff. Equ. Appl., 31 (1997), 45-93.

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