# American Institute of Mathematical Sciences

May  2014, 34(5): 2105-2133. doi: 10.3934/dcds.2014.34.2105

## On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients

 1 Basque Center for Applied Mathematics (BCAM), Bizkaia Technology Park, Building 500, E-48160 Derio, Basque Country, Spain

Received  March 2013 Revised  August 2013 Published  October 2013

We study an optimal boundary control problem (OCP) associated to a linear elliptic equation $-\mathrm{div}\,\left(\nabla y+A(x)\nabla y\right)=f$. The characteristic feature of this equation is the fact that the matrix $A(x)=[a_{ij}(x)]_{i,j=1,\dots,N}$ is skew-symmetric, $a_{ij}(x)=-a_{ji}(x)$, measurable, and 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--- namely, they have approximable solutions as well as another type of weak solutions that can not be obtained through an approximation of matrix $A$, the corresponding OCP is well-possed and admits a unique solution. At the same time, an optimal solution to such problem can inherit a singular character of the original matrix $A$. We indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that each of that optimal solutions can be attainable by solutions of special optimal boundary control problems.
Citation: Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
##### References:
 [1] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [2] 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. [3] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999. [4] 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. [5] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Second edition. Advances in Design and Control, 22. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. [6] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. [7] M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence, Probab. Theory and Related Fields, 105 (1996), 279-334. doi: 10.1007/BF01192211. [8] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Theory and applications. Translated from the 1999 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, 187. American Mathematical Society, Providence, RI, 2000. [9] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [10] P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Systems & Control: Foundations & Applications. Birkhäuser, Boston, 2011. doi: 10.1007/978-0-8176-8149-4. [11] 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. [12] 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. [13] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. [14] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 183. Springer-Verlag, New York-Heidelberg, 1973. [15] 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. [16] J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387. [17] 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. [18] V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58; translation in Sb. Math., 189 (1998), 1139-1170. doi: 10.1070/SM1998v189n08ABEH000344. [19] V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783. [20] 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.

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##### References:
 [1] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [2] 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. [3] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999. [4] 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. [5] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Second edition. Advances in Design and Control, 22. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. [6] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. [7] M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence, Probab. Theory and Related Fields, 105 (1996), 279-334. doi: 10.1007/BF01192211. [8] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Theory and applications. Translated from the 1999 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, 187. American Mathematical Society, Providence, RI, 2000. [9] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [10] P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Systems & Control: Foundations & Applications. Birkhäuser, Boston, 2011. doi: 10.1007/978-0-8176-8149-4. [11] 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. [12] 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. [13] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. [14] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 183. Springer-Verlag, New York-Heidelberg, 1973. [15] 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. [16] J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387. [17] 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. [18] V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58; translation in Sb. Math., 189 (1998), 1139-1170. doi: 10.1070/SM1998v189n08ABEH000344. [19] V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783. [20] 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.
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