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 & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
References:
[1]

R. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975). Google Scholar

[2]

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

[3]

D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999). Google Scholar

[4]

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

[5]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization,, Second edition. Advances in Design and Control, (2011). Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics. CRC Press, (1992). Google Scholar

[7]

M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence,, Probab. Theory and Related Fields, 105 (1996), 279. doi: 10.1007/BF01192211. Google Scholar

[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, (1999). Google Scholar

[9]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Academic Press, (1980). Google Scholar

[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, (2011). doi: 10.1007/978-0-8176-8149-4. Google Scholar

[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. doi: 10.1007/s10957-011-9840-4. Google Scholar

[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. doi: 10.4171/ZAA/1447. Google Scholar

[13]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar

[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, (1973). Google Scholar

[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. doi: 10.1016/j.crma.2009.05.008. Google Scholar

[16]

J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385. Google Scholar

[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. doi: 10.1006/jfan.1999.3556. Google Scholar

[18]

V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189 (1998), 27. doi: 10.1070/SM1998v189n08ABEH000344. Google Scholar

[19]

V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156. doi: 10.1007/BF02465783. Google Scholar

[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. doi: 10.1023/B:FAIA.0000042802.86050.5e. Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975). Google Scholar

[2]

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

[3]

D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999). Google Scholar

[4]

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

[5]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization,, Second edition. Advances in Design and Control, (2011). Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics. CRC Press, (1992). Google Scholar

[7]

M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence,, Probab. Theory and Related Fields, 105 (1996), 279. doi: 10.1007/BF01192211. Google Scholar

[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, (1999). Google Scholar

[9]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Academic Press, (1980). Google Scholar

[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, (2011). doi: 10.1007/978-0-8176-8149-4. Google Scholar

[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. doi: 10.1007/s10957-011-9840-4. Google Scholar

[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. doi: 10.4171/ZAA/1447. Google Scholar

[13]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar

[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, (1973). Google Scholar

[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. doi: 10.1016/j.crma.2009.05.008. Google Scholar

[16]

J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385. Google Scholar

[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. doi: 10.1006/jfan.1999.3556. Google Scholar

[18]

V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189 (1998), 27. doi: 10.1070/SM1998v189n08ABEH000344. Google Scholar

[19]

V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156. doi: 10.1007/BF02465783. Google Scholar

[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. doi: 10.1023/B:FAIA.0000042802.86050.5e. Google Scholar

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