Optimal L^2-control problem in coefficients for a linear elliptic equation. I. Existence result

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March 2015, 5(1): 73-96. doi: 10.3934/mcrf.2015.5.73

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result

Thierry Horsin ^1, and Peter I. Kogut ^2,

1.	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

Received March 2013 Revised May 2014 Published January 2015

Abstract
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In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain $\Omega$. The matrix-valued coefficients $A$ of such systems is our control in $\Omega$ and will be taken in $L^2(\Omega;\mathbb{R}^{N\times N})$ which in particular may comprises the case of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may exhibit non-uniqueness of weak solutions--- namely, approximable solutions as well as another type of weak solutions that can not be obtained through the $L^\infty$-approximation of matrix $A$. Following the direct method in the calculus of variations, we show that the given OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can not be attainable through the $L^\infty$-approximation of the original problem.

Keywords: variational convergence, variational solutions, existence result., Control in coefficients.

Mathematics Subject Classification: Primary: 49J20, 35J57; Secondary: 49J45, 35J7.

Citation: Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

References:

[1]	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
[2]	D. Cioranescu and F. Murat, A strange term coming from nowhere,, in Topics in the Mathematical Modelling of Composite Materials, (1997), 45. Google Scholar
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[4]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). Google Scholar
[5]	M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence,, Probab. Theory and Related Fields, 105 (1996), 279. doi: 10.1007/BF01192211. Google Scholar
[6]	T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems,, Bulletin of Dniproperovsk National University, 22 (2014), 3. Google Scholar
[7]	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*
[8]	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
[9]	P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis,, Systems & Control: Foundations & Applications, (2011). doi: 10.1007/978-0-8176-8149-4. Google Scholar
[10]	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*
[11]	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
[12]	P. I. Kogut, O. P. Kupenko and G. Leugering, Optimal control in matrix-valued coefficients for nonlinear monotone problems: Optimality conditions. Part I,, Zeitschrift für Analysis und ihre Anwendungen, (2014). Google Scholar
[13]	P. I. Kogut, O. P. Kupenko and G. Leugering, Optimal control in matrix-valued coefficients for nonlinear monotone problems: Optimality conditions. Part II,, Zeitschrift für Analysis und ihre Anwendungen, (2014). Google Scholar
[14]	J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar
[15]	J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385. Google Scholar
[16]	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*
[17]	V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156. doi: 10.1007/BF02465783. Google Scholar
[18]	V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189* (1998), 27. doi: 10.1070/SM1998v189n08ABEH000344. Google Scholar*
[19]	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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References:

[1]	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
[2]	D. Cioranescu and F. Murat, A strange term coming from nowhere,, in Topics in the Mathematical Modelling of Composite Materials, (1997), 45. Google Scholar
[3]	J.-M. Coron, J.-M. Ghidaglia and F. Hélein, eds., Nematics,, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, (1991). doi: 10.1007/978-94-011-3428-6. Google Scholar
[4]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). Google Scholar
[5]	M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence,, Probab. Theory and Related Fields, 105 (1996), 279. doi: 10.1007/BF01192211. Google Scholar
[6]	T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems,, Bulletin of Dniproperovsk National University, 22 (2014), 3. Google Scholar
[7]	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*
[8]	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
[9]	P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis,, Systems & Control: Foundations & Applications, (2011). doi: 10.1007/978-0-8176-8149-4. Google Scholar
[10]	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*
[11]	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
[12]	P. I. Kogut, O. P. Kupenko and G. Leugering, Optimal control in matrix-valued coefficients for nonlinear monotone problems: Optimality conditions. Part I,, Zeitschrift für Analysis und ihre Anwendungen, (2014). Google Scholar
[13]	P. I. Kogut, O. P. Kupenko and G. Leugering, Optimal control in matrix-valued coefficients for nonlinear monotone problems: Optimality conditions. Part II,, Zeitschrift für Analysis und ihre Anwendungen, (2014). Google Scholar
[14]	J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar
[15]	J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385. Google Scholar
[16]	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*
[17]	V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156. doi: 10.1007/BF02465783. Google Scholar
[18]	V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189* (1998), 27. doi: 10.1070/SM1998v189n08ABEH000344. Google Scholar*
[19]	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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