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 and 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), 83-94. doi: 10.1007/s13163-010-0030-y.
[2]	D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topics in the Mathematical Modelling of Composite Materials, Prog. Nonlinear Diff. Equ. Appl., 31, Birkhäuser, Boston, 1997, 45-93.
[3]	J.-M. Coron, J.-M. Ghidaglia and F. Hélein, eds., Nematics, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 332, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3428-6.
[4]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
[5]	M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence, Probab. Theory and Related Fields, 105 (1996), 279-334. doi: 10.1007/BF01192211.
[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, Series: Mathematical Modelling, 22 (2014), 3-38.
[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), 773-778. doi: 10.1016/j.crma.2009.05.008.
[8]	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.
[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, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8149-4.
[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), 205-232. doi: 10.1007/s10957-011-9840-4.
[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), 31-53. doi: 10.4171/ZAA/1447.
[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, (to appear).
[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, (to appear).
[14]	J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
[15]	J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387.
[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), 103-153. doi: 10.1006/jfan.1999.3556.
[17]	V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783.
[18]	V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. doi: 10.1070/SM1998v189n08ABEH000344.
[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), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.

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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), 83-94. doi: 10.1007/s13163-010-0030-y.
[2]	D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topics in the Mathematical Modelling of Composite Materials, Prog. Nonlinear Diff. Equ. Appl., 31, Birkhäuser, Boston, 1997, 45-93.
[3]	J.-M. Coron, J.-M. Ghidaglia and F. Hélein, eds., Nematics, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 332, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3428-6.
[4]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
[5]	M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence, Probab. Theory and Related Fields, 105 (1996), 279-334. doi: 10.1007/BF01192211.
[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, Series: Mathematical Modelling, 22 (2014), 3-38.
[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), 773-778. doi: 10.1016/j.crma.2009.05.008.
[8]	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.
[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, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8149-4.
[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), 205-232. doi: 10.1007/s10957-011-9840-4.
[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), 31-53. doi: 10.4171/ZAA/1447.
[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, (to appear).
[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, (to appear).
[14]	J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
[15]	J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387.
[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), 103-153. doi: 10.1006/jfan.1999.3556.
[17]	V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783.
[18]	V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. doi: 10.1070/SM1998v189n08ABEH000344.
[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), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.

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