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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

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), 83-94. 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, Prog. Nonlinear Diff. Equ. Appl., 31, Birkhäuser, Boston, 1997, 45-93.  Google Scholar

[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.  Google Scholar

[4]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar

[5]

M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence, Probab. Theory and Related Fields, 105 (1996), 279-334. 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, Series: Mathematical Modelling, 22 (2014), 3-38. 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), 773-778. 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, Series A, 34 (2014), 2105-2133. 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, Birkhäuser/Springer, New York, 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), 205-232. 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), 31-53. 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, (to appear). 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, (to appear). Google Scholar

[14]

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

[15]

J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387.  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), 103-153. doi: 10.1006/jfan.1999.3556.  Google Scholar

[17]

V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783.  Google Scholar

[18]

V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. 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), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar

[5]

M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence, Probab. Theory and Related Fields, 105 (1996), 279-334. 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, Series: Mathematical Modelling, 22 (2014), 3-38. 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), 773-778. 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, Series A, 34 (2014), 2105-2133. 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, Birkhäuser/Springer, New York, 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), 205-232. 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), 31-53. 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, (to appear). 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, (to appear). Google Scholar

[14]

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

[15]

J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387.  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), 103-153. doi: 10.1006/jfan.1999.3556.  Google Scholar

[17]

V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783.  Google Scholar

[18]

V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58. 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), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.  Google Scholar

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