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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
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show all references

References:
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Revista Matematica Complutense, 24 (2011), 83-94. doi: 10.1007/s13163-010-0030-y.  Google Scholar

[2]

in Topics in the Mathematical Modelling of Composite Materials, Prog. Nonlinear Diff. Equ. Appl., 31, Birkhäuser, Boston, 1997, 45-93.  Google Scholar

[3]

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]

CRC Press, Boca Raton, 1992.  Google Scholar

[5]

Probab. Theory and Related Fields, 105 (1996), 279-334. doi: 10.1007/BF01192211.  Google Scholar

[6]

Bulletin of Dniproperovsk National University, Series: Mathematical Modelling, 22 (2014), 3-38. Google Scholar

[7]

C. R. Math. Acad. Sci. Paris, 347 (2009), 773-778. doi: 10.1016/j.crma.2009.05.008.  Google Scholar

[8]

Descrete and Continuous Dynamical System, Series A, 34 (2014), 2105-2133. doi: 10.3934/dcds.2014.34.2105.  Google Scholar

[9]

Systems & Control: Foundations & Applications, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8149-4.  Google Scholar

[10]

Journal of Optimization Theory and Applications, 150 (2011), 205-232. doi: 10.1007/s10957-011-9840-4.  Google Scholar

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Zeitschrift für Analysis und ihre Anwendungen, 31 (2012), 31-53. doi: 10.4171/ZAA/1447.  Google Scholar

[12]

Zeitschrift für Analysis und ihre Anwendungen, 2014, (to appear). Google Scholar

[13]

Zeitschrift für Analysis und ihre Anwendungen, 2014, (to appear). Google Scholar

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Springer-Verlag, Berlin, 1971.  Google Scholar

[15]

Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385-387.  Google Scholar

[16]

J. of Functional Analysis, 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.  Google Scholar

[17]

Functional Analysis and Its Applications, 31 (1997), 156-166. doi: 10.1007/BF02465783.  Google Scholar

[18]

Sbornik: Mathematics, 189 (1998), 27-58. doi: 10.1070/SM1998v189n08ABEH000344.  Google Scholar

[19]

Functional Analysis and Its Applications, 38 (2004), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.  Google Scholar

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