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Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result
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 |
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. |
[2] |
D. Cioranescu and F. Murat, A strange term coming from nowhere,, in Topics in the Mathematical Modelling of Composite Materials, (1997), 45.
|
[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. |
[4] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).
|
[5] |
M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence,, Probab. Theory and Related Fields, 105 (1996), 279.
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, 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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[15] |
J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385.
|
[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. |
[17] |
V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156.
doi: 10.1007/BF02465783. |
[18] |
V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189 (1998), 27.
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.
doi: 10.1023/B:FAIA.0000042802.86050.5e. |
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.
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, (1997), 45.
|
[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. |
[4] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).
|
[5] |
M. A. Fannjiang and G. C. Papanicolaou, Diffusion in turbulence,, Probab. Theory and Related Fields, 105 (1996), 279.
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, 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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[15] |
J. Serrin, Pathological solutions of elliptic differential equations,, Ann. Scuola Norm. Sup. Pisa, 18 (1964), 385.
|
[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. |
[17] |
V. V. Zhikov, Diffusion in incompressible random flow,, Functional Analysis and Its Applications, 31 (1997), 156.
doi: 10.1007/BF02465783. |
[18] |
V. V. Zhikov, Weighted Sobolev spaces,, Sbornik: Mathematics, 189 (1998), 27.
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.
doi: 10.1023/B:FAIA.0000042802.86050.5e. |
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