-
Previous Article
An infinite time horizon portfolio optimization model with delays
- MCRF Home
- This Issue
-
Next Article
Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity
Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions
1. | heSam Université, 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 |
3. | heSam Université Conservatoire National des Arts et Métiers, M2N, Case 2D 5000, 292 rue Saint-Martin, 75003 Paris, France |
References:
[1] | |
[2] |
M. Briane and J. Casado-Diaz, Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients, J. of Diff. Equa., 245 (2008), 2038-2054.
doi: 10.1016/j.jde.2008.07.027. |
[3] |
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. |
[4] |
D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999. |
[5] |
A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, AMS, Providence, RI, 2000. |
[6] |
A. D. Ioffe and V. M. Tichomirov, Theory of Extremal Problems, North-Holland, Amsterdam-New York, 1979. |
[7] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2000. |
[8] |
T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188.
doi: 10.3233/ASY-161365. |
[9] |
T. Horsin and P. I. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. i. existence results, Math. Control Relat. Fields, 5 (2015), 73-96.
doi: 10.3934/mcrf.2015.5.73. |
[10] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980. |
[11] |
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. |
[12] |
P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011.
doi: 10.1007/978-0-8176-8149-4. |
[13] |
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. |
[14] |
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. |
[15] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972. |
[16] |
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. |
[17] |
J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 3 (1964), 385-387. |
[18] |
J. L. Vazquez and N. B. Zographopoulos, Functional aspects of the Hardy inequlity. Appearance of a hidden energy, Journal of Evolution Equations, 12 (2012), 713-739.
doi: 10.1007/s00028-012-0151-5. |
[19] |
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. |
[20] |
V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166.
doi: 10.1007/BF02465783. |
[21] |
V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58.
doi: 10.1070/SM1998v189n08ABEH000344. |
[22] |
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. |
[23] | |
[24] |
Ph. Destuynder, Analyse, Traitement et Synthèse D'images Numériques, Hermes, Paris, 2006. |
[25] |
O. Faugeras, Three-Dimensional Computer Vision, MIT Press, Cambridge, 1993. |
[26] |
D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topic in the Math. Modelling of Composit Materials, Boston, Birkhäuser, Prog. Non-linear Diff. Equ. Appl., 31 (1997), 45-93. |
show all references
References:
[1] | |
[2] |
M. Briane and J. Casado-Diaz, Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients, J. of Diff. Equa., 245 (2008), 2038-2054.
doi: 10.1016/j.jde.2008.07.027. |
[3] |
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. |
[4] |
D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999. |
[5] |
A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, AMS, Providence, RI, 2000. |
[6] |
A. D. Ioffe and V. M. Tichomirov, Theory of Extremal Problems, North-Holland, Amsterdam-New York, 1979. |
[7] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2000. |
[8] |
T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188.
doi: 10.3233/ASY-161365. |
[9] |
T. Horsin and P. I. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. i. existence results, Math. Control Relat. Fields, 5 (2015), 73-96.
doi: 10.3934/mcrf.2015.5.73. |
[10] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980. |
[11] |
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. |
[12] |
P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis, Birkhäuser, Boston, 2011.
doi: 10.1007/978-0-8176-8149-4. |
[13] |
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. |
[14] |
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. |
[15] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972. |
[16] |
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. |
[17] |
J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa, 3 (1964), 385-387. |
[18] |
J. L. Vazquez and N. B. Zographopoulos, Functional aspects of the Hardy inequlity. Appearance of a hidden energy, Journal of Evolution Equations, 12 (2012), 713-739.
doi: 10.1007/s00028-012-0151-5. |
[19] |
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. |
[20] |
V. V. Zhikov, Diffusion in incompressible random flow, Functional Analysis and Its Applications, 31 (1997), 156-166.
doi: 10.1007/BF02465783. |
[21] |
V. V. Zhikov, Weighted Sobolev spaces, Sbornik: Mathematics, 189 (1998), 27-58.
doi: 10.1070/SM1998v189n08ABEH000344. |
[22] |
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. |
[23] | |
[24] |
Ph. Destuynder, Analyse, Traitement et Synthèse D'images Numériques, Hermes, Paris, 2006. |
[25] |
O. Faugeras, Three-Dimensional Computer Vision, MIT Press, Cambridge, 1993. |
[26] |
D. Cioranescu and F. Murat, A strange term coming from nowhere, in Topic in the Math. Modelling of Composit Materials, Boston, Birkhäuser, Prog. Non-linear Diff. Equ. Appl., 31 (1997), 45-93. |
[1] |
Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial and Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 |
[2] |
Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003 |
[3] |
Laurent Pfeiffer. Optimality conditions in variational form for non-linear constrained stochastic control problems. Mathematical Control and Related Fields, 2020, 10 (3) : 493-526. doi: 10.3934/mcrf.2020008 |
[4] |
Stanisław Migórski, Biao Zeng. Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4477-4498. doi: 10.3934/dcdsb.2018172 |
[5] |
Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064 |
[6] |
Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619 |
[7] |
Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4193-4223. doi: 10.3934/dcds.2015.35.4193 |
[8] |
Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164 |
[9] |
Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331 |
[10] |
Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101 |
[11] |
Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451 |
[12] |
Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control and Related Fields, 2021, 11 (3) : 479-498. doi: 10.3934/mcrf.2021009 |
[13] |
Vyacheslav Maksimov. The method of extremal shift in control problems for evolution variational inequalities under disturbances. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021048 |
[14] |
Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347 |
[15] |
Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure and Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311 |
[16] |
Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014 |
[17] |
Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001 |
[18] |
Sarita Sharma, Anurag Jayswal, Sarita Choudhury. Sufficiency and mixed type duality for multiobjective variational control problems involving α-V-univexity. Evolution Equations and Control Theory, 2017, 6 (1) : 93-109. doi: 10.3934/eect.2017006 |
[19] |
EL Hassene Osmani, Mounir Haddou, Naceurdine Bensalem. A new relaxation method for optimal control of semilinear elliptic variational inequalities obstacle problems. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021061 |
[20] |
Evelyn Herberg, Michael Hinze. Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022013 |
2020 Impact Factor: 1.284
Tools
Metrics
Other articles
by authors
[Back to Top]