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Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces
1. | Université d'Orléans, Bâtiment de Mathématiques (MAPMO), B.P. 6759, 45067 Orléans cedex 2 |
References:
[1] |
F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators and uniform controllability of semi discretized parabolic equations, J. Math. Pur. Appl., 93 (2010), 240-276.
doi: 10.1016/j.matpur.2009.11.003. |
[2] |
F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim, 48 (2010), 5357-5397.
doi: 10.1137/100784278. |
[3] |
F. Boyer, F. Hubert and J. Le Rousseau, Uniform null-controllability properties for space/time-discretized parabolic equations, Numer. Math, 118 (2011), 601-661.
doi: 10.1007/s00211-011-0368-1. |
[4] |
J. Bramble, A. Shatz, V. Thomee and L. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Num. Anal., 14 (1977), 218-241.
doi: 10.1137/0714015. |
[5] |
C. Carthel, R. Glowinski and J. L. Lions, On exact and approximate Boundary Controllabilities for the heat equation: a numerical approach, J. Optimal. Theory Appl., 82 (1994), 429-484.
doi: 10.1007/BF02192213. |
[6] |
Y. Chitour and E. Trélat, Controllability of partial differential equations, Advanced topics in control systems theory, Lecture Notes in Control and Inform. Sci, Spinger, London, 328 (2006), 171-198.
doi: 10.1007/11583592_5. |
[7] |
I. Ekeland and R. Temam, Convex Analysic and Variational Problems, Classics in Applied Mathematics 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1999.
doi: 10.1137/1.9781611971088. |
[8] |
S. Ervedoza and J. Valein, On the observability of abstract time-discrete linear parabolic equations, Revista Matematica Complutense, 23 (2010), 163-190.
doi: 10.1007/s13163-009-0014-y. |
[9] |
S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems, Journal of Functional Analysis, 254 (2008), 3037-3078.
doi: 10.1016/j.jfa.2008.03.005. |
[10] |
H. O. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems, Appl. Math. Optim., 15 (1987), 141-185.
doi: 10.1007/BF01442651. |
[11] |
H. O. Fattorini and H. Frankowska, Necessary conditions for infinite dimensional problems, Mathematics of Control Signals, and Systems, 4 (1991), 41-67.
doi: 10.1007/BF02551380. |
[12] |
A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture notes, vol 34, Seoul National University, Korea, 1996. |
[13] |
R. Glowinski, J. L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematical and its Applications, 117. Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511721595. |
[14] |
J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation, M2AN Math. Model. Number. Anal., 33 (1999), 407-438.
doi: 10.1051/m2an:1999123. |
[15] |
S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control system, Systems and Control Letters, 55 (2006), 597-609.
doi: 10.1016/j.sysconle.2006.01.004. |
[16] |
S. Labbé and E. Trélat, Generalization of the finite difference method in distributions spaces, Preprint HAL, ccsd-00097806, 2006. |
[17] |
I. Lasiecka and R. Triggiani, Control theory for partial differential equation: Continuous and approximation theories. I. Abstract parabolic systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. |
[18] |
G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[19] |
L. Leon and E. Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation, ESAIM Control Optim., Calc. Var., 8 (2002), 827-862.
doi: 10.1051/cocv:2002025. |
[20] |
X. Li and Y. L. Yao, Maximum Principle of distributed parameter systems with time lags, Lecture Notes in Control and Information Sciences, Spinger- Verlag, New York, 75 (1985), 410-427.
doi: 10.1007/BFb0005665. |
[21] |
X. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908.
doi: 10.1137/0329049. |
[22] |
J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[23] |
A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1-D heat equation, Sem. EDP, Ecole Polytechnique, VIII (1998), 1-22. |
[24] |
A. Munch and E. Zuazua, Numerical approximation of the null controls for the heat equation through transmutation, Inverse Problems, 26 (2010), 085018, 39 pp.
doi: 10.1088/0266-5611/26/8/085018. |
[25] |
M. Negreanu and E. Zuazua, Uniform boundary controllability of discre 1-D wave equation. Optimization and control of distributed systems, Systems Control Lett., 48 (2003), 261-279.
doi: 10.1016/S0167-6911(02)00271-2. |
[26] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44, Springer-Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[27] |
L. S. Pontryagin, et al., The Mathematical Theory of Optimal Processes, vol. 4. Interscience, 1962. |
[28] |
O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematical and its Applications, 103. Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197. |
[29] |
E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics, Vuibert, Paris, 2005, 246 pp. |
[30] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Advanced Texts Basler Lehrbucher, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[31] |
E. Zuazua, Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square, J. Math. Pures Appl., 78 (1999), 523-563.
doi: 10.1016/S0021-7824(98)00008-7. |
[32] |
E. Zuazua, Controllability of the partial differential equations and its semi-discrete approximations, Discrete Contin. Dyn. Syst., 8 (2002), 469-513.
doi: 10.3934/dcds.2002.8.469. |
[33] |
E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for 1-D wave equation, Rendiconti di Matematica VIII, 24 (2004), 201-237. |
[34] |
E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Wave approximated by finite-difference method, SIAM Review, 47 (2005), 197-243.
doi: 10.1137/S0036144503432862. |
[35] |
E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, III (2006), 1389-1417. |
show all references
References:
[1] |
F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators and uniform controllability of semi discretized parabolic equations, J. Math. Pur. Appl., 93 (2010), 240-276.
doi: 10.1016/j.matpur.2009.11.003. |
[2] |
F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for the elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim, 48 (2010), 5357-5397.
doi: 10.1137/100784278. |
[3] |
F. Boyer, F. Hubert and J. Le Rousseau, Uniform null-controllability properties for space/time-discretized parabolic equations, Numer. Math, 118 (2011), 601-661.
doi: 10.1007/s00211-011-0368-1. |
[4] |
J. Bramble, A. Shatz, V. Thomee and L. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Num. Anal., 14 (1977), 218-241.
doi: 10.1137/0714015. |
[5] |
C. Carthel, R. Glowinski and J. L. Lions, On exact and approximate Boundary Controllabilities for the heat equation: a numerical approach, J. Optimal. Theory Appl., 82 (1994), 429-484.
doi: 10.1007/BF02192213. |
[6] |
Y. Chitour and E. Trélat, Controllability of partial differential equations, Advanced topics in control systems theory, Lecture Notes in Control and Inform. Sci, Spinger, London, 328 (2006), 171-198.
doi: 10.1007/11583592_5. |
[7] |
I. Ekeland and R. Temam, Convex Analysic and Variational Problems, Classics in Applied Mathematics 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1999.
doi: 10.1137/1.9781611971088. |
[8] |
S. Ervedoza and J. Valein, On the observability of abstract time-discrete linear parabolic equations, Revista Matematica Complutense, 23 (2010), 163-190.
doi: 10.1007/s13163-009-0014-y. |
[9] |
S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems, Journal of Functional Analysis, 254 (2008), 3037-3078.
doi: 10.1016/j.jfa.2008.03.005. |
[10] |
H. O. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems, Appl. Math. Optim., 15 (1987), 141-185.
doi: 10.1007/BF01442651. |
[11] |
H. O. Fattorini and H. Frankowska, Necessary conditions for infinite dimensional problems, Mathematics of Control Signals, and Systems, 4 (1991), 41-67.
doi: 10.1007/BF02551380. |
[12] |
A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture notes, vol 34, Seoul National University, Korea, 1996. |
[13] |
R. Glowinski, J. L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematical and its Applications, 117. Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511721595. |
[14] |
J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation, M2AN Math. Model. Number. Anal., 33 (1999), 407-438.
doi: 10.1051/m2an:1999123. |
[15] |
S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control system, Systems and Control Letters, 55 (2006), 597-609.
doi: 10.1016/j.sysconle.2006.01.004. |
[16] |
S. Labbé and E. Trélat, Generalization of the finite difference method in distributions spaces, Preprint HAL, ccsd-00097806, 2006. |
[17] |
I. Lasiecka and R. Triggiani, Control theory for partial differential equation: Continuous and approximation theories. I. Abstract parabolic systems, Encyclopedia of Mathematics and its Applications, 74. Cambridge University Press, Cambridge, 2000. |
[18] |
G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.
doi: 10.1080/03605309508821097. |
[19] |
L. Leon and E. Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation, ESAIM Control Optim., Calc. Var., 8 (2002), 827-862.
doi: 10.1051/cocv:2002025. |
[20] |
X. Li and Y. L. Yao, Maximum Principle of distributed parameter systems with time lags, Lecture Notes in Control and Information Sciences, Spinger- Verlag, New York, 75 (1985), 410-427.
doi: 10.1007/BFb0005665. |
[21] |
X. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908.
doi: 10.1137/0329049. |
[22] |
J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[23] |
A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1-D heat equation, Sem. EDP, Ecole Polytechnique, VIII (1998), 1-22. |
[24] |
A. Munch and E. Zuazua, Numerical approximation of the null controls for the heat equation through transmutation, Inverse Problems, 26 (2010), 085018, 39 pp.
doi: 10.1088/0266-5611/26/8/085018. |
[25] |
M. Negreanu and E. Zuazua, Uniform boundary controllability of discre 1-D wave equation. Optimization and control of distributed systems, Systems Control Lett., 48 (2003), 261-279.
doi: 10.1016/S0167-6911(02)00271-2. |
[26] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44, Springer-Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[27] |
L. S. Pontryagin, et al., The Mathematical Theory of Optimal Processes, vol. 4. Interscience, 1962. |
[28] |
O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematical and its Applications, 103. Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197. |
[29] |
E. Trélat, (French version), Optimal control: Theory and applications, Concrete Mathematics, Vuibert, Paris, 2005, 246 pp. |
[30] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Advanced Texts Basler Lehrbucher, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[31] |
E. Zuazua, Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square, J. Math. Pures Appl., 78 (1999), 523-563.
doi: 10.1016/S0021-7824(98)00008-7. |
[32] |
E. Zuazua, Controllability of the partial differential equations and its semi-discrete approximations, Discrete Contin. Dyn. Syst., 8 (2002), 469-513.
doi: 10.3934/dcds.2002.8.469. |
[33] |
E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for 1-D wave equation, Rendiconti di Matematica VIII, 24 (2004), 201-237. |
[34] |
E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Wave approximated by finite-difference method, SIAM Review, 47 (2005), 197-243.
doi: 10.1137/S0036144503432862. |
[35] |
E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, III (2006), 1389-1417. |
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