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Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition
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On the influence of the coupling on the dynamics of single-observed cascade systems of PDE's
1. | Institut Elie Cartan de Lorraine, UMR-CNRS 7502, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 1, France |
References:
[1] |
F. Alabau-Boussouira, Indirect boundary observability of a weakly coupled wave system, C. R. Acad. Sci. Paris, Série I, 333 (2001), 645-650.
doi: 10.1016/S0764-4442(01)02076-6. |
[2] |
F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Opt., 42 (2003), 871-906.
doi: 10.1137/S0363012902402608. |
[3] |
F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Acad. Sci. Paris, Série I, 349 (2011), 395-400.
doi: 10.1016/j.crma.2011.02.004. |
[4] |
F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, Journal de Mathématiques Pures et Appliquées, 99 (2013), 544-576.
doi: 10.1016/j.matpur.2012.09.012. |
[5] |
F. Alabau-Boussouira, Insensitizing controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE's by a single control, Mathematics of Control, Signals, and Systems, 26 (2014), 1-46.
doi: 10.1007/s00498-013-0112-8. |
[6] |
F. Alabau-Boussouira, Controllability of cascade coupled systems of multi-dimensional evolution PDE's by a reduced number of controls, C. R. Acad. Sci. Paris, Série I, 350 (2012), 577-582.
doi: 10.1016/j.crma.2012.05.009. |
[7] |
F. Alabau-Boussouira, A hierarchic multi-levels energy method for the control of bi-diagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls, Adv. in Differential Equations, 18 (2013), 1005-1072. |
[8] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.
doi: 10.3934/mcrf.2011.1.267. |
[9] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time of controllability of two parabolic equations with disjoint control and coupling domains, C. R. Acad. Sci. Paris, Série I, 352 (2014), 391-396.
doi: 10.1016/j.crma.2014.03.004. |
[10] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Opt., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[11] |
F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space- dependent coefficients, Math. Control Relat. Fields, 4 (2014), 263-287.
doi: 10.3934/mcrf.2014.4.263. |
[12] |
J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. |
[13] |
J.-M. Coron, S. Guerrero and L. Rosier, Null controllability of a parabolic system with a cubic coupling term, SIAM J. Control Optim., 48 (2010), 5629-5653.
doi: 10.1137/100784539. |
[14] |
R. Dáger, Insensitizing controls for the 1-D wave equation, SIAM J. Control Opt., 45 (2006), 1758-1768.
doi: 10.1137/060654372. |
[15] |
B. Dehman, J. Le Rousseau and M. Léautaud, Controllability of two coupled wave equations on a compact manifold, ARMA, 211 (2014), 113-187.
doi: 10.1007/s00205-013-0670-4. |
[16] |
H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Progress of Theoretical Physics, 69 (1983), 32-47.
doi: 10.1143/PTP.69.32. |
[17] |
O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM COCV, 16 (2010), 247-274.
doi: 10.1051/cocv/2008077. |
[18] |
L. Kocarev, Z. Tasev, T. Stojanovski and U. Parlitz, Synchronizing spatiotemporal chaos, Chaos, 7 (1997), 635-643.
doi: 10.1063/1.166263. |
[19] |
V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Collection RMA, 36, Masson-John Wiley, Paris-Chicester, 1994. |
[20] |
T. Li and B. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, C. R. Acad. Sci. Paris, 351 (2013), 687-693.
doi: 10.1016/j.crma.2013.09.013. |
[21] |
T. Li and B. Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math. Ser. B, 34 (2013), 139-160.
doi: 10.1007/s11401-012-0754-8. |
[22] |
J. L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Vol. 1-2, Masson, Paris, 1988. |
[23] |
J. L. Lions, Remarques préliminaires sur le contrôle des systèmes à données incomplètes, in Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, 1989, 43-54. |
[24] |
G. Olive, Contrôlabilité de Systèmes Paraboliques Linéaires Couplés, Thèse de doctorat de l'université d'Aix-Marseille, 2013. |
[25] |
L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 291-296.
doi: 10.1016/j.crma.2011.01.014. |
[26] |
L. Tebou, Locally distributed desensitizing controls for the wave equation, C. R. Acad. Sci. Paris, Série I, 346 (2008), 407-412.
doi: 10.1016/j.crma.2008.02.019. |
[27] |
L. Tebou, Some results on the controllability of coupled semilinear wave equations: the desensitizing control case, SIAM J. Control Opt., 49 (2011), 1221-1238.
doi: 10.1137/100803080. |
[28] |
L. de Teresa, Insensitizing controls for a semilinear heat equation, CPDE, 25 (2000), 39-72.
doi: 10.1080/03605300008821507. |
[29] |
L. de Teresa and E. Zuazua, Identification of the class of initial data for the insensitizing control of the heat equation, CPAA, 8 (2009), 457-471.
doi: 10.3934/cpaa.2009.8.457. |
[30] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[31] |
C. W. Wu and O. L. Chua, A unified framework for synchronization and control of dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 979-998.
doi: 10.1142/S0218127494000691. |
show all references
References:
[1] |
F. Alabau-Boussouira, Indirect boundary observability of a weakly coupled wave system, C. R. Acad. Sci. Paris, Série I, 333 (2001), 645-650.
doi: 10.1016/S0764-4442(01)02076-6. |
[2] |
F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Opt., 42 (2003), 871-906.
doi: 10.1137/S0363012902402608. |
[3] |
F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Acad. Sci. Paris, Série I, 349 (2011), 395-400.
doi: 10.1016/j.crma.2011.02.004. |
[4] |
F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, Journal de Mathématiques Pures et Appliquées, 99 (2013), 544-576.
doi: 10.1016/j.matpur.2012.09.012. |
[5] |
F. Alabau-Boussouira, Insensitizing controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE's by a single control, Mathematics of Control, Signals, and Systems, 26 (2014), 1-46.
doi: 10.1007/s00498-013-0112-8. |
[6] |
F. Alabau-Boussouira, Controllability of cascade coupled systems of multi-dimensional evolution PDE's by a reduced number of controls, C. R. Acad. Sci. Paris, Série I, 350 (2012), 577-582.
doi: 10.1016/j.crma.2012.05.009. |
[7] |
F. Alabau-Boussouira, A hierarchic multi-levels energy method for the control of bi-diagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls, Adv. in Differential Equations, 18 (2013), 1005-1072. |
[8] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.
doi: 10.3934/mcrf.2011.1.267. |
[9] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time of controllability of two parabolic equations with disjoint control and coupling domains, C. R. Acad. Sci. Paris, Série I, 352 (2014), 391-396.
doi: 10.1016/j.crma.2014.03.004. |
[10] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Opt., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[11] |
F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space- dependent coefficients, Math. Control Relat. Fields, 4 (2014), 263-287.
doi: 10.3934/mcrf.2014.4.263. |
[12] |
J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. |
[13] |
J.-M. Coron, S. Guerrero and L. Rosier, Null controllability of a parabolic system with a cubic coupling term, SIAM J. Control Optim., 48 (2010), 5629-5653.
doi: 10.1137/100784539. |
[14] |
R. Dáger, Insensitizing controls for the 1-D wave equation, SIAM J. Control Opt., 45 (2006), 1758-1768.
doi: 10.1137/060654372. |
[15] |
B. Dehman, J. Le Rousseau and M. Léautaud, Controllability of two coupled wave equations on a compact manifold, ARMA, 211 (2014), 113-187.
doi: 10.1007/s00205-013-0670-4. |
[16] |
H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Progress of Theoretical Physics, 69 (1983), 32-47.
doi: 10.1143/PTP.69.32. |
[17] |
O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM COCV, 16 (2010), 247-274.
doi: 10.1051/cocv/2008077. |
[18] |
L. Kocarev, Z. Tasev, T. Stojanovski and U. Parlitz, Synchronizing spatiotemporal chaos, Chaos, 7 (1997), 635-643.
doi: 10.1063/1.166263. |
[19] |
V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Collection RMA, 36, Masson-John Wiley, Paris-Chicester, 1994. |
[20] |
T. Li and B. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, C. R. Acad. Sci. Paris, 351 (2013), 687-693.
doi: 10.1016/j.crma.2013.09.013. |
[21] |
T. Li and B. Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math. Ser. B, 34 (2013), 139-160.
doi: 10.1007/s11401-012-0754-8. |
[22] |
J. L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Vol. 1-2, Masson, Paris, 1988. |
[23] |
J. L. Lions, Remarques préliminaires sur le contrôle des systèmes à données incomplètes, in Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, 1989, 43-54. |
[24] |
G. Olive, Contrôlabilité de Systèmes Paraboliques Linéaires Couplés, Thèse de doctorat de l'université d'Aix-Marseille, 2013. |
[25] |
L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 291-296.
doi: 10.1016/j.crma.2011.01.014. |
[26] |
L. Tebou, Locally distributed desensitizing controls for the wave equation, C. R. Acad. Sci. Paris, Série I, 346 (2008), 407-412.
doi: 10.1016/j.crma.2008.02.019. |
[27] |
L. Tebou, Some results on the controllability of coupled semilinear wave equations: the desensitizing control case, SIAM J. Control Opt., 49 (2011), 1221-1238.
doi: 10.1137/100803080. |
[28] |
L. de Teresa, Insensitizing controls for a semilinear heat equation, CPDE, 25 (2000), 39-72.
doi: 10.1080/03605300008821507. |
[29] |
L. de Teresa and E. Zuazua, Identification of the class of initial data for the insensitizing control of the heat equation, CPAA, 8 (2009), 457-471.
doi: 10.3934/cpaa.2009.8.457. |
[30] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[31] |
C. W. Wu and O. L. Chua, A unified framework for synchronization and control of dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 979-998.
doi: 10.1142/S0218127494000691. |
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