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March  2015, 5(1): 1-30. doi: 10.3934/mcrf.2015.5.1

On the influence of the coupling on the dynamics of single-observed cascade systems of PDE's

Fatiha Alabau-Boussouira 1,

1. 

Institut Elie Cartan de Lorraine, UMR-CNRS 7502, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 1, France

Received  December 2013 Revised  June 2014 Published  January 2015

We consider single-observed cascade systems of hyperbolic equations. We first consider the class of bounded operators that satisfy a non negativity property $(NNP)$. Within this class, we give a necessary and sufficient condition for observability of the cascade system by a single observation. We further show that if the coupling operator does not satisfy $(NNP)$ (contrarily to [5], or also e.g.[3,4] for symmetrically coupled systems), the usual observability inequality through a single component may still occur in a general framework, under some smallness conditions, but it may also be violated. When the coupling operator is a multiplication operator, $(NNP)$ is violated whenever the coupling coefficient changes sign in the spatial domain. We give explicit constructive examples of such coupling operators for which unique continuation may fail for an infinite dimensional set of initial data, that we characterize explicitly. We also exhibit examples of couplings and initial data for which the observability inequality holds but in weaker norms. These examples extend to parabolic systems. Finally, we show that the two-level energy method [1,2] which involves different levels of energies for the observed and unobserved component, may involve the same levels of energies of these respective components, if the differential order of the coupling is higher (operating here through velocities instead of displacements). We further give an application to controlled systems coupled in velocities. This shows that the answer to observability and unique continuation questions for single-observed cascade systems is much more involved in the case of coupling operators that violate $(NNP)$ or of higher order coupling operators, and that the mathematical properties of the coupling operator greatly influence the dynamics of the observed system even though it operates through lower order differential terms. We indicate several extensions and future directions of research.
Keywords: observability, Control theory, hyperbolic equations, cascade systems., wave equation.
Mathematics Subject Classification: Primary: 93B05, 93B07, 35L05, 35L10, 35L5.
Citation: Fatiha Alabau-Boussouira. On the influence of the coupling on the dynamics of single-observed cascade systems of PDE's. Mathematical Control & Related Fields, 2015, 5 (1) : 1-30. doi: 10.3934/mcrf.2015.5.1
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[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.  Google Scholar

[14]

R. Dáger, Insensitizing controls for the 1-D wave equation, SIAM J. Control Opt., 45 (2006), 1758-1768. doi: 10.1137/060654372.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. Kocarev, Z. Tasev, T. Stojanovski and U. Parlitz, Synchronizing spatiotemporal chaos, Chaos, 7 (1997), 635-643. doi: 10.1063/1.166263.  Google Scholar

[19]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Collection RMA, 36, Masson-John Wiley, Paris-Chicester, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Vol. 1-2, Masson, Paris, 1988. Google Scholar

[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. Google Scholar

[24]

G. Olive, Contrôlabilité de Systèmes Paraboliques Linéaires Couplés, Thèse de doctorat de l'université d'Aix-Marseille, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

L. de Teresa, Insensitizing controls for a semilinear heat equation, CPDE, 25 (2000), 39-72. doi: 10.1080/03605300008821507.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[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.  Google Scholar

[14]

R. Dáger, Insensitizing controls for the 1-D wave equation, SIAM J. Control Opt., 45 (2006), 1758-1768. doi: 10.1137/060654372.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. Kocarev, Z. Tasev, T. Stojanovski and U. Parlitz, Synchronizing spatiotemporal chaos, Chaos, 7 (1997), 635-643. doi: 10.1063/1.166263.  Google Scholar

[19]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Collection RMA, 36, Masson-John Wiley, Paris-Chicester, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués, Vol. 1-2, Masson, Paris, 1988. Google Scholar

[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. Google Scholar

[24]

G. Olive, Contrôlabilité de Systèmes Paraboliques Linéaires Couplés, Thèse de doctorat de l'université d'Aix-Marseille, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

L. de Teresa, Insensitizing controls for a semilinear heat equation, CPDE, 25 (2000), 39-72. doi: 10.1080/03605300008821507.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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