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

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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

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

Abstract
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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, 333* (2001), 645. 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. 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, 349* (2011), 395. 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. 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, 26 (2014), 1. 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, 350* (2012), 577. 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.
[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. 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, 352* (2014), 391. 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. 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. doi: 10.3934/mcrf.2014.4.263.
[12]	J.-M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs, (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. doi: 10.1137/100784539.
[14]	R. Dáger, Insensitizing controls for the 1-D wave equation,, SIAM J. Control Opt., 45 (2006), 1758. 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. 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. 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. doi: 10.1051/cocv/2008077.
[18]	L. Kocarev, Z. Tasev, T. Stojanovski and U. Parlitz, Synchronizing spatiotemporal chaos,, Chaos, 7 (1997), 635. doi: 10.1063/1.166263.
[19]	V. Komornik, Exact Controllability and Stabilization, The Multiplier Method,, Collection RMA, (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. 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. doi: 10.1007/s11401-012-0754-8.
[22]	J. L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués,, Vol. 1-2, (1988), 1.
[23]	J. L. Lions, Remarques préliminaires sur le contrôle des systèmes à données incomplètes,, in Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), (1989), 43.
[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, 349* (2011), 291. doi: 10.1016/j.crma.2011.01.014.*
[26]	L. Tebou, Locally distributed desensitizing controls for the wave equation,, C. R. Acad. Sci. Paris, 346* (2008), 407. 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. doi: 10.1137/100803080.
[28]	L. de Teresa, Insensitizing controls for a semilinear heat equation,, CPDE, 25 (2000), 39. 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. doi: 10.3934/cpaa.2009.8.457.
[30]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Advanced Texts: Basler Lehrbücher, (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. 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, 333* (2001), 645. 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. 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, 349* (2011), 395. 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. 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, 26 (2014), 1. 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, 350* (2012), 577. 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.
[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. 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, 352* (2014), 391. 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. 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. doi: 10.3934/mcrf.2014.4.263.
[12]	J.-M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs, (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. doi: 10.1137/100784539.
[14]	R. Dáger, Insensitizing controls for the 1-D wave equation,, SIAM J. Control Opt., 45 (2006), 1758. 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. 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. 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. doi: 10.1051/cocv/2008077.
[18]	L. Kocarev, Z. Tasev, T. Stojanovski and U. Parlitz, Synchronizing spatiotemporal chaos,, Chaos, 7 (1997), 635. doi: 10.1063/1.166263.
[19]	V. Komornik, Exact Controllability and Stabilization, The Multiplier Method,, Collection RMA, (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. 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. doi: 10.1007/s11401-012-0754-8.
[22]	J. L. Lions, Contrôlabilité Exacte et Stabilisation de Systèmes Distribués,, Vol. 1-2, (1988), 1.
[23]	J. L. Lions, Remarques préliminaires sur le contrôle des systèmes à données incomplètes,, in Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), (1989), 43.
[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, 349* (2011), 291. doi: 10.1016/j.crma.2011.01.014.*
[26]	L. Tebou, Locally distributed desensitizing controls for the wave equation,, C. R. Acad. Sci. Paris, 346* (2008), 407. 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. doi: 10.1137/100803080.
[28]	L. de Teresa, Insensitizing controls for a semilinear heat equation,, CPDE, 25 (2000), 39. 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. doi: 10.3934/cpaa.2009.8.457.
[30]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Advanced Texts: Basler Lehrbücher, (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. doi: 10.1142/S0218127494000691.

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