American Institute of Mathematical Sciences

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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
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show all references

References:
[1]

C. R. Acad. Sci. Paris, Série I, 333 (2001), 645-650. doi: 10.1016/S0764-4442(01)02076-6.  Google Scholar

[2]

SIAM J. Control Opt., 42 (2003), 871-906. doi: 10.1137/S0363012902402608.  Google Scholar

[3]

C. R. Acad. Sci. Paris, Série I, 349 (2011), 395-400. doi: 10.1016/j.crma.2011.02.004.  Google Scholar

[4]

Journal de Mathématiques Pures et Appliquées, 99 (2013), 544-576. doi: 10.1016/j.matpur.2012.09.012.  Google Scholar

[5]

Mathematics of Control, Signals, and Systems, 26 (2014), 1-46. doi: 10.1007/s00498-013-0112-8.  Google Scholar

[6]

C. R. Acad. Sci. Paris, Série I, 350 (2012), 577-582. doi: 10.1016/j.crma.2012.05.009.  Google Scholar

[7]

Adv. in Differential Equations, 18 (2013), 1005-1072.  Google Scholar

[8]

Mathematical Control and Related Fields, 1 (2011), 267-306. doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[9]

C. R. Acad. Sci. Paris, Série I, 352 (2014), 391-396. doi: 10.1016/j.crma.2014.03.004.  Google Scholar

[10]

SIAM J. Control Opt., 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[11]

Math. Control Relat. Fields, 4 (2014), 263-287. doi: 10.3934/mcrf.2014.4.263.  Google Scholar

[12]

Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[13]

SIAM J. Control Optim., 48 (2010), 5629-5653. doi: 10.1137/100784539.  Google Scholar

[14]

SIAM J. Control Opt., 45 (2006), 1758-1768. doi: 10.1137/060654372.  Google Scholar

[15]

ARMA, 211 (2014), 113-187. doi: 10.1007/s00205-013-0670-4.  Google Scholar

[16]

Progress of Theoretical Physics, 69 (1983), 32-47. doi: 10.1143/PTP.69.32.  Google Scholar

[17]

ESAIM COCV, 16 (2010), 247-274. doi: 10.1051/cocv/2008077.  Google Scholar

[18]

Chaos, 7 (1997), 635-643. doi: 10.1063/1.166263.  Google Scholar

[19]

Collection RMA, 36, Masson-John Wiley, Paris-Chicester, 1994.  Google Scholar

[20]

C. R. Acad. Sci. Paris, 351 (2013), 687-693. doi: 10.1016/j.crma.2013.09.013.  Google Scholar

[21]

Chin. Ann. Math. Ser. B, 34 (2013), 139-160. doi: 10.1007/s11401-012-0754-8.  Google Scholar

[22]

Vol. 1-2, Masson, Paris, 1988. Google Scholar

[23]

in Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, 1989, 43-54. Google Scholar

[24]

Thèse de doctorat de l'université d'Aix-Marseille, 2013. Google Scholar

[25]

C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 291-296. doi: 10.1016/j.crma.2011.01.014.  Google Scholar

[26]

C. R. Acad. Sci. Paris, Série I, 346 (2008), 407-412. doi: 10.1016/j.crma.2008.02.019.  Google Scholar

[27]

SIAM J. Control Opt., 49 (2011), 1221-1238. doi: 10.1137/100803080.  Google Scholar

[28]

CPDE, 25 (2000), 39-72. doi: 10.1080/03605300008821507.  Google Scholar

[29]

CPAA, 8 (2009), 457-471. doi: 10.3934/cpaa.2009.8.457.  Google Scholar

[30]

Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[31]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 979-998. doi: 10.1142/S0218127494000691.  Google Scholar

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