Stability and controllability of a wave equation with dynamical boundary control

Bopeng Rao; Laila Toufayli; Ali Wehbe

doi:10.3934/mcrf.2015.5.305

Article Contents

2015, Volume 5, Issue 2: 305-320. Doi: 10.3934/mcrf.2015.5.305

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Stability and controllability of a wave equation with dynamical boundary control

1.
Université de Strasbourg, Institut de Recherche Mathématique Avancée, 7 rue René Descartes, 67084 Strasbourg
2.
Université Libanaise, EDST et Faculté des sciences I, Equipe EDP-AN, Hadath, Beyrouth
3.
Université Libanaise, Sciences 1 et EDST, Equipe EDP-AN, Hadath, Beyrouth

Received: November 30, 2013

Revised: May 31, 2014

Published: May 2015

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Abstract

In this work, we consider the stabilization and the exact controllability of a wave equation with dynamical boundary control. We first prove the strong stability of the system and establish a polynomial decay rate for smooth solutions. We next show the exact controllability by means of a singular dynamical boundary control.

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Mathematics Subject Classification: 35B35, 93B05, 93C20.

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References

[1]	W. Arendt and C. J. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.doi: 10.1090/S0002-9947-1988-0933321-3.
[2]	C. D. Benchimol, A note on weak stabilization of contraction semi-groups, SIAM J. Control Optim., 16 (1978), 373-379.doi: 10.1137/0316023.
[3]	H. Brezis, Analyse Fonctionelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.
[4]	N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.doi: 10.1002/cpa.3160400305.
[5]	F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs., 1 (1985), 43-56.
[6]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, 1994.
[7]	J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Vol. I, Masson, Paris, 1988.
[8]	J.-L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.doi: 10.1137/1030001.
[9]	J.-L. Lions and E. Magenes, Problemes aux Limites Non-Homogenes et Applications, Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.
[10]	Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, Chapman & Hall/CRC, 1999.
[11]	Ö. Morgül, Dynamic boundary control of a Euler-Bernoulli beam, IEEE Transactions on Automatic Control, 37 (1992), 639-642.doi: 10.1109/9.135504.
[12]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 1983.doi: 10.1007/978-1-4612-5561-1.
[13]	J. Prüss, On the spectrum of $C^0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.doi: 10.2307/1999112.
[14]	B. Rao, Stabilization of elastic plates with dynamical boundary control, SIAM J. Control Optim., 36 (1998), 148-163.doi: 10.1137/S0363012996300975.
[15]	B. Rao, Exact boundary controllability of a hybrid system of elasticity by the HUM method, ESIAM: COCV, 6 (2001), 183-199.doi: 10.1051/cocv:2001107.
[16]	B. Rao and A. Wehbe, Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent, J. Evol. Equ., 5 (2005), 137-152.doi: 10.1007/s00028-005-0171-5.
[17]	D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.doi: 10.1006/jmaa.1993.1071.
[18]	M. Slemrod, Feedback stabilization of a linear system in Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2 (1989), 265-285.doi: 10.1007/BF02551387.
[19]	A. Wehbe, Rational energy decay rate in a wave equation with dynamical control, Appl. Math. Letters, 16 (2003), 357-364.doi: 10.1016/S0893-9659(03)80057-5.
[20]	A. Wehbe, Observability and controllability for a vibrating string with dynamical boundary control, Electr. J. Diff. Equ., (2010), 1-13.
[21]	A. Wehbe, Optimal energy decay rate in the Rayleigh beam equation with boundary dynamical controls, Bull. Belg. Math. Soc., 13 (2006), 385-400.