June  2015, 5(2): 305-320. doi: 10.3934/mcrf.2015.5.305

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, France

2. 

Université Libanaise, EDST et Faculté des sciences I, Equipe EDP-AN, Hadath, Beyrouth, Lebanon

3. 

Université Libanaise, Sciences 1 et EDST, Equipe EDP-AN, Hadath, Beyrouth

Received  December 2013 Revised  June 2014 Published  April 2015

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.
Citation: Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
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.

show all references

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.

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