June  2016, 5(2): 235-250. doi: 10.3934/eect.2016003

Exponential stability of a coupled system with Wentzell conditions

1. 

Laboratoire AMNEDP, Mathematics Department, USTHB, BP 32 El-Alia, Bab-Ezzouar, Algiers, Algeria, Algeria

Received  February 2016 Revised  April 2016 Published  June 2016

A coupled system of hyperbolic equations in Maxwell/wave with Wentzell conditions in a bounded domain of $\mathbb{R}^3$ is considered. Under suitable assumptions, we show the exponential stability of the system. Our method is based on an identity with multipliers that allows to show an appropriate stability estimate.
Citation: Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations and Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
References:
[1]

R. Bey, A. Heminna and J. P. Loheac, Boundary stabilization of a linear elastodynamic system with variable coefficients, Electronic Journal of Differential Equations, 78 (2001), 1-23.

[2]

M. Eller, J. E. Lagnese and S. Nicaise, Stabilization of heteregeneous Maxwell's equations by linear or nonlinear boundary feedbacks, Electronic journal of differential equations, 21 (2002), 1-26.

[3]

M. Eller, J. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comp. and Apppl. Math., 21 (2002), 135-165.

[4]

A. Heminna, Contrôlabilité Exacte et Stabilisation Frontière de Divers Problèmes aux Limites Modélisant des Jonctions de Multi-structures, Thesis, U.S.T.H.B, Alger, 2000.

[5]

A. Heminna, Contrôlabilité exacte d'un problème avec conditions de Ventcel evolutives pour le système linéaire de l'elasticité, Revista Matemàtica Complutense, 14 (2001), 231-270. doi: 10.5209/rev_REMA.2001.v14.n1.17061.

[6]

A. Heminna, Stabilisation frontière de l'équation des ondes avec condition de Ventcel, Maghreb Math. Rev, 11 (2002), 165-196.

[7]

A. Heminna, Stabilisation frontière de problèmes de Ventcel, C. R. Acad. Sci. Paris Sèr. I Math, 328 (1999), 1171-1174. doi: 10.1016/S0764-4442(99)80434-0.

[8]

A. Heminna, Stabilisation Frontière de Problèmes de Ventcel, ESAIM Control Optim. Calc. Var, 5 (2000), 591-622. doi: 10.1051/cocv:2000123.

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985.

[10]

B. V. Kapitanov and M. A. Raupp, Exact boundary controllability in problems of transmission for the system of electromagneto-elastic, Math. Meth. Appl. Sci, 24 (2001), 193-207. doi: 10.1002/mma.205.

[11]

V. Komornik, Boundary stabilization, observation and control of Maxwell's equations, PanAm. Math. J, 4 (1994), 47-61.

[12]

V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, RAM 36, Masson, Paris, 1994.

[13]

J. E. Lagnese, Exact controllability of Maxwell's equations in a general region, SIAM J. Control Optim, 27 (1989), 374-388. doi: 10.1137/0327019.

[14]

K. Laoubi and S. Nicaise, Polynomial stabilization of the wave equation with Ventcel's boundary conditions, Math. Nachr, 283 (2010), 1428-1438. doi: 10.1002/mana.200710162.

[15]

K. Lemrabet, Etude de Divers Problèmes aux Limites de Ventcel D'origine Physique ou Mécanique dans des Domaines non Réguliers, Thesis, U.S.T.H.B, Alger, 1987.

[16]

K. Lemrabet, Problème aux limites de Ventcel dans un domaine non régulier, C. R. Acad. Sci. Paris Sér. I Math, 300 (1985), 531-534.

[17]

J. L. Lions, Contrôlabilité Exacte, Perturbation et Stabilisation de Syst\`eme Distribués, tome1, Masson, 1988.

[18]

S. Nicaise, Stability and controllability of the electromagneto-elastic system, Portugaliae mathematica, 60 (2003), 37-70.

[19]

S. Nicaise, Exact boundary controllability of Maxwell's equations in heteregeneous media and an application to an inverse source problem, SIAM J. Control and Opt, 38 (2000), 1145-1170. doi: 10.1137/S0363012998344373.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys and Monographs, 49, AMS, 1997.

[22]

A. D. Wentzell (Ventcel), On boundary conditions for multi-dimensional diffusion processes, Theor. Probab. Appl, 4 (1959), 164-177. doi: 10.1137/1104014.

show all references

References:
[1]

R. Bey, A. Heminna and J. P. Loheac, Boundary stabilization of a linear elastodynamic system with variable coefficients, Electronic Journal of Differential Equations, 78 (2001), 1-23.

[2]

M. Eller, J. E. Lagnese and S. Nicaise, Stabilization of heteregeneous Maxwell's equations by linear or nonlinear boundary feedbacks, Electronic journal of differential equations, 21 (2002), 1-26.

[3]

M. Eller, J. E. Lagnese and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comp. and Apppl. Math., 21 (2002), 135-165.

[4]

A. Heminna, Contrôlabilité Exacte et Stabilisation Frontière de Divers Problèmes aux Limites Modélisant des Jonctions de Multi-structures, Thesis, U.S.T.H.B, Alger, 2000.

[5]

A. Heminna, Contrôlabilité exacte d'un problème avec conditions de Ventcel evolutives pour le système linéaire de l'elasticité, Revista Matemàtica Complutense, 14 (2001), 231-270. doi: 10.5209/rev_REMA.2001.v14.n1.17061.

[6]

A. Heminna, Stabilisation frontière de l'équation des ondes avec condition de Ventcel, Maghreb Math. Rev, 11 (2002), 165-196.

[7]

A. Heminna, Stabilisation frontière de problèmes de Ventcel, C. R. Acad. Sci. Paris Sèr. I Math, 328 (1999), 1171-1174. doi: 10.1016/S0764-4442(99)80434-0.

[8]

A. Heminna, Stabilisation Frontière de Problèmes de Ventcel, ESAIM Control Optim. Calc. Var, 5 (2000), 591-622. doi: 10.1051/cocv:2000123.

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985.

[10]

B. V. Kapitanov and M. A. Raupp, Exact boundary controllability in problems of transmission for the system of electromagneto-elastic, Math. Meth. Appl. Sci, 24 (2001), 193-207. doi: 10.1002/mma.205.

[11]

V. Komornik, Boundary stabilization, observation and control of Maxwell's equations, PanAm. Math. J, 4 (1994), 47-61.

[12]

V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, RAM 36, Masson, Paris, 1994.

[13]

J. E. Lagnese, Exact controllability of Maxwell's equations in a general region, SIAM J. Control Optim, 27 (1989), 374-388. doi: 10.1137/0327019.

[14]

K. Laoubi and S. Nicaise, Polynomial stabilization of the wave equation with Ventcel's boundary conditions, Math. Nachr, 283 (2010), 1428-1438. doi: 10.1002/mana.200710162.

[15]

K. Lemrabet, Etude de Divers Problèmes aux Limites de Ventcel D'origine Physique ou Mécanique dans des Domaines non Réguliers, Thesis, U.S.T.H.B, Alger, 1987.

[16]

K. Lemrabet, Problème aux limites de Ventcel dans un domaine non régulier, C. R. Acad. Sci. Paris Sér. I Math, 300 (1985), 531-534.

[17]

J. L. Lions, Contrôlabilité Exacte, Perturbation et Stabilisation de Syst\`eme Distribués, tome1, Masson, 1988.

[18]

S. Nicaise, Stability and controllability of the electromagneto-elastic system, Portugaliae mathematica, 60 (2003), 37-70.

[19]

S. Nicaise, Exact boundary controllability of Maxwell's equations in heteregeneous media and an application to an inverse source problem, SIAM J. Control and Opt, 38 (2000), 1145-1170. doi: 10.1137/S0363012998344373.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys and Monographs, 49, AMS, 1997.

[22]

A. D. Wentzell (Ventcel), On boundary conditions for multi-dimensional diffusion processes, Theor. Probab. Appl, 4 (1959), 164-177. doi: 10.1137/1104014.

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