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November  2014, 34(11): 4371-4388. doi: 10.3934/dcds.2014.34.4371

Polynomial stabilization of some dissipative hyperbolic systems

Kais Ammari 1, , Eduard Feireisl 2, and  Serge Nicaise 3,

1. 

UR Analyse et Contrôle des Edp (05/UR/15-01), Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, 5019 Monastir, Tunisia
2. 

Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1
3. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV and FR CNRS 2956, Le Mont Houy, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9

Received  September 2013 Revised  January 2014 Published  May 2014

We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
Keywords: observability inequality, polynomial stability, dissipative hyberbolic system, acoustic equation., Exponential stability, resolvent estimate.
Mathematics Subject Classification: Primary: 35L04, 93B07; Secondary: 93B52, 74H5.
Citation: Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4371-4388. doi: 10.3934/dcds.2014.34.4371
References:
[1]

G. Allaire, Homogenization of the Navier-Stokes equations and derivation of Brinkman's law, In Mathématiques appliquées aux sciences de l'ingénieur (Santiago, 1989), pages 7-20. Cépaduès, Toulouse, 1991.  Google Scholar

[2]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386. doi: 10.1051/cocv:2001114.  Google Scholar

[3]

P. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math., 81 (1999), 497-520. doi: 10.1007/s002110050401.  Google Scholar

[4]

W. Arendt and C. J. K. 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.  Google Scholar

[5]

C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Rend. Sem. Mat. Univ. Politec. Torino 1988, (Special Issue), (1989), 11-31. Nonlinear hyperbolic equations in applied sciences.  Google Scholar

[6]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[7]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. doi: 10.1002/mana.200410429.  Google Scholar

[8]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.  Google Scholar

[9]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.  Google Scholar

[10]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math., 46 (1989), 245-258.  Google Scholar

[11]

J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978.  Google Scholar

[12]

M. J. Lighthill, On sound generated aerodynamically. I. General theory, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564-587. doi: 10.1098/rspa.1952.0060.  Google Scholar

[13]

M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1-32. doi: 10.1098/rspa.1954.0049.  Google Scholar

[14]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.  Google Scholar

[15]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928.  Google Scholar

[16]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.  Google Scholar

[17]

K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Differential Equations, 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[18]

L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 57-62. doi: 10.1016/j.crma.2011.12.001.  Google Scholar

[19]

E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asymptotic Anal., 1 (1988), 161-185.  Google Scholar

[20]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl. (9), 70 (1991), 513-529.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization of the Navier-Stokes equations and derivation of Brinkman's law, In Mathématiques appliquées aux sciences de l'ingénieur (Santiago, 1989), pages 7-20. Cépaduès, Toulouse, 1991.  Google Scholar

[2]

K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386. doi: 10.1051/cocv:2001114.  Google Scholar

[3]

P. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math., 81 (1999), 497-520. doi: 10.1007/s002110050401.  Google Scholar

[4]

W. Arendt and C. J. K. 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.  Google Scholar

[5]

C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Rend. Sem. Mat. Univ. Politec. Torino 1988, (Special Issue), (1989), 11-31. Nonlinear hyperbolic equations in applied sciences.  Google Scholar

[6]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[7]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. doi: 10.1002/mana.200410429.  Google Scholar

[8]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.  Google Scholar

[9]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.  Google Scholar

[10]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math., 46 (1989), 245-258.  Google Scholar

[11]

J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978.  Google Scholar

[12]

M. J. Lighthill, On sound generated aerodynamically. I. General theory, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564-587. doi: 10.1098/rspa.1952.0060.  Google Scholar

[13]

M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1-32. doi: 10.1098/rspa.1954.0049.  Google Scholar

[14]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.  Google Scholar

[15]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928.  Google Scholar

[16]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.  Google Scholar

[17]

K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Differential Equations, 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[18]

L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 57-62. doi: 10.1016/j.crma.2011.12.001.  Google Scholar

[19]

E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asymptotic Anal., 1 (1988), 161-185.  Google Scholar

[20]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl. (9), 70 (1991), 513-529.  Google Scholar

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