# American Institute of Mathematical Sciences

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

## Polynomial stabilization of some dissipative hyperbolic systems

 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.
Citation: Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 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), 7. [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. doi: 10.1051/cocv:2001114. [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. doi: 10.1007/s002110050401. [4] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837. doi: 10.1090/S0002-9947-1988-0933321-3. [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, (1989), 11. [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. doi: 10.1137/0330055. [7] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425. doi: 10.1002/mana.200410429. [8] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. [9] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. [10] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math., 46 (1989), 245. [11] J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978). [12] M. J. Lighthill, On sound generated aerodynamically. I. General theory,, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564. doi: 10.1098/rspa.1952.0060. [13] M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound,, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1. doi: 10.1098/rspa.1954.0049. [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, (1988). [15] K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control Optim., 35 (1997), 1574. doi: 10.1137/S0363012995284928. [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. doi: 10.1007/s00033-004-3073-4. [17] K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Differential Equations, 240 (2007), 92. doi: 10.1016/j.jde.2007.05.016. [18] L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Math. Acad. Sci. Paris, 350 (2012), 57. doi: 10.1016/j.crma.2011.12.001. [19] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asymptotic Anal., 1 (1988), 161. [20] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures Appl. (9), 70 (1991), 513.

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), 7. [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. doi: 10.1051/cocv:2001114. [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. doi: 10.1007/s002110050401. [4] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837. doi: 10.1090/S0002-9947-1988-0933321-3. [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, (1989), 11. [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. doi: 10.1137/0330055. [7] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425. doi: 10.1002/mana.200410429. [8] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. [9] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. [10] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math., 46 (1989), 245. [11] J. Lighthill, Waves in Fluids,, Cambridge University Press, (1978). [12] M. J. Lighthill, On sound generated aerodynamically. I. General theory,, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564. doi: 10.1098/rspa.1952.0060. [13] M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound,, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1. doi: 10.1098/rspa.1954.0049. [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, (1988). [15] K. Liu, Locally distributed control and damping for the conservative systems,, SIAM J. Control Optim., 35 (1997), 1574. doi: 10.1137/S0363012995284928. [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. doi: 10.1007/s00033-004-3073-4. [17] K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Differential Equations, 240 (2007), 92. doi: 10.1016/j.jde.2007.05.016. [18] L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Math. Acad. Sci. Paris, 350 (2012), 57. doi: 10.1016/j.crma.2011.12.001. [19] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems,, Asymptotic Anal., 1 (1988), 161. [20] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures Appl. (9), 70 (1991), 513.
 [1] Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1 [2] Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003 [3] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [4] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673 [5] George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207 [6] Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321 [7] Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014 [8] Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801 [9] Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219 [10] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [11] Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733 [12] Giorgio Menegatti, Luca Rondi. Stability for the acoustic scattering problem for sound-hard scatterers. Inverse Problems & Imaging, 2013, 7 (4) : 1307-1329. doi: 10.3934/ipi.2013.7.1307 [13] Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367 [14] Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219 [15] S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 [16] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424 [17] Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471 [18] Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 [19] Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028 [20] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

2016 Impact Factor: 1.099