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March  2012, 2(1): 45-60. doi: 10.3934/mcrf.2012.2.45

## Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms

 1 Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received  August 2011 Revised  December 2011 Published  January 2012

We consider the Euler-Bernoulli equation coupled with a wave equation in a bounded domain. The Euler-Bernoulli has clamped boundary conditions and the wave equation has Dirichlet boundary conditions. The damping which is distributed everywhere in the domain under consideration acts through one of the equations only; its effect is transmitted to the other equation through the coupling. First we consider the case where the dissipation acts through the Euler-Bernoulli equation. We show that in this case the coupled system is not exponentially stable. Next, using a frequency domain approach combined with the multiplier techniques, and a recent result of Borichev and Tomilov on polynomial decay characterization of bounded semigroups, we provide precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this new system is not exponentially stable, and we provide precise polynomial decay estimates for its energy. The results obtained complement those existing in the literature involving the hinged Euler-Bernoulli equation.
Citation: Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control and Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
##### References:
 [1] Fatiha Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368. [2] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems, J. Evolution Equations, 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0. [3] G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z. [4] G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292. doi: 10.1007/PL00005977. [5] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, Fluids and waves, 15-54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007. [6] 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. [7] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization, 16 (1978), 373-379. doi: 10.1137/0316023. [8] 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. [9] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316. doi: 10.1137/0318022. [10] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [11] V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method," RAM, Masson & John Wiley, Paris, 1994. [12] J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989. [13] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [14] J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués," Vol. 1, RMA 8, Masson, Paris, 1988. [15] 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. [16] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [17] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.1090/S0002-9947-1984-0743749-9. [18] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl. (9), 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [19] D. L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl., 40 (1972), 336-368. doi: 10.1016/0022-247X(72)90055-8. [20] 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. [21] L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations, DCDS B, 14 (2010), 1601-1620. doi: 10.3934/dcdsb.2010.14.1601. [22] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.1090/S0002-9939-1989-0953013-0. [23] X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Commun. Contemp. Math., 5 (2003), 25-83. doi: 10.1142/S0219199703000896. [24] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x.

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##### References:
 [1] Fatiha Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. doi: 10.1137/S0363012901385368. [2] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems, J. Evolution Equations, 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0. [3] G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z. [4] G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292. doi: 10.1007/PL00005977. [5] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, Fluids and waves, 15-54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007. [6] 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. [7] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization, 16 (1978), 373-379. doi: 10.1137/0316023. [8] 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. [9] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316. doi: 10.1137/0318022. [10] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [11] V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method," RAM, Masson & John Wiley, Paris, 1994. [12] J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989. [13] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [14] J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués," Vol. 1, RMA 8, Masson, Paris, 1988. [15] 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. [16] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [17] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.1090/S0002-9947-1984-0743749-9. [18] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl. (9), 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [19] D. L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl., 40 (1972), 336-368. doi: 10.1016/0022-247X(72)90055-8. [20] 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. [21] L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations, DCDS B, 14 (2010), 1601-1620. doi: 10.3934/dcdsb.2010.14.1601. [22] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.1090/S0002-9939-1989-0953013-0. [23] X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Commun. Contemp. Math., 5 (2003), 25-83. doi: 10.1142/S0219199703000896. [24] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x.
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