• Previous Article
    Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints
  • MCRF Home
  • This Issue
  • Next Article
    Extension of the $\nu$-metric for stabilizable plants over $H^\infty$
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 & 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.  doi: 10.1137/S0363012901385368.  Google Scholar

[2]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems,, J. Evolution Equations, 2 (2002), 127.  doi: 10.1007/s00028-002-8083-0.  Google Scholar

[3]

G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163.  doi: 10.1007/s00245-006-0884-z.  Google Scholar

[4]

G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system,, Semigroup Forum, 57 (1998), 278.  doi: 10.1007/PL00005977.  Google Scholar

[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, (2007), 15.   Google Scholar

[6]

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.  Google Scholar

[7]

C. D. Benchimol, A note on weak stabilizability of contraction semigroups,, SIAM J. Control Optimization, 16 (1978), 373.  doi: 10.1137/0316023.  Google Scholar

[8]

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.  Google Scholar

[9]

J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311.  doi: 10.1137/0318022.  Google Scholar

[10]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar

[11]

V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method,", RAM, (1994).   Google Scholar

[12]

J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Stud. Appl. Math. 10, (1989).   Google Scholar

[13]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Ration. Mech. Anal., 148 (1999), 179.  doi: 10.1007/s002050050160.  Google Scholar

[14]

J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués,", Vol. 1, (1988).   Google Scholar

[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.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[16]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar

[17]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.1090/S0002-9947-1984-0743749-9.  Google Scholar

[18]

J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system,, J. Math. Pures Appl. (9), 84 (2005), 407.  doi: 10.1016/j.matpur.2004.09.006.  Google Scholar

[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.  doi: 10.1016/0022-247X(72)90055-8.  Google Scholar

[20]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems,, J. Math. Anal. Appl., 173 (1993), 339.  doi: 10.1006/jmaa.1993.1071.  Google Scholar

[21]

L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations,, DCDS B, 14 (2010), 1601.  doi: 10.3934/dcdsb.2010.14.1601.  Google Scholar

[22]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation,, Proc. Amer. Math. Soc., 105 (1989), 375.  doi: 10.1090/S0002-9939-1989-0953013-0.  Google Scholar

[23]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, Commun. Contemp. Math., 5 (2003), 25.  doi: 10.1142/S0219199703000896.  Google Scholar

[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.  doi: 10.1007/s00205-006-0020-x.  Google Scholar

show all references

References:
[1]

Fatiha Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems,, SIAM J. Control Optim., 41 (2002), 511.  doi: 10.1137/S0363012901385368.  Google Scholar

[2]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems,, J. Evolution Equations, 2 (2002), 127.  doi: 10.1007/s00028-002-8083-0.  Google Scholar

[3]

G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163.  doi: 10.1007/s00245-006-0884-z.  Google Scholar

[4]

G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system,, Semigroup Forum, 57 (1998), 278.  doi: 10.1007/PL00005977.  Google Scholar

[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, (2007), 15.   Google Scholar

[6]

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.  Google Scholar

[7]

C. D. Benchimol, A note on weak stabilizability of contraction semigroups,, SIAM J. Control Optimization, 16 (1978), 373.  doi: 10.1137/0316023.  Google Scholar

[8]

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.  Google Scholar

[9]

J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311.  doi: 10.1137/0318022.  Google Scholar

[10]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar

[11]

V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method,", RAM, (1994).   Google Scholar

[12]

J. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Stud. Appl. Math. 10, (1989).   Google Scholar

[13]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Ration. Mech. Anal., 148 (1999), 179.  doi: 10.1007/s002050050160.  Google Scholar

[14]

J.-L. Lions, "Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués,", Vol. 1, (1988).   Google Scholar

[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.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[16]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar

[17]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.1090/S0002-9947-1984-0743749-9.  Google Scholar

[18]

J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system,, J. Math. Pures Appl. (9), 84 (2005), 407.  doi: 10.1016/j.matpur.2004.09.006.  Google Scholar

[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.  doi: 10.1016/0022-247X(72)90055-8.  Google Scholar

[20]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems,, J. Math. Anal. Appl., 173 (1993), 339.  doi: 10.1006/jmaa.1993.1071.  Google Scholar

[21]

L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations,, DCDS B, 14 (2010), 1601.  doi: 10.3934/dcdsb.2010.14.1601.  Google Scholar

[22]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation,, Proc. Amer. Math. Soc., 105 (1989), 375.  doi: 10.1090/S0002-9939-1989-0953013-0.  Google Scholar

[23]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, Commun. Contemp. Math., 5 (2003), 25.  doi: 10.1142/S0219199703000896.  Google Scholar

[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.  doi: 10.1007/s00205-006-0020-x.  Google Scholar

[1]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[2]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425

[3]

Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675

[4]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37

[5]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

[6]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029

[7]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251

[8]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[9]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[10]

Yongqin Liu, Shuichi Kawashima. Decay property for a plate equation with memory-type dissipation. Kinetic & Related Models, 2011, 4 (2) : 531-547. doi: 10.3934/krm.2011.4.531

[11]

Mohammad M. Al-Gharabli, Aissa Guesmia, Salim A. Messaoudi. Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 159-180. doi: 10.3934/cpaa.2019009

[12]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

[13]

Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987

[14]

Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483

[15]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057

[16]

Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic & Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605

[17]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

[18]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017

[19]

Adnan H. Sabuwala, Doreen De Leon. Particular solution to the Euler-Cauchy equation with polynomial non-homegeneities. Conference Publications, 2011, 2011 (Special) : 1271-1278. doi: 10.3934/proc.2011.2011.1271

[20]

Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]