November  2003, 9(6): 1625-1639. doi: 10.3934/dcds.2003.9.1625

Global stability for damped Timoshenko systems

1. 

Department of Research and Development, National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitandinha, CEP 25651-070, Petrópolis, RJ and UFRJ, Rio de Janeiro, Brazil

2. 

Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany

Received  July 2002 Revised  September 2003 Published  September 2003

We consider a nonlinear Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. The system has a dissipative mechanism through frictional damping being present only in the equation for the rotation angle. We first give an alternative proof for a sufficient and necessary condition for exponential stability for the linear case. Polynomial stability is proved in general. The global existence of small, smooth solutions and the exponential stability is investigated for the nonlinear case.
Citation: J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625
[1]

Yi Cheng, Zhihui Dong, Donal O' Regan. Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021230

[2]

Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control and Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321

[3]

Xiu-Fang Liu, Gen-Qi Xu. Exponential stabilization of Timoshenko beam with input and output delays. Mathematical Control and Related Fields, 2016, 6 (2) : 271-292. doi: 10.3934/mcrf.2016004

[4]

Filippo Dell'Oro, Marcio A. Jorge Silva, Sandro B. Pinheiro. Exponential stability of Timoshenko-Gurtin-Pipkin systems with full thermal coupling. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2189-2207. doi: 10.3934/dcdss.2022050

[5]

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

[6]

M. Grasselli, Vittorino Pata, Giovanni Prouse. Longtime behavior of a viscoelastic Timoshenko beam. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 337-348. doi: 10.3934/dcds.2004.10.337

[7]

Mohammad Al-Gharabli, Mohamed Balegh, Baowei Feng, Zayd Hajjej, Salim A. Messaoudi. Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021038

[8]

Manoel J. Dos Santos, João C. P. Fortes, Marcos L. Cardoso. Exponential stability for a piezoelectric beam with a magnetic effect and past history. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021283

[9]

Jeongho Ahn, David E. Stewart. A viscoelastic Timoshenko beam with dynamic frictionless impact. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 1-22. doi: 10.3934/dcdsb.2009.12.1

[10]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447

[11]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations and Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021

[12]

Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261

[13]

Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733

[14]

Bruce Geist and Joyce R. McLaughlin. Eigenvalue formulas for the uniform Timoshenko beam: the free-free problem. Electronic Research Announcements, 1998, 4: 12-17.

[15]

Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations and Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713

[16]

Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075

[17]

Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control and Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035

[18]

Genni Fragnelli, Dimitri Mugnai. Stability of solutions for nonlinear wave equations with a positive--negative damping. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 615-622. doi: 10.3934/dcdss.2011.4.615

[19]

Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations and Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041

[20]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (112)

Other articles
by authors

[Back to Top]