Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T22:21:22.006Z Has data issue: false hasContentIssue false
  • English
  • Français

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Published online by Cambridge University Press:  27 September 2011

Bao-Zhu Guo  and
Guo-Dong Zhang
Bao-Zhu Guo
Affiliation:
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, P.R. China. bzguo@iss.ac.cn School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R. China
Guo-Dong Zhang
Affiliation:
School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa School of Mathematical Science, Heilongjiang University, Harbin 150080, P.R. China
Get access

Abstract

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the latter case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded. The results of this paper expositorily demonstrate the proper modeling the elastic systems with Boltzmann damping.

Keywords

Wave equationspectrumRiesz basisstabilityBoltzmann damping
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 889 - 913
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amendola, G., Fabrizio, M., Golden, J.M. and Lazzari, B., Free energies and asymptotic behaviour for incompressible viscoelastic fluids. Appl. Anal. 88 (2009) 789–805. Google Scholar
H.T. Banks, G.A. Pinter, L.K. Potter, B.C. Munoz and L.C. Yanyo, Estimation and control related issues in smart material structures and fluids, The 4th International Conference on Optimization: Techniques and Applications. Perth, Australia (1998) 19–34.
Banks, H.T., Hood, J.B. and Medhin, N.G., A molecular based model for polymer viscoelasticity: intra- and inter-molecular variability. Appl. Math. Model. 32 (2008) 2753–2767. Google Scholar
Chen, G., Fulling, S.A., Narcowich, F.J. and Sun, S., Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Control Optim. 51 (1991) 266–301. Google Scholar
Chen, S.P., Liu, K.S. and Liu, Z.Y., Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping. SIAM J. Appl. Math. 59 (1999) 651–668. Google Scholar
Dafermos, C.M., An abstract Volterra equation with application to linear viscoelasticity. J. Differential Equations 7 (1970) 554–569. Google Scholar
Fabrizio, M. and Lazzari, B., On the existence and asymptotic stability of solutions for linearly viscoelastic solids. Arch. Ration. Mech. Anal. 116 (1991) 139–152. Google Scholar
Fabrizio, M. and Polidoro, S., Asymptotic decay for some diferential systems with fading memory. Appl. Anal. 81 (2002) 1245–1264. Google Scholar
I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Math. Monographs 18. AMS Providence (1969).
Guo, B.Z., Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim. 39 (2001) 1736–1747. Google Scholar
Guo, B.Z., Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients. SIAM J. Control Optim. 40 (2002) 1905–1923. Google Scholar
B.Z. Guo and H. Zwart, Riesz Spectral System. Preprint, University of Twenty, the Netherlands (2001).
Guo, B.Z., Wang, J.M. and Zhang, G.D., Spectral analysis of a wave equation with Kelvin–Voigt damping. Z. Angew. Math. Mech. 90 (2010) 323–342. Google Scholar
Jacob, B., Trunk, C. and Winklmeier, M., Analyticity and Riesz basis property of semigroups associated to damped vibrations. J. Evol. Equ. 8 (2008) 263–281. Google Scholar
Liu, K.S. and Liu, Z.Y., On the type of C 0-semigroup associated with the abstract linear viscoelastic system. Z. Angew. Math. Phys. 47 (1996) 1–15. Google Scholar
Liu, K.S. and Liu, Z.Y., Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping. SIAM J. Control Optim. 36 (1998) 1086–1098. Google Scholar
Liu, K.S. and Liu, Z.Y., Exponential decay of energy of vibrating strings with local viscoelasticity. Z. Angew. Math. Phys. 53 (2002) 265–280. Google Scholar
Liu, Y.Z. and Zheng, S.M., On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math. 54 (1996) 21–31. Google Scholar
B.P. Rao, Optimal energy decay rate in a damped Rayleigh beam, Contemporary Mathematics. RI, Providence 209 (1997) 221–229.
Renardy, M., On localized Kelvin–Voigt damping. Z. Angew. Math. Mech. 84 (2004) 280–283. Google Scholar
Rivera, J.E.M. and Oquendo, H.P., The transmission problem of viscoelastic waves. Acta Appl. Math. 62 (2000) 1–21. Google Scholar
Tzou, H.S. and Ding, J.H., Optimal control of precision paraboloidal shell structronic systems. J. Sound Vib. 276 (2004) 273–291. Google Scholar
Wang, J.M., Guo, B.Z. and Fu, M.Y., Dynamic behavior of a heat equation with memory. Math. Methods Appl. Sci. 32 (2009) 1287–1310. Google Scholar
Zhao, H.L., Liu, K.S. and Liu, Z.Y., A note on the exponential decay of energy of a Euler–Bernoulli beam with local viscoelasticity. J. Elasticity 74 (2004) 175–183. Google Scholar
Zhao, H.L., Liu, K.S. and Zhang, C.G., Stability for the Timoshenko beam system with local Kelvin-Voigt damping. Acta Math. Sinica (Engl. Ser.) 21 (2005) 655–666. Google Scholar
Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equations 15 (1990) 205–235. Google Scholar