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

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 3, page 889-913
- ISSN: 1292-8119

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topGuo, Bao-Zhu, and Zhang, Guo-Dong. "On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 889-913. <http://eudml.org/doc/276368>.

@article{Guo2012,

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.},

author = {Guo, Bao-Zhu, Zhang, Guo-Dong},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Wave equation; spectrum; Riesz basis; stability; Boltzmann damping; memory of materials; generalized eigenfunctions},

language = {eng},

month = {11},

number = {3},

pages = {889-913},

publisher = {EDP Sciences},

title = {On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗},

url = {http://eudml.org/doc/276368},

volume = {18},

year = {2012},

}

TY - JOUR

AU - Guo, Bao-Zhu

AU - Zhang, Guo-Dong

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2012/11//

PB - EDP Sciences

VL - 18

IS - 3

SP - 889

EP - 913

AB - 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.

LA - eng

KW - Wave equation; spectrum; Riesz basis; stability; Boltzmann damping; memory of materials; generalized eigenfunctions

UR - http://eudml.org/doc/276368

ER -

## References

top- G. Amendola, M. Fabrizio, J.M. Golden and B. Lazzari, Free energies and asymptotic behaviour for incompressible viscoelastic fluids. Appl. Anal.88 (2009) 789–805. Zbl1170.76007
- 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.
- H.T. Banks, J.B. Hood and N.G. Medhin, A molecular based model for polymer viscoelasticity: intra- and inter-molecular variability. Appl. Math. Model.32 (2008) 2753–2767. Zbl1167.74321
- G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Control Optim.51 (1991) 266–301. Zbl0734.35009
- S.P. Chen, K.S. Liu and Z.Y. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping. SIAM J. Appl. Math.59 (1999) 651–668. Zbl0924.35018
- C.M. Dafermos, An abstract Volterra equation with application to linear viscoelasticity. J. Differential Equations7 (1970) 554–569. Zbl0212.45302
- M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linearly viscoelastic solids. Arch. Ration. Mech. Anal.116 (1991) 139–152. Zbl0766.73013
- M. Fabrizio and S. Polidoro, Asymptotic decay for some diferential systems with fading memory. Appl. Anal.81 (2002) 1245–1264. Zbl1035.45006
- I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Math. Monographs18. AMS Providence (1969). Zbl0181.13504
- B.Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim.39 (2001) 1736–1747. Zbl1183.93111
- B.Z. Guo, Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients. SIAM J. Control Optim.40 (2002) 1905–1923. Zbl1015.93025
- B.Z. Guo and H. Zwart, Riesz Spectral System. Preprint, University of Twenty, the Netherlands (2001).
- B.Z. Guo, J.M. Wang and G.D. Zhang, Spectral analysis of a wave equation with Kelvin–Voigt damping. Z. Angew. Math. Mech.90 (2010) 323–342. Zbl1279.35085
- B. Jacob, C. Trunk and M. Winklmeier, Analyticity and Riesz basis property of semigroups associated to damped vibrations. J. Evol. Equ.8 (2008) 263–281. Zbl1156.47038
- K.S. Liu and Z.Y. Liu, On the type of C0-semigroup associated with the abstract linear viscoelastic system. Z. Angew. Math. Phys.47 (1996) 1–15. Zbl0841.73026
- K.S. Liu and Z.Y. Liu, Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping. SIAM J. Control Optim.36 (1998) 1086–1098. Zbl0909.35018
- K.S. Liu and Z.Y. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity. Z. Angew. Math. Phys.53 (2002) 265–280. Zbl0999.35012
- Y.Z. Liu and S.M. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quart. Appl. Math.54 (1996) 21–31. Zbl0868.35011
- B.P. Rao, Optimal energy decay rate in a damped Rayleigh beam, Contemporary Mathematics. RI, Providence 209 (1997) 221–229. Zbl0910.35013
- M. Renardy, On localized Kelvin–Voigt damping. Z. Angew. Math. Mech.84 (2004) 280–283. Zbl1058.35143
- J.E.M. Rivera and H.P. Oquendo, The transmission problem of viscoelastic waves. Acta Appl. Math.62 (2000) 1–21. Zbl0980.74033
- H.S. Tzou and J.H. Ding, Optimal control of precision paraboloidal shell structronic systems. J. Sound Vib.276 (2004) 273–291. Zbl1236.74210
- J.M. Wang, B.Z. Guo and M.Y. Fu, Dynamic behavior of a heat equation with memory. Math. Methods Appl. Sci.32 (2009) 1287–1310. Zbl1180.35099
- H.L. Zhao, K.S. Liu and Z.Y. Liu, A note on the exponential decay of energy of a Euler–Bernoulli beam with local viscoelasticity. J. Elasticity74 (2004) 175–183. Zbl1058.74050
- H.L. Zhao, K.S. Liu and C.G. Zhang, Stability for the Timoshenko beam system with local Kelvin-Voigt damping. Acta Math. Sinica (Engl. Ser.)21 (2005) 655–666. Zbl1084.35013
- E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equations15 (1990) 205–235. Zbl0716.35010

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