Abstract
The stability of a kind of one-dimensional thermo-elastic system of type II is considered. This system consists of two strongly coupled wave equations. Suppose that there exists an viscoelastic damping at one end of the 1-d domain. If without coupling, this damping always makes one of these two wave equations (the corresponding pure elastic system) achieve exponential stability and the other wave system (heat equation of type II) be conservative. Whether the coupling can pass the damping effect from the dissipative elastic system to the conservative heat system of type II is discussed. However, by a detailed spectral analysis, it is proved that this thermo-elastic system is at most asymptotically stable but not exponentially stable. A numerical simulation is given to support these results obtained in this paper.
Similar content being viewed by others
References
Komornik V., Zuazua E.: A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69, 33–54 (1990)
Komornik, V.: Exact controllability and stabilization: the multiplier method. Research in Applied Mathematics, vol. 36, Wiley-Masson (1994)
Lagnese J.: Note on boundary stabilization of wave equations. SIAM J. Control Optim. 26, 1250–1256 (1988)
Chen G.: Control and stabilization for the wave equation in a bounded domain, Part I. SIAM J. Control Optim. 17, 66–81 (1979)
Lasiecka I., Triggiani R.: Uniform exponential decay in a bounded region with L 2(0, T; L 2(∑)) feedback control in the Dirichlet boundary conditions. J. Differ. Equ. 66, 340–390 (1987)
Dafermos C.M.: On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity. Arch. Ration. Mech. Anal. 29, 241–271 (1968)
Muñoz Rivera J.E.: Energy decay rates in linear thermoelasticty. Funkcialaj Ekvacioj 35, 19–30 (1992)
Ignaczak J., Ostoja-Starzewski M.: Thermoelasticity with Finite Wave Speeds, Oxford mathematical Monographs. Oxford University Press, New York (2010)
Lord H.W., Shulman Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)
Racke R.: Thermoelasticity with second sound-Exponential stability in linear and non-linear 1-d. Math. Methods Appl. Sci. 25, 409–441 (2002)
Green A.E., Naghdi P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)
Green A.E., Naghdi P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. Ser. A 432, 171–194 (1991)
Green A.E., Naghdi P.M.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15, 253–264 (1992)
Green A.E., Naghdi P.M.: A unified pocedure for contruction of theories of deformable media, I. Clasical continuum physics. Proc. R. Soc. Lond. Ser. A 448, 335–356 (1995)
Chirita S., Ciarletta M.: Spatial behavior for some non-standard problems in linear thermoelasticity without energy dissipation. J. Math. Anal. Appl. 367, 58–68 (2010)
Messaoudi S.A., Said-Houari B.: Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl. 348, 298–307 (2008)
Quintanilla R., Racke R.: Stability in thermoelasticity of type III. Discrete Contin. Dyn. Syst. Ser. B 3, 383–400 (2003)
Liu Z.Y., Quintanilla R.: Energy decay rate of a mixed type II and type III thermoelastic system. Discrete Contin. Dyn. Syst. Ser. B 14, 1433–1444 (2010)
Leseduarte M.C., Magana A., Quintanilla R.: On the time decay of solutions in porous-thermo-elasticity of type II. Discrete Contin. Dyn. Syst. Ser. B 13, 375–391 (2010)
Liu Z., Zheng S.: Semigroups Associated with Dissipative Systems. Chapman&Hall/CRC, Boca Raton (1999)
Zhang X., Zuazua E.: Decay of solutions of the system of thermoelasticity of type III. Commun. Contemp. Math. 5, 1–59 (2003)
Djebabla A., Tatar N.: Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping. J. Dyn. Control Syst. 16, 189–210 (2010)
Liu Z.Y., Rao B.P.: Frequency domain approach for the polynomial stability of a system of partially damped wave equations. J. Math. Anal. Appl. 335, 860–881 (2007)
Xu G.Q., Han Z.J., Yung S.P.: Riesz basis property of serially connected Timoshenko beams. Int. J. Control 80, 470–485 (2007)
Han Z.J., Xu G.Q.: Stabilization and Riesz basis property of two serially connected Timoshenko beams system. Z. Angew. Math. Mech. 89, 962–980 (2009)
Guo B.Z., Xie Y.: A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks. SIAM J. Control Optim. 43, 1234–1252 (2004)
Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)
Lyubich Yu.I., Phóng V.Q.: Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88, 34–37 (1988)
Young, R.M.: An Introduction to Nonharmonic Fourier Series, pp. 80–84, Theorem 10. Academic Press, London (1980)
Avdonin S.A., Ivanov S.A.: Families of Exponentials. The Method of Moments in Controllability Problems For Distributed Parameter Systems. Cambridge University Press, Cambridge (1995)
Pan, C.D., Pan, C.B.: Elementary Number Theory, 3rd edn. Peking University Press, Beijing (2003, in Chinese)
Han Z.J., Xu G.Q.: Dynamical behavior of a hybrid system of nonhomogeneous timoshenko beam with partial non-collocated inputs. J. Dyn. Control Syst. 17, 77–121 (2011)
Xu G.Q., Yung S.P.: The expansion of semigroup and criterion of Riesz basis. J. Differ. Equ. 210, 1–24 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the Natural Science Foundation of China grant NSFC-61104130, 61174080 and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20110032120074).
Rights and permissions
About this article
Cite this article
Han, ZJ., Xu, GQ. & Tang, XQ. Stability analysis of a thermo-elastic system of type II with boundary viscoelastic damping. Z. Angew. Math. Phys. 63, 675–689 (2012). https://doi.org/10.1007/s00033-011-0184-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-011-0184-6