Exponential stability of serially connected thermoelastic system of type II with nodal damping

References

• Dafermos CM. On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity. Arch. Rational Mech. Anal. 1968;29:241–271.
   Web of Science ®Google Scholar
• Muñoz Rivera JE. Energy decay rates in linear thermoelasticty. Funkcialaj Ekvacioj. 1992;35: 19–30.
   Google Scholar
• Muñoz Rivera JE. Asymptotic behaviour in inhomogeneous linear thermoelasticity. Applicable Anal. 1994;53:55–66.
   Google Scholar
• Casas Pablo S, Quintanilla R. Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 2005;32:652–658.
   Web of Science ®Google Scholar
• Racke R. Initial boundary value problems in one-dimensional nonlinear thermoelasticity. Math. Methods Appl. Sci. 1988;10:517–529.
   Web of Science ®Google Scholar
• Racke R. Exponential decay for a class of initial boundary value problems in thermoelasticity. Comput. Appl. Math. 1993;12:67–80.
   Google Scholar
• Ignaczak J, Ostoja-Starzewski M. Thermoelasticity with finite wave speeds. New York: Oxford University Press; 2010.
   Google Scholar
• Green AE, Naghdi PM. A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. Ser. A. 1991;432:171–194.
   Web of Science ®Google Scholar
• Green AE, Naghdi PM. On undamped heat waves in an elastic solid. J. Thermal Stresses. 1992;15:253–264.
   Web of Science ®Google Scholar
• Green AE, Naghdi PM. Thermoelasticity without energy dissipation. J. Elasticity. 1993;31: 189–208.
   Google Scholar
• Green AE, Naghdi PM. A unified pocedure for contruction of theories of deformable media, I. Clasical continuum physics. Proc. R. Soc. Lond. Ser. A. 1995;448:335–356.
   Web of Science ®Google Scholar
• Chandrasekharaiah DS. Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 1998;51:705–729.
   Google Scholar
• Quintanilla R. Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation. Continuum Mech. Thermodyn. 2001;13:121–129.
   Web of Science ®Google Scholar
• Liu ZY, Quintanilla R. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete Contin. Dyn. Syst. Ser. B. 2010;14:1433–1444.
   Web of Science ®Google Scholar
• Leseduarte MC, Magana A, Quintanilla R. On the time decay of solutions in porous-thermo-elasticity of type II. Discrete Contin. Dyn. Syst. Ser. B. 2010;13:375–391.
   Web of Science ®Google Scholar
• Zhang X, Zuazua E. Decay of solutions of the system of thermoelasticity of type III. Commun. Contemporary Math. 2003;5:1–59.
   Web of Science ®Google Scholar
• Messaoudi SA, Said-Houari B. Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl. 2008;348:298–307.
   Web of Science ®Google Scholar
• Messaoudi SA, Fareh A. Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds. Arabian J. Math. 2013;2:199–207.
   Google Scholar
• Messaoudi SA, Said-Houari B. Energy decay in a Timoshenko-type system with history in thermoelasticity of type III. Advanced Differential Equ. 2009;4:375–400.
   Google Scholar
• Wang JM, Guo BZ. On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type. J. Franklin Institute-Engineering Applied Math. 2007;344:75–96.
   Web of Science ®Google Scholar
• Aouadi M, Soufyane A. Polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory. Applicable Anal. 2010;89:935–948.
   Web of Science ®Google Scholar
• Djebabla A, Tatar N. Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping. J. Dynamical Control Sys. 2010;16:189–210.
   Web of Science ®Google Scholar
• Lazzari B, Nibbi R. On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary. J. Math. Anal. Appl. 2008;338:317–329.
   Web of Science ®Google Scholar
• Muñoz Rivera JE, Oquendo HP. The transmission problem for thermoelastic beams. J. Thermal Stresses. 2001;24:1137–1158.
   Web of Science ®Google Scholar
• Messaoudi SA, Said-Houari B. Energy decay in a transmission problem in thermoelasticity of type III. IMA J. Appl. Math. 2009;74:344–360.
   Web of Science ®Google Scholar
• Shel F. Exponential stability of a network of elastic and thermoelastic materials. Math. Methods Applied Sci. 2013; 36:869–879; doi:10.1002/mma.2644.
   Google Scholar
• Quintanilla R, Racke R. Stability in thermoelasticity of type III. Discrete Contin. Dyn. Syst. Ser. B. 2003;3:383–400.
   Web of Science ®Google Scholar
• Marzocchi A, Muñoz Rivera JE, Naso MG. Transmission problem in thermoelasticity with symmetry. IMA J. Appl. Math. 2002;68:23–46.
   Web of Science ®Google Scholar
• Aouadi M. Exponential stability in hyperbolic thermoelastic diffusion problem with second sound. Int. J. Differential Equ. 2011. Article ID 274843. doi:10.1155/2011/274843.
   Google Scholar
• Chen G, Delfour M, Krall A, Payre G. Modeling, stabilization and control of serially connected beams. SIAM J. Control Opt. 1987;25:526–546.
   Web of Science ®Google Scholar
• Guo BZ, Xie Y. A sufficient condition on Riesz basis with parentheses of non-selfadjoint operator and application to a serially connected string system under joint feedbacks. SIAM J. Control Optim. 2004;43:1234–1252.
   Web of Science ®Google Scholar
• Liu KS, Huang FL, Chen G. Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage. SIAM J. Applied Math. 1989;49:1694–1707.
   Web of Science ®Google Scholar
• Xu GQ, Han ZJ, Yung SP. Riesz basis property of serially connected Timoshenko beams. Int. J. Control. 2007;80:470–485.
   Web of Science ®Google Scholar
• Han ZJ, Xu GQ. Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement. Asian J. Control. 2012;14:95–108.
   Web of Science ®Google Scholar
• Mercier D. Spectrum analysis of a serially connected Euler-Bernouilli beams problem. Netw. Heterog. Media. 2009;4:709–730.
   Web of Science ®Google Scholar
• Ammari K, Mercier D, Regnier V, Valein J. Spectral analysis and stabilization of a chain of serially Euler-Bernoulli beams and strings. Commun. Pure Applied Anal. 2012;11:785–807.
   Web of Science ®Google Scholar
• Han ZJ, Xu GQ, Tang XQ. Stability analysis of a thermo-elastic system of type II with boundary viscoelastic damping. Zeitschrift fur Angewandte Mathematik und Physik. 2012;63:675–689.
   Web of Science ®Google Scholar
• Gearhart LM. Spectral theory for contraction semigroups on Hilbert space. Trans. American Math. Soci. 1978;236:385–394.
   Web of Science ®Google Scholar
• Prüss J. On the spectrum of C0-semigroups. Trans. American Math. Soci. 1984;284:847–857.
   Web of Science ®Google Scholar
• Liu Z, Zheng S. Semigroups associated with dissipative systems. Boca Raton: Chapman & Hall/CRC; 1999.
   Google Scholar
• Pazy A. Semigroups of Linear Operators and applications to partial differential equations. Berlin: Springer-Verlag; 1983.
   Google Scholar
• Lyubich YuI, Phóng VQ. Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 1988;88:34–37.
   Web of Science ®Google Scholar