October  2015, 20(8): 2733-2750. doi: 10.3934/dcdsb.2015.20.2733

Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping

1. 

Department of Mathematics, Tianjin University of Technology, Tianjin 300384, China

2. 

Department of Mathematics, Tianjin University, Tianjin 300072

Received  October 2014 Revised  December 2014 Published  August 2015

In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as ``thermoelasticity of type II''. This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that the energy of this system generally decays to zero exponentially. Furthermore, by showing the spectrum of the system is empty under certain condition and estimating the norm of the resolvent operator, we give a sufficient condition on the super-stability of this thermoelastic system. Under this condition, the solution to the system is identical to zero after finite time. Moreover, we also estimate the maximum existence time of the nonzero part of the solution. Finally, we give some numerical simulations.
Citation: Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733
References:
[1]

M. Aouadi, Exponential stability in hyperbolic thermoelastic diffusion problem with second sound,, International Journal of Differential Equations, 2011 (2011).  doi: 10.1155/2011/274843.  Google Scholar

[2]

A. V. Balakrishnan, Smart structures and super-stability,, in Computational Science for 21st Century, (1997), 660.   Google Scholar

[3]

A. V. Balakrishnan, On superstable semigroup of operators,, Dynamic Systems and Applications, 5 (1996), 371.   Google Scholar

[4]

A. V. Balakrishnan, On superstability of semigroups, systems modelling and optimization,, in Proceedings of the 18th IFIP Conference on System Modelling and Optimization (eds. M. P. Polis, (1999), 12.   Google Scholar

[5]

A. V. Balakrishnan, Superstability of systems,, Appl. Math. Comput., 164 (2005), 321.  doi: 10.1016/j.amc.2004.06.052.  Google Scholar

[6]

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature,, Appl. Mech. Rev., 51 (1998), 705.   Google Scholar

[7]

D. S. Chandrasekharaiah, A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipations,, J. Elasticity, 43 (1996), 279.  doi: 10.1080/01495739608946173.  Google Scholar

[8]

D. S. Chandrasekharaiah, Complete solutions in the theory of thermoelasticity without energy dissipations,, Mech. Res. Comm., 24 (1997), 625.  doi: 10.1016/S0093-6413(97)00080-3.  Google Scholar

[9]

C. M. Dafermos, On the existence and the asymptotic stability of solution to the equation of linear thermoelasticity,, Arch. Rational Mech. Anal., 29 (1968), 241.  doi: 10.1007/BF00276727.  Google Scholar

[10]

A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping,, Journal of Dynamical and Control Systems, 16 (2010), 189.  doi: 10.1007/s10883-010-9089-5.  Google Scholar

[11]

L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, IMA Journal of Applied Mathematics, 75 (2010), 881.  doi: 10.1093/imamat/hxq038.  Google Scholar

[12]

H. D. Fernández Sare and J. E. Muñoz Rivera, Optimal rates of decay in 2-d thermoelasticity with second sound,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.4734239.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. R. Soc. Lond. Ser. A, 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, Journal of Elasticity, 31 (1993), 189.  doi: 10.1007/BF00044969.  Google Scholar

[15]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253.  doi: 10.1080/01495739208946136.  Google Scholar

[16]

A. E. Green and P. M. Naghdi, A unified pocedure for contruction of theories of deformable media, I. Clasical continuum physics,, Proc. R. Soc. Lond. Ser. A, 448 (1995), 335.  doi: 10.1098/rspa.1995.0020.  Google Scholar

[17]

L. M. Gearhart, Spectral theory for contraction semigroups on Hilbert space,, Transactions of the American Mathematical Society, 236 (1978), 385.  doi: 10.1090/S0002-9947-1978-0461206-1.  Google Scholar

[18]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar

[19]

J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds,, Oxford mathematical Monographs, (2010).   Google Scholar

[20]

S. Jiang and R. Racke, Evolution Equations in Thermoelasticity,, Monographs and Surveys in Pure and Applied Mathematics, (2000).   Google Scholar

[21]

Z. Liu and R. Quintanilla, Energy decay rate of a mixed type II and type III thermoelastic system,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433.  doi: 10.3934/dcdsb.2010.14.1433.  Google Scholar

[22]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems,, CRC Research Notes in Mathematics, (1999).   Google Scholar

[23]

B. Lazzari and R. Nibbi, On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary,, J. Math. Anal. Appl., 338 (2008), 317.  doi: 10.1016/j.jmaa.2007.05.017.  Google Scholar

[24]

M. C. Leseduarte, A. Magana and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II,, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 375.  doi: 10.3934/dcdsb.2010.13.375.  Google Scholar

[25]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system,, Discrete Contin. Dyn. Sys., 33 (2013), 5189.  doi: 10.3934/dcds.2013.33.5189.  Google Scholar

[26]

I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system,, Nonlinear Differential Equations and Applications, 15 (2008), 689.  doi: 10.1007/s00030-008-0011-8.  Google Scholar

[27]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III,, J. Math. Anal. Appl., 348 (2008), 298.  doi: 10.1016/j.jmaa.2008.07.036.  Google Scholar

[28]

S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds,, Arabian J. Math., 2 (2013), 199.  doi: 10.1007/s40065-012-0061-y.  Google Scholar

[29]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III,, Advanced in Differential Equations, 14 (2009), 375.   Google Scholar

[30]

J. E. Muñoz Rivera, Energy decay rates in linear thermoelasticty,, Funkcialaj Ekvacioj, 35 (1992), 19.   Google Scholar

[31]

J. E. Muñoz Rivera, Asymptotic behaviour in inhomogeneous linear thermoelasticity,, Applicable Analysis, 53 (1994), 55.  doi: 10.1080/00036819408840243.  Google Scholar

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[33]

J. Prüss, On the spectrum of $C_0$-semigroups,, Transactions of the American Mathematical Society, 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

[34]

R. Quintanilla, Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation,, Continuum Mech. Thermodyn, 13 (2001), 121.  doi: 10.1007/s001610100044.  Google Scholar

[35]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type III,, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 383.  doi: 10.3934/dcdsb.2003.3.383.  Google Scholar

[36]

Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks,, IMA Journal of Mathematical Control and Information, 31 (2014), 73.  doi: 10.1093/imamci/dnt003.  Google Scholar

[37]

L. N. Trefethen, Spectral Methods in Matlab,, SIAM, (2000).  doi: 10.1137/1.9780898719598.  Google Scholar

[38]

F. E. Udwadia, Boundary control, quiet boundaries, super-stability and super-instability,, Appl. Math. Comput., 164 (2005), 327.  doi: 10.1016/j.amc.2004.06.040.  Google Scholar

[39]

F. E. Udwadia, On the longitudinal vibrations of a bar with viscous boundaries: Super-stability, superinstability, and loss of damping,, Internat. J. Eng. Sci., 50 (2012), 79.  doi: 10.1016/j.ijengsci.2011.09.001.  Google Scholar

[40]

J. M. Wang and B. Z. Guo, On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type,, Journal of the Franklin Institute-Engineering and Applied Mathematics, 344 (2007), 75.  doi: 10.1016/j.jfranklin.2005.10.003.  Google Scholar

[41]

G. Q. Xu, Stabilization of string system with linear boundary feedback,, Nonlinear Anal.: Hybrid Syst., 1 (2007), 383.  doi: 10.1016/j.nahs.2006.07.003.  Google Scholar

[42]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, Communications in Contemporary Mathematics, 5 (2003), 25.  doi: 10.1142/S0219199703000896.  Google Scholar

[43]

Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network,, Acta Appl. Math., 124 (2013), 19.  doi: 10.1007/s10440-012-9768-1.  Google Scholar

show all references

References:
[1]

M. Aouadi, Exponential stability in hyperbolic thermoelastic diffusion problem with second sound,, International Journal of Differential Equations, 2011 (2011).  doi: 10.1155/2011/274843.  Google Scholar

[2]

A. V. Balakrishnan, Smart structures and super-stability,, in Computational Science for 21st Century, (1997), 660.   Google Scholar

[3]

A. V. Balakrishnan, On superstable semigroup of operators,, Dynamic Systems and Applications, 5 (1996), 371.   Google Scholar

[4]

A. V. Balakrishnan, On superstability of semigroups, systems modelling and optimization,, in Proceedings of the 18th IFIP Conference on System Modelling and Optimization (eds. M. P. Polis, (1999), 12.   Google Scholar

[5]

A. V. Balakrishnan, Superstability of systems,, Appl. Math. Comput., 164 (2005), 321.  doi: 10.1016/j.amc.2004.06.052.  Google Scholar

[6]

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature,, Appl. Mech. Rev., 51 (1998), 705.   Google Scholar

[7]

D. S. Chandrasekharaiah, A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipations,, J. Elasticity, 43 (1996), 279.  doi: 10.1080/01495739608946173.  Google Scholar

[8]

D. S. Chandrasekharaiah, Complete solutions in the theory of thermoelasticity without energy dissipations,, Mech. Res. Comm., 24 (1997), 625.  doi: 10.1016/S0093-6413(97)00080-3.  Google Scholar

[9]

C. M. Dafermos, On the existence and the asymptotic stability of solution to the equation of linear thermoelasticity,, Arch. Rational Mech. Anal., 29 (1968), 241.  doi: 10.1007/BF00276727.  Google Scholar

[10]

A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping,, Journal of Dynamical and Control Systems, 16 (2010), 189.  doi: 10.1007/s10883-010-9089-5.  Google Scholar

[11]

L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, IMA Journal of Applied Mathematics, 75 (2010), 881.  doi: 10.1093/imamat/hxq038.  Google Scholar

[12]

H. D. Fernández Sare and J. E. Muñoz Rivera, Optimal rates of decay in 2-d thermoelasticity with second sound,, J. Math. Phys., 53 (2012).  doi: 10.1063/1.4734239.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. R. Soc. Lond. Ser. A, 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, Journal of Elasticity, 31 (1993), 189.  doi: 10.1007/BF00044969.  Google Scholar

[15]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253.  doi: 10.1080/01495739208946136.  Google Scholar

[16]

A. E. Green and P. M. Naghdi, A unified pocedure for contruction of theories of deformable media, I. Clasical continuum physics,, Proc. R. Soc. Lond. Ser. A, 448 (1995), 335.  doi: 10.1098/rspa.1995.0020.  Google Scholar

[17]

L. M. Gearhart, Spectral theory for contraction semigroups on Hilbert space,, Transactions of the American Mathematical Society, 236 (1978), 385.  doi: 10.1090/S0002-9947-1978-0461206-1.  Google Scholar

[18]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar

[19]

J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds,, Oxford mathematical Monographs, (2010).   Google Scholar

[20]

S. Jiang and R. Racke, Evolution Equations in Thermoelasticity,, Monographs and Surveys in Pure and Applied Mathematics, (2000).   Google Scholar

[21]

Z. Liu and R. Quintanilla, Energy decay rate of a mixed type II and type III thermoelastic system,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433.  doi: 10.3934/dcdsb.2010.14.1433.  Google Scholar

[22]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems,, CRC Research Notes in Mathematics, (1999).   Google Scholar

[23]

B. Lazzari and R. Nibbi, On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary,, J. Math. Anal. Appl., 338 (2008), 317.  doi: 10.1016/j.jmaa.2007.05.017.  Google Scholar

[24]

M. C. Leseduarte, A. Magana and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II,, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 375.  doi: 10.3934/dcdsb.2010.13.375.  Google Scholar

[25]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system,, Discrete Contin. Dyn. Sys., 33 (2013), 5189.  doi: 10.3934/dcds.2013.33.5189.  Google Scholar

[26]

I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system,, Nonlinear Differential Equations and Applications, 15 (2008), 689.  doi: 10.1007/s00030-008-0011-8.  Google Scholar

[27]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III,, J. Math. Anal. Appl., 348 (2008), 298.  doi: 10.1016/j.jmaa.2008.07.036.  Google Scholar

[28]

S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds,, Arabian J. Math., 2 (2013), 199.  doi: 10.1007/s40065-012-0061-y.  Google Scholar

[29]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III,, Advanced in Differential Equations, 14 (2009), 375.   Google Scholar

[30]

J. E. Muñoz Rivera, Energy decay rates in linear thermoelasticty,, Funkcialaj Ekvacioj, 35 (1992), 19.   Google Scholar

[31]

J. E. Muñoz Rivera, Asymptotic behaviour in inhomogeneous linear thermoelasticity,, Applicable Analysis, 53 (1994), 55.  doi: 10.1080/00036819408840243.  Google Scholar

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[33]

J. Prüss, On the spectrum of $C_0$-semigroups,, Transactions of the American Mathematical Society, 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

[34]

R. Quintanilla, Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation,, Continuum Mech. Thermodyn, 13 (2001), 121.  doi: 10.1007/s001610100044.  Google Scholar

[35]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type III,, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 383.  doi: 10.3934/dcdsb.2003.3.383.  Google Scholar

[36]

Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks,, IMA Journal of Mathematical Control and Information, 31 (2014), 73.  doi: 10.1093/imamci/dnt003.  Google Scholar

[37]

L. N. Trefethen, Spectral Methods in Matlab,, SIAM, (2000).  doi: 10.1137/1.9780898719598.  Google Scholar

[38]

F. E. Udwadia, Boundary control, quiet boundaries, super-stability and super-instability,, Appl. Math. Comput., 164 (2005), 327.  doi: 10.1016/j.amc.2004.06.040.  Google Scholar

[39]

F. E. Udwadia, On the longitudinal vibrations of a bar with viscous boundaries: Super-stability, superinstability, and loss of damping,, Internat. J. Eng. Sci., 50 (2012), 79.  doi: 10.1016/j.ijengsci.2011.09.001.  Google Scholar

[40]

J. M. Wang and B. Z. Guo, On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type,, Journal of the Franklin Institute-Engineering and Applied Mathematics, 344 (2007), 75.  doi: 10.1016/j.jfranklin.2005.10.003.  Google Scholar

[41]

G. Q. Xu, Stabilization of string system with linear boundary feedback,, Nonlinear Anal.: Hybrid Syst., 1 (2007), 383.  doi: 10.1016/j.nahs.2006.07.003.  Google Scholar

[42]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, Communications in Contemporary Mathematics, 5 (2003), 25.  doi: 10.1142/S0219199703000896.  Google Scholar

[43]

Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network,, Acta Appl. Math., 124 (2013), 19.  doi: 10.1007/s10440-012-9768-1.  Google Scholar

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