March  2012, 2(1): 1-16. doi: 10.3934/mcrf.2012.2.1

Exponential stability of a general network of 1-d thermoelastic rods

1. 

Université de Sfax, Institut Supérieur d’Informatique et du Multimédia de Sfax, Pôle technologique, Route de Tunis, km 10, B.P. 242, Sfax 3021, Tunisia

2. 

Université de Sfax, Institut Supérieur d’Informatique et du Multimédia de Sfax, Route Manzel Chaker, Km 0.5, B.P. 1172, Sfax 3018, Tunisia

Received  November 2010 Revised  December 2011 Published  January 2012

We consider a finite planar network of 1-$d$ thermoelastic rods using Fourier's law or Cattaneo's law for heat conduction, we show that the system is exponentially stable in the two cases.
Citation: Abdallah Ben Abdallah, Farhat Shel. Exponential stability of a general network of 1-d thermoelastic rods. Mathematical Control & Related Fields, 2012, 2 (1) : 1-16. doi: 10.3934/mcrf.2012.2.1
References:
[1]

K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback,, Math. Control Signals Systems, 15 (2002), 229.  doi: 10.1007/s004980200009.  Google Scholar

[2]

D. E. Carlson, Linear thermoelasticity,, in, (1972), 297.   Google Scholar

[3]

C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation,, Comput. Rendus, 247 (1958), 431.   Google Scholar

[4]

R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures,", Mathématiques & Applications, 50 (2006).   Google Scholar

[5]

J. E. Muñoz Rivera, F. Ammar Khodja, A. Benabdallah and R. Racke, Energy decay for Timoshenko system of memory type,, J. Differential Equations, 194 (2003), 82.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[6]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space,, Trans. Amer. Math. Soc., 236 (1978), 385.  doi: 10.1090/S0002-9947-1978-0461206-1.  Google Scholar

[7]

Y. N. Guo and G. Q. Xu, Stability and Riesz basis property for general network of strings,, J. Dynamical and Control Systems, 15 (2009), 223.  doi: 10.1007/s10883-009-9064-1.  Google Scholar

[8]

S. Jiang and R. Racke, "Evolution Equation in Thermoelasticity,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 112 (2000).   Google Scholar

[9]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams,, Mathematical Methods in the Applied Sciences, 16 (1993), 327.  doi: 10.1002/mma.1670160503.  Google Scholar

[10]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity,, Arch. Rat. Mech. Anal., 141 (1998), 297.  doi: 10.1007/s002050050078.  Google Scholar

[11]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Archives Rat. Mech. Anal., 148 (1999), 179.  doi: 10.1007/s002050050160.  Google Scholar

[12]

Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems,", Chapman & Hall/CRC Research Notes in Mathematics, 398 (1999).   Google Scholar

[13]

A. Marzocchi, J. E. Muñoz Rivera and M. G. Naso, Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity,, Math. Meth. Appl. Sci., 25 (2002), 955.  doi: 10.1002/mma.323.  Google Scholar

[14]

A. Pazy, "Semigroup of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar

[15]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.1090/S0002-9947-1984-0743749-9.  Google Scholar

[16]

R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d,, Math. Meth. Appl. Sci., 25 (2002), 409.  doi: 10.1002/mma.298.  Google Scholar

[17]

R. Racke, J. E. M. Rivera and H. F. Sare, Stability for a transmission problem in thermoelasticity with second sound,, Journal of Thermal Stresses, 31 (2008), 1170.  doi: 10.1080/01495730802508004.  Google Scholar

[18]

Y. Saad, "Iterative Methods for Sparse Linear Systems,", Second edition, (2003).   Google Scholar

[19]

Hugo D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law,, Arch. Rational Mech. Anal., 194 (2009), 221.  doi: 10.1007/s00205-009-0220-2.  Google Scholar

[20]

J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks,, Lin. Algebra Appl., 71 (1985), 309.  doi: 10.1016/0024-3795(85)90258-7.  Google Scholar

show all references

References:
[1]

K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback,, Math. Control Signals Systems, 15 (2002), 229.  doi: 10.1007/s004980200009.  Google Scholar

[2]

D. E. Carlson, Linear thermoelasticity,, in, (1972), 297.   Google Scholar

[3]

C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation,, Comput. Rendus, 247 (1958), 431.   Google Scholar

[4]

R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures,", Mathématiques & Applications, 50 (2006).   Google Scholar

[5]

J. E. Muñoz Rivera, F. Ammar Khodja, A. Benabdallah and R. Racke, Energy decay for Timoshenko system of memory type,, J. Differential Equations, 194 (2003), 82.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar

[6]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space,, Trans. Amer. Math. Soc., 236 (1978), 385.  doi: 10.1090/S0002-9947-1978-0461206-1.  Google Scholar

[7]

Y. N. Guo and G. Q. Xu, Stability and Riesz basis property for general network of strings,, J. Dynamical and Control Systems, 15 (2009), 223.  doi: 10.1007/s10883-009-9064-1.  Google Scholar

[8]

S. Jiang and R. Racke, "Evolution Equation in Thermoelasticity,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 112 (2000).   Google Scholar

[9]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams,, Mathematical Methods in the Applied Sciences, 16 (1993), 327.  doi: 10.1002/mma.1670160503.  Google Scholar

[10]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity,, Arch. Rat. Mech. Anal., 141 (1998), 297.  doi: 10.1007/s002050050078.  Google Scholar

[11]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Archives Rat. Mech. Anal., 148 (1999), 179.  doi: 10.1007/s002050050160.  Google Scholar

[12]

Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems,", Chapman & Hall/CRC Research Notes in Mathematics, 398 (1999).   Google Scholar

[13]

A. Marzocchi, J. E. Muñoz Rivera and M. G. Naso, Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity,, Math. Meth. Appl. Sci., 25 (2002), 955.  doi: 10.1002/mma.323.  Google Scholar

[14]

A. Pazy, "Semigroup of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar

[15]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.1090/S0002-9947-1984-0743749-9.  Google Scholar

[16]

R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d,, Math. Meth. Appl. Sci., 25 (2002), 409.  doi: 10.1002/mma.298.  Google Scholar

[17]

R. Racke, J. E. M. Rivera and H. F. Sare, Stability for a transmission problem in thermoelasticity with second sound,, Journal of Thermal Stresses, 31 (2008), 1170.  doi: 10.1080/01495730802508004.  Google Scholar

[18]

Y. Saad, "Iterative Methods for Sparse Linear Systems,", Second edition, (2003).   Google Scholar

[19]

Hugo D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law,, Arch. Rational Mech. Anal., 194 (2009), 221.  doi: 10.1007/s00205-009-0220-2.  Google Scholar

[20]

J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks,, Lin. Algebra Appl., 71 (1985), 309.  doi: 10.1016/0024-3795(85)90258-7.  Google Scholar

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