# American Institute of Mathematical Sciences

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 and 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), 229-255. doi: 10.1007/s004980200009. [2] D. E. Carlson, Linear thermoelasticity, in "Handbuch der Physik," Springer-Verlag, Berlin, (1972), 297-346. [3] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Comput. Rendus, 247 (1958), 431-433. [4] R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures," Mathématiques & Applications, 50, Springer-Verlag, Berlin, 2006. [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), 82-115. doi: 10.1016/S0022-0396(03)00185-2. [6] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. [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), 223-245. doi: 10.1007/s10883-009-9064-1. [8] S. Jiang and R. Racke, "Evolution Equation in Thermoelasticity," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 112, Chapman & Hall/CRC, Boca Raton, FL, 2000. [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), 327-358. doi: 10.1002/mma.1670160503. [10] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rat. Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078. [11] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Archives Rat. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [12] Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. [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), 955-980. doi: 10.1002/mma.323. [14] A. Pazy, "Semigroup of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, 1983. [15] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.1090/S0002-9947-1984-0743749-9. [16] R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. [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), 1170-1189. doi: 10.1080/01495730802508004. [18] Y. Saad, "Iterative Methods for Sparse Linear Systems," Second edition, SIAM, Philadelphia, PA, 2003. [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), 221-251. doi: 10.1007/s00205-009-0220-2. [20] J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks, Lin. Algebra Appl., 71 (1985), 309-325. doi: 10.1016/0024-3795(85)90258-7.

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##### 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), 229-255. doi: 10.1007/s004980200009. [2] D. E. Carlson, Linear thermoelasticity, in "Handbuch der Physik," Springer-Verlag, Berlin, (1972), 297-346. [3] C. Cattaneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Comput. Rendus, 247 (1958), 431-433. [4] R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures," Mathématiques & Applications, 50, Springer-Verlag, Berlin, 2006. [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), 82-115. doi: 10.1016/S0022-0396(03)00185-2. [6] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. [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), 223-245. doi: 10.1007/s10883-009-9064-1. [8] S. Jiang and R. Racke, "Evolution Equation in Thermoelasticity," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 112, Chapman & Hall/CRC, Boca Raton, FL, 2000. [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), 327-358. doi: 10.1002/mma.1670160503. [10] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rat. Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078. [11] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Archives Rat. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [12] Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems," Chapman & Hall/CRC Research Notes in Mathematics, 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. [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), 955-980. doi: 10.1002/mma.323. [14] A. Pazy, "Semigroup of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, 1983. [15] J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.1090/S0002-9947-1984-0743749-9. [16] R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. [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), 1170-1189. doi: 10.1080/01495730802508004. [18] Y. Saad, "Iterative Methods for Sparse Linear Systems," Second edition, SIAM, Philadelphia, PA, 2003. [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), 221-251. doi: 10.1007/s00205-009-0220-2. [20] J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks, Lin. Algebra Appl., 71 (1985), 309-325. doi: 10.1016/0024-3795(85)90258-7.
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