# American Institute of Mathematical Sciences

May  2022, 15(5): 1183-1220. doi: 10.3934/dcdss.2021142

## Exponential and polynomial stability results for networks of elastic and thermo-elastic rods

 1 Université Polytechnique, Hauts-de-France, LAMAV, FR CNRS 2037, 59313 Valenciennes Cedex 9, France 2 Lebanese University, Faculty of Sciences 1, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon

* Corresponding author: Serge Nicaise

Received  February 2021 Revised  September 2021 Published  May 2022 Early access  December 2021

In this paper, we investigate a network of elastic and thermo-elastic materials. On each thermo-elastic edge, we consider two coupled wave equations such that one of them is damped via a coupling with a heat equation. On each elastic edge (undamped), we consider two coupled conservative wave equations. Under some conditions, we prove that the thermal damping is enough to stabilize the whole system. If the two waves propagate with the same speed on each thermo-elastic edge, we show that the energy of the system decays exponentially. Otherwise, a polynomial energy decay is attained. Finally, we present some other boundary conditions and show that under sufficient conditions on the lengths of some elastic edges, the energy of the system decays exponentially on some particular networks similar to the ones considered in [18].

Citation: Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe. Exponential and polynomial stability results for networks of elastic and thermo-elastic rods. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1183-1220. doi: 10.3934/dcdss.2021142
##### References:
 [1] A. B. Abdallah and F. Shel, Exponential stability of a general network of 1-d thermoelastic rods, Math. Control Relat. Fields, 2 (2012), 1-16.  doi: 10.3934/mcrf.2012.2.1. [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3. [3] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2009), 455-478.  doi: 10.1007/s00208-009-0439-0. [4] J. Burns, Z. Liu and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl., 179 (1993), 574-591.  doi: 10.1006/jmaa.1993.1370. [5] C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271.  doi: 10.1007/BF00276727. [6] L. Fatori, E. Lueders and J. Rivera, Transmission problem for hyperbolic thermoelastic systems, J. Thermal Stresses, 26 (2003), 739-763.  doi: 10.1080/713855994. [7] Z.-J. Han and E. Zuazua, Decay rates for elastic-thermoelastic star-shaped networks, Netw. Heterog. Media, 12 (2017), 461-488.  doi: 10.3934/nhm.2017020. [8] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442.  doi: 10.1016/0022-247X(92)90217-2. [9] A. Hayek, S. Nicaise, Z. Salloum and A. Wehbe, A transmission problem of a system of weakly coupled wave equations with Kelvin–Voigt dampings and non-smooth coefficient at the interface, SeMA, 77 (2020), 305-338.  doi: 10.1007/s40324-020-00218-x. [10] F. L. Huang, Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [11] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231.  doi: 10.1007/s002050050160. [12] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, FL, 1999. [13] Z. Liu and S. M. Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math., 51 (1993), 535-545.  doi: 10.1090/qam/1233528. [14] A. Marzocchi, J. E. M. Rivera and M. G. Naso, Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity, Math. Methods Appl. Sci., 25 (2002), 955-980.  doi: 10.1002/mma.323. [15] J. C. Oliveira and R. C. Charão, Stabilization of a locally damped thermoelastic system, Comput. Appl. Math., 27 (2008), 319-357. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [17] J. Prüss, On the spectrum of ${C}_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112. [18] F. Shel, Exponential stability of a network of elastic and thermoelastic materials, Math. Methods Appl. Sci., 36 (2013), 869-879.  doi: 10.1002/mma.2644. [19] F. Shel, Exponential stability of a network of beams, J. Dyn. Control Syst., 21 (2015), 443-460.  doi: 10.1007/s10883-014-9257-0. [20] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Control Optim., 48 (2009), 2771-2797.  doi: 10.1137/080733590.

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##### References:
 [1] A. B. Abdallah and F. Shel, Exponential stability of a general network of 1-d thermoelastic rods, Math. Control Relat. Fields, 2 (2012), 1-16.  doi: 10.3934/mcrf.2012.2.1. [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3. [3] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2009), 455-478.  doi: 10.1007/s00208-009-0439-0. [4] J. Burns, Z. Liu and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl., 179 (1993), 574-591.  doi: 10.1006/jmaa.1993.1370. [5] C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271.  doi: 10.1007/BF00276727. [6] L. Fatori, E. Lueders and J. Rivera, Transmission problem for hyperbolic thermoelastic systems, J. Thermal Stresses, 26 (2003), 739-763.  doi: 10.1080/713855994. [7] Z.-J. Han and E. Zuazua, Decay rates for elastic-thermoelastic star-shaped networks, Netw. Heterog. Media, 12 (2017), 461-488.  doi: 10.3934/nhm.2017020. [8] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442.  doi: 10.1016/0022-247X(92)90217-2. [9] A. Hayek, S. Nicaise, Z. Salloum and A. Wehbe, A transmission problem of a system of weakly coupled wave equations with Kelvin–Voigt dampings and non-smooth coefficient at the interface, SeMA, 77 (2020), 305-338.  doi: 10.1007/s40324-020-00218-x. [10] F. L. Huang, Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [11] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231.  doi: 10.1007/s002050050160. [12] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, FL, 1999. [13] Z. Liu and S. M. Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math., 51 (1993), 535-545.  doi: 10.1090/qam/1233528. [14] A. Marzocchi, J. E. M. Rivera and M. G. Naso, Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity, Math. Methods Appl. Sci., 25 (2002), 955-980.  doi: 10.1002/mma.323. [15] J. C. Oliveira and R. C. Charão, Stabilization of a locally damped thermoelastic system, Comput. Appl. Math., 27 (2008), 319-357. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [17] J. Prüss, On the spectrum of ${C}_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112. [18] F. Shel, Exponential stability of a network of elastic and thermoelastic materials, Math. Methods Appl. Sci., 36 (2013), 869-879.  doi: 10.1002/mma.2644. [19] F. Shel, Exponential stability of a network of beams, J. Dyn. Control Syst., 21 (2015), 443-460.  doi: 10.1007/s10883-014-9257-0. [20] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Control Optim., 48 (2009), 2771-2797.  doi: 10.1137/080733590.
A thermoelastic rod
An elastic/thermo-elastic transmission problem
An elastic/thermo-elastic transmission problem
Elastic/therm-elastic networks
Elastic/thermo-elastic star shaped network
Elastic/thermo-elastic networks
A circuit and its parametrizations: $\; {\pi_{1}(0) = a_{1}, \; \pi_{2}(0) = a_{2}, \; {\rm{and}}\; \pi_{3}(0) = a_{3}}$
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