doi: 10.3934/dcdss.2021142
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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 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 & Continuous Dynamical Systems - S, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. BurnsZ. 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.  Google Scholar

[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.  Google Scholar

[6]

L. FatoriE. Lueders and J. Rivera, Transmission problem for hyperbolic thermoelastic systems, J. Thermal Stresses, 26 (2003), 739-763.  doi: 10.1080/713855994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. HayekS. NicaiseZ. 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.  Google Scholar

[10]

F. L. Huang, Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar

[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.  Google Scholar

[12]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[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.  Google Scholar

[14]

A. MarzocchiJ. 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.  Google Scholar

[15]

J. C. Oliveira and R. C. Charão, Stabilization of a locally damped thermoelastic system, Comput. Appl. Math., 27 (2008), 319-357.   Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.  Google Scholar

[17]

J. Prüss, On the spectrum of $ {C}_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. BurnsZ. 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.  Google Scholar

[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.  Google Scholar

[6]

L. FatoriE. Lueders and J. Rivera, Transmission problem for hyperbolic thermoelastic systems, J. Thermal Stresses, 26 (2003), 739-763.  doi: 10.1080/713855994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. HayekS. NicaiseZ. 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.  Google Scholar

[10]

F. L. Huang, Characteristics conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar

[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.  Google Scholar

[12]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[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.  Google Scholar

[14]

A. MarzocchiJ. 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.  Google Scholar

[15]

J. C. Oliveira and R. C. Charão, Stabilization of a locally damped thermoelastic system, Comput. Appl. Math., 27 (2008), 319-357.   Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.  Google Scholar

[17]

J. Prüss, On the spectrum of $ {C}_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  A thermoelastic rod
Figure 2.  An elastic/thermo-elastic transmission problem
Figure 3.  An elastic/thermo-elastic transmission problem
Figure 4.  Elastic/therm-elastic networks
Figure 5.  Elastic/thermo-elastic star shaped network
Figure 6.  Elastic/thermo-elastic networks
Figure 7.  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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