Exponential stability to a laminated beam in thermoelasticity of type III with delay

Madani Douib; Salah Zitouni; Abdelhak Djebabla

doi:10.26637/mjm1001/002

Abstract

In this paper, we study the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III with delay term in the fourth equation. We first give the well-posedness of the system by using semigroup method and Lumer-Philips theorem. Then, by using the perturbed energy method and construct some Lyapunov functionals, we obtain the exponential decay result for the case of equal wave speeds.

Keywords:

Laminated beam, thermoelasticity of type III, delay, well-posedness, exponential stability

Mathematics Subject Classification:

35B40, 35L56, 93D20, 74F05

Authors Imprint References Funding Related Articles Downloads

Madani Douib Department of Mathematics, Faculty of Sciences, University of Annaba, Algeria. Department of Mathematics, Teachers Higher College of Laghouat, Algeria.
Salah Zitouni Department of Mathematics, Teachers Higher College of Laghouat, Algeria. https://orcid.org/0000-0002-9949-7939
Abdelhak Djebabla Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras 41000, Algeria.

Pages: 20-35
Date Published: 01-01-2022
Vol. 10 No. 01 (2022): Malaya Journal of Matematik (MJM)

T. A. Apalara, Uniform stability of a laminated beam with structural damping and second sound, Z. Angew. Math. Phys., 68(2)(2017), 1-16.

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev., 51(12)(1998), 705-729.

M. M. ChEN, W. J. Liu AND W. C. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms, Adv. Nonlinear Anal., 7(4)(2018), 547-569.

R. Datкo, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26(3)(1988), 697-713.

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24(1986), 152-156.

A. Djebabla and N. E. TATAR, Exponential stabilization of the timoshenko system by a thermo-viscoelastic damping, J. Dyn. Control Syst., 16(2010), 189-210.

M. Douib, S. Zitouni, And A. Djebabla, Well-posedness and exponential decay for a laminated beam in thermoelasticity of type III with delay term, Mathematica, 62(2021), 58-76.

S. Drabla, S. A. Messaoudi and F. Boulanouar, A general decay result for a multidimensional weakly damped thermoelastic system with second sound, Discrete Contin. Dyn. Syst. Ser. B, 22(4)(2017), 13291339 .

A. Fareh and S. A. MeSsaoudi, Stabilization of a type III thermoelastic timoshenko system in the presence of a time-distributed delay, Math. Nachr., 290(7)(2017), 1017-1032.

J. A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.

A. E. Green And P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. London Ser. A, 432(1885)(1991), 171-194.

A. E. Green and P. M. NAghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15(2)(1992), 253-264.

A. E. Green And P. M. NAghdi, Thermoelasticity without energy dissipation, J Elasticity, 31(1993), 189208 .

A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., 30(4)(2013), 507-526.

S. W. HAnsen, A model for a two-layered plate with interfacial slip, Control and estimation of distributed parameter systems: nonlinear phenomena (Vorau, 1993), Internat. Ser. Numer. Math., Birkhauser, Basel, (1994), 143-170.

S. W. Hansen and R. Spies, Structural damping in a laminated beam due to interfacial slip, J. Sound Vibration, 204(2)(1997), 183-202.

J. HAO AND P. WANG, symptotical stability for memory-type porous thermoelastic system of type III with constant time delay, Math. Methods Appl. Sci. 39(2016), 3855-3865.

M. Kafini, S. A. Messaoudi, M. I. Mustafa and T. Apalara, Well-posedness and stability results in a timoshenko-type system of thermoelasticity of type III with delay, Z. Angew. Math. Phys., 66 (4)(2015), 1499-1517.

G. Li, X. Y. Kong And W. J. LIU, General decay for a laminated beam with structural damping and memory: the case of non equal wave speeds, J. Integral Equ. Appl., 30(1)(2018), 95-116.

W. J. LiU, K. W. CHEN AND J. Yu, Existence and general decay for the full von Kármán beam with a thermoviscoelastic damping, frictional dampings and a delay term, IMA J. Math. Control Inform., 34 (2)(2017), 521-542.

W. J. LiU And W. F. Zhao, Stabilization of a thermoelastic laminated beam with past history, Appl. Math. Optim., 80(1)(2019), 103-133.

A. Lo And N. E. TATAR, Stabilization of laminated beams with interfacial slip, Electron. J. Differential Equations, 2015(129)(2015), 1-14.

W. LiU, Y. LuAn, Y. LiU AND G. Li, Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III, Math. Methods Appl. Sci., 43(6)(2020), 3148-3166.

S. A. Messaoudi and T. A. Apalara, General stability result in a memory-type porous thermoelasticity system of type III, Arab J. Math. Sci., 20(2)(2014), 213-232.

S. A. MEssaOUdi And A. FAREh, Energy decay in a timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arab. J. Math. (Springer), 2(2)(2013), 199-207.

J. E. MuÑoz Rivera And R. RACKe, Mildly dissipative nonlinear timoshenko systems- global existence and exponential stability, J. Math. Anal. Appl., 276 (1)(2002), 248-278.

M. I. Mustafa, On the decay rates for thermoviscoelastic systems of type III, Appl. Math. Comput., $mathbf{2 3 9}(2014)$, 29-37.

S. NiCAISE AND C. PignotTi, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45(5)(2006), 1561-1558.

S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary timevarying delay, Discrete Contin. Dyn. Syst. Ser. S, 4(3)(2011), 693-722.

A. PAZy, Semigroups of Linear Operators and Applications to Partial Differential Equations, SpringerVerlag, New York, 1983.

N. E. TATAR, Stabilization of a laminated beam with interfacial slip by boundary controls, Bound. Value Probl., 2015(2015), 1-15.

G. Q. XU, S. P. Yung And L. K. LI, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (4)(2006), 770-785.