UNIFORM DECAY IN WEAKLY DISSIPATIVE TIMOSHENKO SYSTEM WITH INTERNAL DISTRIBUTED DELAY FEEDBACKS

doi:10.1016/S0252-9602(16)30042-X

摘要/Abstract

摘要：

In this paper we consider one-dimensional Timoshenko system with linear frictional damping and a distributed delay acting on the displacement equation. Under suitable assumptions on the weight of the delay and the wave speeds, we establish the well-posedness of the system and show that the dissipation through the frictional damping is strong enough to uniformly stabilize the system even in the presence of delay.

关键词: Timoshenko system, well-posedness, uniform stability, distributed delay

Abstract:

Key words: Timoshenko system, well-posedness, uniform stability, distributed delay

中图分类号:

35B40

Tijani A. APALARA. UNIFORM DECAY IN WEAKLY DISSIPATIVE TIMOSHENKO SYSTEM WITH INTERNAL DISTRIBUTED DELAY FEEDBACKS[J]. 数学物理学报(英文版), 2016, 36(3): 815-830.

Tijani A. APALARA. UNIFORM DECAY IN WEAKLY DISSIPATIVE TIMOSHENKO SYSTEM WITH INTERNAL DISTRIBUTED DELAY FEEDBACKS[J]. Acta mathematica scientia,Series B, 2016, 36(3): 815-830.

参考文献

[1] Abdallah C, Dorato P, Benitez-Read J, Byrne R. Delayed Positive Feedback Can Stabilize Oscillatory System. San Francisco: ACC, 1993: 3106-3107
[2] Almeida Júnior D S, Santos M L, Muñoz Rivera J E. Stability to weakly dissipative Timoshenko systems. Math Meth Appl Sci, 2013, 36(14): 1965-1976
[3] Apalara T A, Messaoudi S A, Mustafa M I, Energy decay in thermoelastic type Ⅲ with viscoelastic damping and delay. Elect J Differ Eqns, 2012, 2012(128): 1-15
[4] Apalara T A, Messaoudi S A. An exponential stability result of a Timoshenko system with Thermoelasticity with second sound and in the presence of delay. Appl Math Optim, 2014: 1-24
[5] Apalara T A. Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay. Elect J Differ Eqns, 2014, 2014(254): 1-15
[6] Apalara T A. Asymptotic behavior of wenkly dissipative Timoshenko system with internal constant delay feedbacks. Appl Anal, 2016, 95(1): 187-202
[7] Beuter A, Bélair J, Labrie C. Feedback and delays in neurological diseases: a modeling study using dynamical systems. Bull Math Bio, 1993, 55(3): 525-541
[8] Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2010
[9] Datko R, Lagnese J, Polis M P. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim, 1986, 24(1): 152-156
[10] Guesmia A, Messaoudi S A. On the control of a viscoelastic damped Timoshenko-type system. Appl Math Compt, 2008, 206(2): 589-597
[11] Guesmia A, Messaoudi S A. General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math Meth Appl Sci, 2009, 32(16): 2102-2122
[12] Guesmia A, Messaoudi S A, Soufyane A. Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-Heat systems. Elect J Diff Equa, 2012, 2012(193): 1-45
[13] Guesmia A, Messaoudi S A. On the stabilization of Timoshenko systems with memory and different speeds of wave propagation. Appl Math Comp, 2013, 219(17): 9424-9437
[14] Guesmia A. Some well-posedness and general stability results in Timoshenko systems with infinite memory and distributed time delay. J Math Phy, 2014, 55(8): 081503
[15] Guesmia A, Tatar N. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Comm Pure Appl Anal, 2015, 14(2): 457-491
[16] Kim J U, Renardy Y. Boundary control of the Timoshenko beam. SIAM J Contr Optim, 1987, 25(6): 1417-1429
[17] Kirane M, Said-Houari B. Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z Angew Math Phys, 2011, 62(6): 1065-1082
[18] Kirane, M., Said-Houari B, Anwar M. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Commun Pure Appl Anal, 2011, 10(2): 667-686
[19] Messaoudi S A, Mustafa M I. On the internal and boundary stabilization of Timoshenko beams. Nonl Differ Eqns Appl, 2008, 15(6): 655-671
[20] Messaoudi S A, MustafaM I. On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math Meth Appl Sci, 2009, 32(4): 454-469
[21] Messaoudi S A, Mustafa M I. A stability result in a memory-type Timoshenko system. Dyn Sys Appl, 2009, 18(3): 457-468
[22] Messaoudi S A, Said-Houari B. Uniform decay in a Timoshenko-type system with past history. J Math Anal Appl, 2009, 360(2): 459-475
[23] Messaoudi S A, Pokojovy M, Said-Houari B. Nonlinear damped Timoshenko systems with second soundglobal existence and exponential stability. Math Meth Appl Sci, 2009, 32(5): 505-534
[24] Messaoudi S A, Apalara T A. Asymptotic stability of thermoelasticity type Ⅲ with delay term and infinite memory. IMA J Math Control Info, 2015, 32(1): 75-95
[25] Muñoz Rivera J E, Racke R. Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J Math Analy Appl, 2002, 276(1): 248-278
[26] Muñoz Rivera J E, Racke R. Timoshenko systems with indefinite damping. J Math Anal Appl, 2008, 341(2): 1068-1083
[27] Muñoz Rivera, J. E. and Fernández Sare, H. D. Stability of Timoshenko systems with past history. J Math Anal Appl, 2008, 339(1): 482-502
[28] Mustafa M I. Uniform stability for thermoelastic systems with boundary time-varying delay. J Math Anal Appl, 2011, 383(2): 490-498
[29] Mustafa M I. Exponential decay in thermoelastic systems with boundary delay. J Abstr Diff Equa, 2011, 2(1): 1-13
[30] MustafaM I, KafiniM. Exponential decay in thermoelastic systems with internal distributed delay. Palestine J Math, 2013, 2(2): 287-299
[31] Mustafa M I. A uniform stability result for thermoelasticity of type Ⅲ with boundary distributed delay. J Abstr Diff Equa Appl, 2014, 2(1): 1-13
[32] Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J Control Optim, 2006, 45(5): 1561-1585
[33] Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Diff Int Equs, 2008, 21(9/10): 935-958
[34] Nicaise S, Valein, J, Fridman E. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete Contin Dyn Syst Ser S, 2009, 2(3): 559-581
[35] Nicaise S, Pignotti C. Interior feedback stabilization of wave equations with time dependent delay. Elect J Differ Eqns, 2011, 2011(41): 1-20
[36] Pignotti C. A note on stabilization of locally damped wave equations with time delay. Sys Control Lett, 2012, 61(1): 92-97
[37] Racke R, Said-Houari B. Global existence and decay property of the Timoshenko system in thermoelasticity with second sound. Nonl Anal Theory Meth Appl, 2012, 75(13): 4957-4973
[38] Racke R. Instability of coupled systems with delay. Comm Pure Appl Anal, 2012, 11(5): 1753-1773
[39] Richard J P. Time-delay systems: an overview of some recent advances and open problems. Automatica, 2003, 39(10): 1667-1694
[40] Raposo C A, Ferreira J, Santos M L, Castro N N O. Exponential stability for the Timoshenko system with two weak dampings. Appl Math Lett, 2005, 18(5): 535-541
[41] Said-Houari B, Kasimov A. Decay property of Timoshenko system in thermoelasticity. Math Meth Appl Sci, 2012, 35(3): 314-333
[42] Said-Houari B, Soufyane A. Stability result of the Timoshenko system with delay and boundary feedback. IMA J Math Contr Info, 2012, 29(3): 383-398
[43] Santos M L, Almeida Júnior D S, Muńoz Rivera J E. The stability number of the Timoshenko system with second sound. J Differ Eqns, 2012, 253(9): 2715-2733
[44] Soufyane A, Whebe A. Uniform stabilization for the Timoshenko beam by a locally distributed damping. Elect J Differ Eqns, 2003, 2003(29): 1-14
[45] Timoshenko S P. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine Series, 1921, 6(41): 245, 744-746