Exponential Stability for a Flexible Structure System with Thermodiffusion Effects and Distributed Delay
DOI:
https://doi.org/10.15407/mag19.03.587Анотація
У статтi дослiджується коректнiсть та асимптотика розв’язкiв для гнучкої структури з ефектами термодифузiї та розподiленого загаювання. За вiдповiдних припущень щодо ваги демпфування та ваги розподiленого загаювання, доведено iснування i єдинiсть розв’язку з використанням теорiї пiвгруп. Далi за допомогою методу збуреної енергiї та побудови деяких функцiоналiв Ляпунова доведено експоненцiальну спаднiсть розв’язку.
Mathematical Subject Classification 2020: 37C75, 93D05.
Ключові слова:
гнучка структура, термодифузiйнi ефекти, розподiлене загаювання, коректнiсть, експоненцiальна стабiльнiстьПосилання
C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory system, 1993 American Control Conference (1993), 3106--3107. https://doi.org/10.23919/ACC.1993.4793475
M.S. Alves, P. Gamboa, G.C. Gorain, A. Rambaud and O. Vera, Asymptotic behavior of a flexible structure with Cattaneo type of thermal effect, Indag. Math. (N.S.) 27 (2016), No. 3, 821--834. https://doi.org/10.1016/j.indag.2016.03.001
M. Aouadi, M. Campo, M.I.M. Copetti, J.R. Fernández, Existence, stability and numerical results for a Timoshenko beam with thermodiffusion effects, Z. Angew. Math. Phys. 70 (2019), No. 4, 117. https://doi.org/10.1007/s00033-019-1161-8
T.A. Apalara, Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay, Electron. J. Differential Equations 2014 (2014), 254.
A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay, Electron. J. Differential Equations 2014 (2014), 212.
A. Beuter, J. Bélair and C. Labrie, Feedback and delays in neurological diseases: a modeling study using dynamical systems, Bull Math Biol. 55 (1993), No. 3, 525--541. https://doi.org/10.1016/S0092-8240(05)80238-1
L. Bouzettouta and A. Djebabla, Exponential stabilization of the full von Kármán beam by a thermal effect and a frictional damping and distributed delay, J. Math. Phys. 60 (2019), No. 4, 041506. https://doi.org/10.1063/1.5043615
J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963), 155--160. https://doi.org/10.1090/qam/160437
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), No. 1, 152--156. https://doi.org/10.1137/0324007
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 63(86) (2021), No. 1, 58--76. https://doi.org/10.24193/mathcluj.2021.1.06
B. Feng, Exponential stabilization of a Timoshenko system with thermodiffusion effects, Z. Angew. Math. Phys. 72 (2021), No. 4, 138. https://doi.org/10.1007/s00033-021-01570-2
L. Gang, L. Yue, Y. Jiangyong and J. Feida, Well-posedness and exponential stability of a flexible structure with second sound and time delay, Appl. Anal. 98 (2019), No. 16, 2903--2915. https://doi.org/10.1080/00036811.2018.1478081
G.C. Gorain, Exponential stabilization of longitudinal vibrations of an inhomogeneous beam, Nonlinear Oscil. 16 (2013), No. 2, 157--164.
S. Hu, M. Dunlavey, S. Guzy and N. Teuscher, A distributed delay approach for modeling delayed outcomes in pharmacokinetics and pharmacodynamics studies, J Pharmacokinet Pharmacodyn. 45 (2018), No. 2, 285--308. https://doi.org/10.1007/s10928-018-9570-4
M. Kafini, S.A. Messaoudi, M.I. Mustafa and T.A. Apalara, Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay, Z. Angew. Math. Phys. 66 (2015), No. 4, 1499--1517. https://doi.org/10.1007/s00033-014-0475-9
H.E. Khochemane, L. Bouzettouta and A. Guerouah, Exponential decay and well-posedness for a one-dimensional porous-elastic system with distributed delay, Appl. Anal. 100 (2021), No. 14, 2950--2964. https://doi.org/10.1080/00036811.2019.1703958
H.E. Khochemane, S. Zitouni and L. Bouzettouta, Stability result for a nonlinear damping porous-elastic system with delay term, Nonlinear Stud. 27 (2020), No. 2, 487--503.
G. Liu, Well-posedness and exponential decay of solutions for a transmission problem with distributed delay, Electron. J. Differential Equations 2017(2017), 174.
W.J. Liu, K.W. Chen and J. Yu, Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA J. Math. Control Inform. 34 (2017), No. 2, 521--542.
S. Misra, M. Alves, G. Gorain and O. Vera, Stability of the vibrations of an inhomogeneous flexible structure with thermal effect, Int. J. Dyn. Control 3 (2015), No. 4, 354--362. https://doi.org/10.1007/s40435-014-0113-6
D.S. Mitrinovic, J.E. Pecaric and A.M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and its Applications (East European Series), 53. Kluwer Academic Publishers Group, Dordrecht, 1991. https://doi.org/10.1007/978-94-011-3562-7_15
K. Mpungu and T.A. Apalara, Exponential stability of laminated beam with constant delay feedback, Math. Model. Anal. 26 (2021), No. 4, 566--581. https://doi.org/10.3846/mma.2021.13759
M.I. Mustafa, A uniform stability result for thermoelasticity of type III with boundary distributed delay, J. Math. Anal. Appl. 415 (2014), No. 1, 148--158. https://doi.org/10.1016/j.jmaa.2014.01.080
S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations 21 (2008), No. 9-10, 935--958. https://doi.org/10.57262/die/1356038593
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1
L.S. Pul'kina, A nonlocal problem with integral conditions for a hyperbolic equation, Differ. Uravn. 40 (2004), No. 7, 887--892 . https://doi.org/10.1023/B:DIEQ.0000047025.64101.16
J.P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica J. IFAC 39 (2003), No. 10, 1667--1694. https://doi.org/10.1016/S0005-1098(03)00167-5
R. Racke, Instability of coupled systems with delay, Commun. Pure Appl. Anal. 11 (2012), No. 5, 1753--1773. https://doi.org/10.3934/cpaa.2012.11.1753
C.A. Raposo, H. Nguyen, J.O. Ribeiro and V. Barros, Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition, Electron. J. Differential Equations 2017 (2017), 279.
S.P. Timoshenko, On the correction for shear of the differential equation for trans-verse vibrationsof prismatic bars, Philos. Mag. 41(1921), 744-746 https://doi.org/10.1080/14786442108636264
