References
- Coleman BD, Noll W. Foundations of linear viscoelasticity. Rev Modern Phys. 1961;33:239–249.
- Coleman BD, Mizel VJ. On the general theory of fading memory. Arch Ration Mech Anal. 1968;29:18–31.
- Dafermos CM. Asymptotic stability in viscoelasticity. Arch Ration Mech Anal. 1970;37:297–308.
- Dafermos CM. On abstract Volterra equations with applications to linear viscoelasticity. J Differ Equ. 1970;7:554–569.
- Fabrizio M, Morro A. Mathematical problems in linear viscoelasticity. Vol 12. Studies in Applied Mathematics. Philadelphia: SIAM; 1992.
- Lasiecka I, Messaoudi SA, Mustafa MI. Note on intrinsic decay rates for abstract wave equations with memory. J Math Phys. 2013;54:031504.
- Murdoch AI. Remarks on the foundations of linear viscoelasticity. J Mech Phys Solids. 1992;40(7):1559–1568.
- Prüss J. Evolutionary integral equations and applications. Basel: Birkhäuser; 1993.
- Abrate S. Vibration of belts and belt drives. Mech Mach Theory. 1992;27:645–659.
- Bapat VA, Srinivasan P. Nonlinear transverse oscillations in traveling strings by the method of harmonic balance. J Appl Mech. 1967;34:775–777.
- Carrier GF. On the nonlinear vibration problem of the elastic string. Q Appl Maths. 1965;3:157–165.
- Mote CD. Dynamic stability of axially moving materials. Shock Vib Dig. 1972;4(4):2–11.
- Mote CD. A study of band saw vibrations. J Franklin Inst. 1965;279(6):430–444.
- Wickert JA, Mote CD. Classical vibration analysis of axially moving continua. J Appl Mech. 1990;57:738–744.
- Wickert JA, Motte CD. Current research on the vibration and stability of axially-moving materials. Shock vib Dig. 1988;20(5):3–13.
- Beikmann RS, Perkins NC, Ulsoy AG. Free vibration of serpentine belt drive systems. J Vib Acoust. 1996;118:406–413.
- Beikmann RS, Perkins NC, Ulsoy AG. Nonlinear coupled vibration response of serpentine belt drive systems. J Vib Acoust. 1996;118:567–574.
- Beikmann RS, Perkins NC, Ulsoy AG. Design and analysis of automotive serpentine belt drive systems for steady state performance. J Mech Des. 1997;119:162–168.
- Fung RF, Shieh JS. Vibration analysis of a non-linear coupled textile-rotor system with synchronous whirling. J Sound Vib. 1997;199:207–221.
- Lee SY, Mote CD. A generalized treatment of the energetics of translating continua. Part I: Strings and second order tensioned pipes, J Sound Vib. 1997;204:717–734.
- Chen JS. Natural frequencies and stability of an axially traveling string in contact with a stationary load system. J Vib Acoust. 1997;119:152–157.
- Fung RF, Tseng CC. Boundary control of an axially moving string via Lyapunov method. J Dyn Syst Meas Control. 1999;121:105–110.
- Li T, Hou Z. Exponential stabilization of an axially moving string with geometrical nonlinearity by linear boundary feedback. J Sound Vib. 2006;296:861–870.
- Shahruz SM. Boundary control of a nonlinear axially moving string. Int J Robust Nonlinear Control. 2000;10(1):17–25.
- Shahruz SM, Kurmaji DA. Vibration suppression of a non-linear axially moving string by boundary control. J Sound Vib. 1997;201(1):145–152.
- Kelleche A, Tatar N-E, Khemmoudj A. Uniform stabilization of an axially moving Kirchhoff string by a boundary control of memory type. J Dyn Control Syst. 2016. DOI:10.1007/s10883-016-9310-2
- Shahruz SM. Boundary control of the axially moving Kirchhoff string. Automatica. 1998;34(10):1273–1277.
- Shahruz SM. Stability of a nonlinear axially moving string with the Kelvin-voigt damping. J Vib Acoust. 2009; 131(1): 014501–014501-4. DOI:10.1115/1.3025835
- Chung CH, Tan CA. Active vibration control of the axially moving string by wave cancellation. J Vib acoust. 1995;117:49–55.
- Lee SY, Mote CD. Vibration control of an axially moving string by boundary control. J Dyn Syst Meas Control. 1996;118:66–74.
- Fung RF, Wu JW. S.L., Wu, Exponential stability of an axially moving string by linear boundary feedback. Automatica. 1999;35(1):177–181.
- Messaoudi S. General decay of solutions of a viscoelastic equation. J Math Anal Appl. 2008;341(2):1457–1467.
- Messaoudi S, Tatar N-E. Exponential decay for a quasilinear viscoelastic equation. Math Nachr. 2009;282(10):1443–1450.
- Pata V. Exponential stability in linear viscoelasticity. Q Appl Math. 2006;LXIV(3):499–513.
- Berkani A, Tatar N-E, Khemmoudj A. Control of a viscoelastic translational Euler-Bernoulli beam. Math Meth Appl Sci. 2017;40:237–254. DOI:10.1002/mma.3985
- Lasiecka I, Wang X. Intrinsic decay rate estimates for semilinear abstract second order equations with memory, new prospects in direct. Inverse Contr Prob Evol Equ. 2014;10:271–303.
- Kelleche A, Tatar N-E, Khemmoudj A. Stability of an axially moving viscoelastic beam. J Dyn Control Syst. 2016. DOI:10.1007/s10883-016-9317-8
- Kelleche A, Tatar N-E. Uniform decay for solutions of an axially moving viscoelastic beam. Appl Math Optim. 2016. DOI:10.1007/s00245-016-9334-8
- Tatar N-E. On a problem arising in isothermal viscoelasticity. Int J Pure Appl Math. 2003;3(1):1–12.
- Tatar N-E. Polynomial stability without polynomial decay of the relaxation function. Math Meth Appl Sci. 2008;31(15):1874–1886.
- Tatar N-E. How far can relaxation functions be increasing in viscoelastic problems. Appl Math Lett. 2009;22(3):336–340.
- Tatar N-E. Exponential decay for a viscoelastic problem with singular kernel. Zeit Angew Math Phys. 2009;60(4):640–650.
- Tatar N-E. On a large class of kernels yielding exponential stability in viscoelasticity. Appl Math Comput. 2009;215(6):2298–2306.
- Tatar N-E. Arbitrary decays in viscoelasticity. J Math Phys. 2011;52(1):013502.
- Tatar N-E. A new class of kernels leading to an arbitrary decay in viscoelasticity. Meditter J Math. 2013;10:213–226.
- Cavalcanti MM, Domingos Cavalcanti VN, Ferreira J. Existence and uniform decay for a non-linear viscoelastic equation with strong damping. Math Meth Appl Sci. 2001;24(14):1043–1053.
- Cavalcanti MM, Domingos Cavalcanti VN, Martinez P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal TMA. 2008;68(1):177–193.
- Cavalcanti MM, Domingos Cavalcanti VN, Lasiecka I, Nascimento FAF. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional damping. Disc Cont Dyn Syst. 2014;19(7):1987–2012.
- Lasiecka I, Fourrier N. Regularity and stability of a wave equation with strong damping and dynamic boundary conditions. Evol Equ Contr Theory. 2013;2(4):631–667.
- Hardy GH, Littlewood JE, Polya G. Inequalities, Cambridge. UK: Cambridge University Press; 1959.
- Airapetyan RG, Ramm AG, Smirnova AB. Continuous methods for solving nonlinear ill-posed problems. Oper Theory Appl. 2000;25:111–138.