References
- S. Abrate. Vibration of belts and belt drives. Mechanism and Machine Theory, 27(6):645–695, 1992. https://doi.org/10.1016/0094-114X(92)90064-O.
- A. Berkani, N.e. Tatar and A. Kelleche. Vibration control of a viscoelastic translational Euler-Bernoulli beam. J. Dyn. Control Sys., pp. 1–33, 2017. https://doi.org/10.1007/s10883-017-9364-9.
- M.M. Cavalcanti, V.N. Domingos Cavalcanti and J. Ferreira. Existence and uniform decay for a non-linear viscoelastic equation with strong damping. Math. Meth. Appl. Sci, 24(14):1043–1053, 2001. https://doi.org/10.1002/mma.250.
- M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. Prates Filho and J.A. Soriano. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Diff. Integral Equ, 14(1):85–116, 2001.
- C.H. Chung and C.A. Tan. Active vibration control of the axially moving string by wave cancellation. J. Vib. Acoust., 117(1):49–55, 1995. https://doi.org/10.1115/1.2873866.
- B.D. Coleman and V.J. Mizel. On the general theory of fading memory. Arch. Ration. Mech. Anal., 29(1):18–31, 1968. https://doi.org/10.1007/BF00256456.
- B.D. Coleman and W. Noll. Foundations of linear viscoelasticity. Rev. Modern. Phys., 33:239–249, 1961. https://doi.org/10.1103/RevModPhys.33.239.
- C.M. Dafermos. Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal., 37(4):297–308, 1970. https://doi.org/10.1007/BF00251609.
- C.M. Dafermos. On abstract Volterra equations with applications to linear viscoelasticity. J. Differential Equations, 7:554–569, 1970. https://doi.org/10.1016/0022-0396(70)90101-4.
- M. Fabrizio and A. Morro. Mathematical Problems in Linear Viscoelasticity. SIAM Stud. Appl. Math. Philadelphia, 1992. https://doi.org/10.1137/1.9781611970807.
- R.F. Fung and C.C. Tseng. Boundary control of an axially moving string via Lyapunov method. J. Dyn. Syst. Measurement Control, 121(1):105–110, 1999. https://doi.org/10.1115/1.2802425.
- R.F. Fung, J.W. Wu, and S.L. Wu. Exponential stabilization of an axially moving string by linear boundary feedback. Automatica, 35(1):177–181, 1999. https://doi.org/10.1016/S0005-1098(98)00173-3.
- G.H. Hardy, J.E. Littlewood and G. Polya. Inequalities. Cambridge, UK: Cambridge University Press, 1959.
- Y.H. Kang, J.Y. Park and J.A. Kim. A memory type boundary stabilization for an Euler-Bernoulli beam under boundary output feedback control. J. Korean Math. Soc., 49(5):947–964, 2012. https://doi.org/10.4134/JKMS.2012.49.5.947.
- A. Kelleche, A. Berkani and N.e. Tatar. Uniform stabilization of a nonlinear axially moving string by a boundary control of memory type. J. Dyn. Control Sys., 2017. https://doi.org/10.1007/s10883-017-9370-y.
- A. Kelleche and N.e. Tatar. Control of an axially moving viscoelastic Kirchhoff string. J. Appl. Anal., 2017. https://doi.org/10.1080/00036811.2016.1277708.
- A. Kelleche and N.e. Tatar. Uniform decay for solutions of an axially moving viscoelastic beam. Appl. Maths. Optim., 75(3):343–364, 2017. https://doi.org/10.1007/s00245-016-9334-8.
- A. Kelleche, N.e. Tatar and A. Khemmoudj. Stability of an axially moving viscoelastic beam. J. Dyn. Control Sys., 23(2):283–299, 2017. https://doi.org/10.1007/s10883-016-9317-8.
- A. Kelleche, N.e. Tatar and A. Khemmoudj. Uniform stabilization of an axially moving Kirchhoff string by a boundary control of memory type. J. Dyn. Control Sys., 23(2):237–247, 2017. https://doi.org/10.1007/s10883-016-9310-2.
- L.G. Leal. Advanced transport phenomena: fluid mechanics and connective transport processes. Cambridge University Press, 2007. ISBN: 9780521179089.
- S.Y. Lee and C.D. Mote. Vibration control of an axially moving string by boundary control. J. Dyn. Systems Measurement Control, 118(1):66–74, 1996. https://doi.org/10.1115/1.2801153.
- Y. Li, D. Aron and C.D. Rahn. Adaptive vibration isolation for axially moving strings:theory and experiment. Automatica, 38(3):379–390, 2002. https://doi.org/10.1016/S0005-1098(01)00219-9.
- Y. Liu, B. Xu, Y. Wu and Y. Hu. Boundary control of an axially moving belt. Proceedings of the 32nd Chinese Control Conference, pp. 1310–1315, 2013.
- Y. Liu, Z. Zhao and W. He. Boundary control of an axially moving accelerated/decelerated belt system. Inter. J. Robust Non. Control, 26(17):38493866, 2016. https://doi.org/10.1002/rnc.3538.
- C.D. Mote. Dynamic stability of axially moving materials. Shock. Vib. Dig., 4(4):2–11, 1972. doi: 10.1177/058310247200400402
- F. Odeh and I. Tadjbakhsh. Uniqueness in the linear theory of viscoelasticity. Arch. Ration. Mech. Anal., 18(3):244–250, 1965. doi: 10.1007/BF00285436
- V. Pata. Exponential stability in linear viscoelasticity. Quart. Appl. Math., 64(3):499–513, 2006. https://doi.org/10.1090/S0033-569X-06-01010-4.
- O. Reynolds. Papers on Mechanical and Physical studies. The Sub-Mechanics of the Universe, Cambridge University Press, 1903.
- S.M. Shahruz. Boundary control of the axially moving Kirchhoff string. Automatica, 34(10):1273–1277, 1998. https://doi.org/10.1016/S0005-1098(98)00074-0.
- N.e. Tatar. Polynomial stability without polynomial decay of the relaxation function. Math. Meth. Appl. Sci., 31(15):1874–1886, 2008. https://doi.org/10.1002/mma.1018.
- N.e. Tatar. On a large class of kernels yielding exponential stability in viscoelasticity. Appl. Math. Comp., 215(6):2298–2306, 2009. https://doi.org/10.1016/j.amc.2009.08.034.
- N.e. Tatar. A new class of kernels leading to an arbitrary decay in viscoelasticity. Meditter. J. Math., 10(1):213–226, 2013. https://doi.org/10.1007/s00009-012-0177-5.
- K.J. Yang, K.S. Hong and F. Matsuno. Robust adaptive boundary control of an axially moving string under a spatiotemporally varying tension. J. Sound. Vib., 273(4-5):1007–1029, 2004. https://doi.org/10.1016/S0022-460X(03)00519-4.
- K.J. Yang, K.S. Hong and F. Matsuno. Robust adaptive control of a cantilevered flexible structure with spatiotemporally varying coefficients and bounded disturbance. JSME Int. J. Ser. C, 47(3):812–822, 2004. https://doi.org/10.1299/jsmec.47.812.