Journal of Applied Nonlinear Dynamics
Existence and Boundary Control of a Nonlinear
Axially Moving String Subject to Disturbances
Journal of Applied Nonlinear Dynamics 11(2) (2022) 343358  DOI:10.5890/JAND.2022.06.006
Abdelkarim Kelleche, Nassereddine Tatar
Facult\'{e} des Sciences et de la Technologie, Universit\'{e} Djilali Boun\^{a}ama, Algeria
\normalsize Department of Mathematics and Statistics, King Fahd University of Petroleum
and Minerals, Dhahran\addressNewline 31261, Saudi Arabia
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Abstract
This paper addresses the stabilization question for a nonlinear model of an
axially moving string. The string is tensioned and is subject to
spatiotemporary varying disturbances. The Hamilton principle of changing
mass is employed to formulate mathematically the problem. By means of the
FaedoGalerkin method, we establish the wellposedness. A boundary control
with a timevarying delay is designed to stabilize uniformly the string.
Then, we derive a decay rate of the solution under the condition that the
retarded term be dominated by the damping one. Some examples are given to
clarify when the rate is exponential or polynomial.
References

[1]  Abrate, S. (1992), Vibrations of belts and belt
drives, Mech Mach Theory, 27, 649659.


[2]  Mote, C.D. (1974), Dynamic stability of axially
moving materials, Shock Vib. Dig., 4(4), 211.


[3]  Wickert, J.A. and Mote, C.D. (1988), Current
research on the vibration and stability of axiallymoving materials,
Shock Vib Dig., 20(5), 313.


[4]  Bapat, V.A. and Srinivasan, P. (1967), Nonlinear
transverse oscillations in traveling strings by the method of harmonic
balance, J. Appl Mech., 34, 775777.


[5]  Carrier, G.F. (1965), On the nonlinear vibration
problem of the elastic string, Quart Appls Maths., 3,
157198.


[6]  Chung, C. and Tan, C.A. (1995), Active vibration
control of the axially moving string by wave cancellation, J. Vib.
Acoust., 117, 4955.


[7]  Kukl{\i}k, V. and Kudlacek, J. (2016),
Hotdip Galvanizing of steel structures, Elsevier.


[8]  McIver, D.B. (1973), Hamilton's principle for
systems of changing mass, J. Engin. Math., 7(3), 249261.


[9]  Yang, K.S and Matsuno, F. (2005), The rate of
change of an energy functional for axially moving continua, IFAC
Proceedings, 38(1), 610615.


[10]  Reynolds, O. (1903), Papers on Mechanical and
Physical studies, The subMechanics of the universe, Cambridge University
Press.


[11]  Quo, Z. (2002), An iterative learning algorithm for boundary
control of a stretched moving string, Automatica, 38(5), 821827.


[12]  Fung, R.F. and Tseng, C.C. (1999), Boundary
control of an axially moving string via Lyapunov method, J. Dyn.
Syst. Meas. Control, 121, 105110.


[13]  Fung, R.F., Wu, J.W., and Wu, S.L. (1999),
Stabilization of an Axially Moving String by Nonlinear Boundary Feedback,
ASME J. Dyn. Syst. Meas. Control, 121, 117121.


[14]  Li, T. and Hou, Z. (2006), Exponential
stabilization of an axially moving string with geometrical nonlinearity by
linear boundary feedback, J. Sound Vib., 296, 861870.


[15]  Shahruz, S.M and Kurmaji, D.A. (1997), Vibration
suppression of a nonlinear axially moving string by boundary control,
J Sound Vib., 201(1), 145152.


[16]  Kelleche, A., Berkani, A., and Tatar, N.E. (2018),
Uniform stabilization of a nonlinear axially moving string by a boundary
control of memory type, J. Dyn. Control Sys., 24(2),
313323.


[17]  Lions, J.L. (1988), Exact controllability,
stabilization and perturbations for distributed parameter system,
SIAM. Rev., 30, 168.


[18]  Xu, G.Q. and Guo, B.Z. (2003), Riesz basis
property of evolution equations in Hilbert spaces and application to a
coupled string equation, SIAM J. Control Optim., 42,
966984.


[19]  Shubov, M.A. (1999), The Riesz basis property
of the system of root vectors for the equation of a nonhomogeneous damped
string: transformation operators method, Methods Appl. Anal.,
6, 571591.


[20]  Datko, R., Lagness, J., and Poilis, M.P. (1986),
An example on the effect of time delays in boundary feedback stabilization
of wave equations, SIAM J. Control Optim., 24, 152156.


[21]  Xu, G.Q., Yung, S.P., and Li, L.K. (2006),
Stabilization of wave systems with input delay in the boundary control,
ESAIM: COCV., 12(4), 770785.


[22]  Nicaise, S. and Pignotti, C. (2006), Stability
and instability results of the wave equation with a delay term in the
boundary or internal feedbacks, SIAM J. Control Optim., 45(5), 15611585.


[23]  Yang, K.S. and Matsuno, F. (2004), Robust
adaptive boundary control of an axially moving string under a
spatiotemporally varying tension, J. Sound. Vib., 273,
10071029.


[24]  Kelleche, A., Tatar, Ne. and Khemmoudj, A.
(2016), Uniform stabilization of an axially moving Kirchhoff string by a
boundary control of memory type, J. Dyn. Control Syst., 23(2), 237247.


[25]  Kelleche, A. and Tatar, Ne. (2018), Control of
an axially moving viscoelastic Kirchhoff string, Applicable Analysis,
97(4), 592609.


[26]  Kelleche, A. and Tatar, Ne. (2018), Existence
and stabilization of a Kirchhoff moving string with a delay in the boundary
or in the internal feedback, Evol. Equ. Control theory, 7(4), 599616.


[27]  Airapetyan, R.G., Ramm, A.G., and Smirnova, A.B.
(2000), Continuous methods for solving nonlinear illposed problems,
Operator Theo. Appl., 25, 111138.


[28]  Lions, J.L. (1969), Quelques m{e}thodes de r{e}solution des probl \`{e}mes aux limites non lin{e}aires, Dunod 1969.


[29]  Gerbi, S. and SaidHouari, B. (2008), Existence
and exponential stability of a damped wave equation with dynamic boundary
conditions and a delay term, Appl. Math. Comp., 218(24),
1190011910.


[30]  Shahruz, S.M. (1998), Boundary control of the
axially moving Kirchhoff string, Automatica, 34(10),
12731277.


[31]  Shahruz, S.M. (2000), Boundary control of a
nonlinear axially moving string, Inter. J. Robust. Nonl. Control,
10(1), 1725.


[32]  Shahruz, S.M. and Kurmaji, D.A. (1997), Vibration
suppression of a nonlinear axially moving string by boundary control,
J. Sound. Vib., 201(1), 145152.
