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December 2018, 7(4): 599-616. doi: 10.3934/eect.2018029

Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback

1. 

Faculé des sciences et de la technologie université Djilali Bounaama, Route Theniet El Had, Soufay 44225 Khemis Miliana, Algeria

2. 

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

* Corresponding author: Abdelkarim Kelleche

Received  June 2017 Revised  August 2018 Published  September 2018

Fund Project: The second author is supported by King Fahd University of Petroleum and Minerals through the Project no.: IN151015

In this paper, we study the effect of an internal or boundary time-delay on the stabilization of a moving string. The models adopted here are nonlinear and of "Kirchhoff" type. The well-posedness of the systems is proven by means of the Faedo-Galerkin method. In both cases, we prove that the solution of the system approaches the equilibrium in an exponential manner in the energy norm. To this end we request that the delayed term be dominated by the damping term. This is established through the multiplier technique.

Citation: Abdelkarim Kelleche, Nasser-Eddine Tatar. Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback. Evolution Equations & Control Theory, 2018, 7 (4) : 599-616. doi: 10.3934/eect.2018029
References:
[1]

L. Q Chen and W.-J. Zhao, The energetics and the stability of axially moving Kirchhoff strings (L), . Acoust. Soc. Am., 117 (2005), 55-88. doi: 10.1121/1.1810310.

[2]

J. Y. ChoiK. S. Hong and K. J. Yang, Exponential Stabilization of an Axially Moving Tensioned Strip by Passive Damping and Boundary, J. Vib. Control, 10 (2004), 661-682. doi: 10.1177/1077546304038103.

[3]

H. R. Clark, Elastic membrane equation in bounded and unbounded domains, EJQTDE, 1 (2002), 21 pp.

[4]

R. DatkoJ. Lagness and M. P. Poilis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156. doi: 10.1137/0324007.

[5]

R. F. Fung and C. C. Tseng, Boundary control of an axially moving string via Lyapunov method, J. Dyn. Syst. Meas. Control, 121 (1999), 105-110. doi: 10.1115/1.2802425.

[6]

F. R. FungJ. W. Wu and S. L. Wu, Stabilization of an Axially Moving String by Nonlinear Boundary Feedback, ASME J. Dyn. Syst. Meas. Control, 121 (1999), 117-121. doi: 10.1115/1.2802428.

[7]

F. R. FungJ. W. Wu and S. L. Wu, Boundary control of the axially moving Kirchhoff string, Automatica, 34 (1998), 1273-1277.

[8]

S. M. R.F. FungJ. W. Wu and S. L. Wu, Exponential stabilization of an axially moving string by linear boundary feedback, Automatica, 35 (1999), 177-181. doi: 10.1016/S0005-1098(98)00173-3.

[9]

S. Gerbi and B. Said-Houari, Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term, Appl. Math. and Comp., 218 (2012), 11900-11910. doi: 10.1016/j.amc.2012.05.055.

[10]

A. KellecheN.-e. Tatar and A. Khemmoudj, Uniform stabilization of an axially moving Kirchhoff string by a boundary control of memory type, J. Dyn. Control Syst., 23 (2017), 237-247. doi: 10.1007/s10883-016-9310-2.

[11]

A. Kelleche and N.-e. Tatar, Control of an axially moving viscoelastic Kirchhoff string, Applicable Analysis, 97 (2018), 592-609. doi: 10.1080/00036811.2016.1277708.

[12]

D. KimY. H. KangJ. B. LeeG. R. Ko and I. H. Jung, Stabilization of a non-linear Kirchhoff equation by boundary feedback control, J. Eng. Math., 77 (2012), 197-209. doi: 10.1007/s10665-012-9547-z.

[13]

I. LasieckaR. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57. doi: 10.1006/jmaa.1999.6348.

[14]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, 1969.

[15]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system, SIAM. Rev., 30 (1988), 1-68. doi: 10.1137/1030001.

[16]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585. doi: 10.1137/060648891.

[17]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, E. J. Diff. Equ, 41 (2011), 1-20.

[18]

S. NicaiseJ. Valein and E. Fridman, Stabilization of the heat and the wave equations with boundary time-varying delays, DCDS-S, 2 (2009), 559-581. doi: 10.3934/dcdss.2009.2.559.

[19]

O. Reynolds, Papers on Mechanical and Physical Studies, vol.3, The sub-Mechanics of the universe, Cambridge University Press, 1903.

[20]

S. M. Shahruz, Boundary control of a nonlinear axially moving string, Inter. J. Robust. Nonl. Control, 10 (2000), 17-25. doi: 10.1002/(SICI)1099-1239(200001)10:1<17::AID-RNC458>3.0.CO;2-9.

[21]

S. M. Shahruz and D. A. Kurmaji, Vibration suppression of a non-linear axially moving string by boundary control, J. Sound. Vib., 201 (1997), 145-152. doi: 10.1006/jsvi.1996.0754.

[22]

M. A. Shubov, 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 (1999), 571-591. doi: 10.4310/MAA.1999.v6.n4.a9.

[23]

G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim., 42 (2003), 966-984. doi: 10.1137/S0363012901400081.

[24]

G. Q. XuS. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: COCV., 12 (2006), 770-785. doi: 10.1051/cocv:2006021.

[25]

K. J. YangK. S. Hong and F. Matsuno, The rate of change of an energy functional for axially moving continua, IFAC Proceedings Volumes, 38 (2005), 610-615. doi: 10.3182/20050703-6-CZ-1902.00502.

show all references

References:
[1]

L. Q Chen and W.-J. Zhao, The energetics and the stability of axially moving Kirchhoff strings (L), . Acoust. Soc. Am., 117 (2005), 55-88. doi: 10.1121/1.1810310.

[2]

J. Y. ChoiK. S. Hong and K. J. Yang, Exponential Stabilization of an Axially Moving Tensioned Strip by Passive Damping and Boundary, J. Vib. Control, 10 (2004), 661-682. doi: 10.1177/1077546304038103.

[3]

H. R. Clark, Elastic membrane equation in bounded and unbounded domains, EJQTDE, 1 (2002), 21 pp.

[4]

R. DatkoJ. Lagness and M. P. Poilis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156. doi: 10.1137/0324007.

[5]

R. F. Fung and C. C. Tseng, Boundary control of an axially moving string via Lyapunov method, J. Dyn. Syst. Meas. Control, 121 (1999), 105-110. doi: 10.1115/1.2802425.

[6]

F. R. FungJ. W. Wu and S. L. Wu, Stabilization of an Axially Moving String by Nonlinear Boundary Feedback, ASME J. Dyn. Syst. Meas. Control, 121 (1999), 117-121. doi: 10.1115/1.2802428.

[7]

F. R. FungJ. W. Wu and S. L. Wu, Boundary control of the axially moving Kirchhoff string, Automatica, 34 (1998), 1273-1277.

[8]

S. M. R.F. FungJ. W. Wu and S. L. Wu, Exponential stabilization of an axially moving string by linear boundary feedback, Automatica, 35 (1999), 177-181. doi: 10.1016/S0005-1098(98)00173-3.

[9]

S. Gerbi and B. Said-Houari, Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term, Appl. Math. and Comp., 218 (2012), 11900-11910. doi: 10.1016/j.amc.2012.05.055.

[10]

A. KellecheN.-e. Tatar and A. Khemmoudj, Uniform stabilization of an axially moving Kirchhoff string by a boundary control of memory type, J. Dyn. Control Syst., 23 (2017), 237-247. doi: 10.1007/s10883-016-9310-2.

[11]

A. Kelleche and N.-e. Tatar, Control of an axially moving viscoelastic Kirchhoff string, Applicable Analysis, 97 (2018), 592-609. doi: 10.1080/00036811.2016.1277708.

[12]

D. KimY. H. KangJ. B. LeeG. R. Ko and I. H. Jung, Stabilization of a non-linear Kirchhoff equation by boundary feedback control, J. Eng. Math., 77 (2012), 197-209. doi: 10.1007/s10665-012-9547-z.

[13]

I. LasieckaR. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57. doi: 10.1006/jmaa.1999.6348.

[14]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, 1969.

[15]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system, SIAM. Rev., 30 (1988), 1-68. doi: 10.1137/1030001.

[16]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585. doi: 10.1137/060648891.

[17]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, E. J. Diff. Equ, 41 (2011), 1-20.

[18]

S. NicaiseJ. Valein and E. Fridman, Stabilization of the heat and the wave equations with boundary time-varying delays, DCDS-S, 2 (2009), 559-581. doi: 10.3934/dcdss.2009.2.559.

[19]

O. Reynolds, Papers on Mechanical and Physical Studies, vol.3, The sub-Mechanics of the universe, Cambridge University Press, 1903.

[20]

S. M. Shahruz, Boundary control of a nonlinear axially moving string, Inter. J. Robust. Nonl. Control, 10 (2000), 17-25. doi: 10.1002/(SICI)1099-1239(200001)10:1<17::AID-RNC458>3.0.CO;2-9.

[21]

S. M. Shahruz and D. A. Kurmaji, Vibration suppression of a non-linear axially moving string by boundary control, J. Sound. Vib., 201 (1997), 145-152. doi: 10.1006/jsvi.1996.0754.

[22]

M. A. Shubov, 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 (1999), 571-591. doi: 10.4310/MAA.1999.v6.n4.a9.

[23]

G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim., 42 (2003), 966-984. doi: 10.1137/S0363012901400081.

[24]

G. Q. XuS. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: COCV., 12 (2006), 770-785. doi: 10.1051/cocv:2006021.

[25]

K. J. YangK. S. Hong and F. Matsuno, The rate of change of an energy functional for axially moving continua, IFAC Proceedings Volumes, 38 (2005), 610-615. doi: 10.3182/20050703-6-CZ-1902.00502.

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