May  2022, 15(5): 1127-1141. doi: 10.3934/dcdss.2021090

Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

1. 

Kuwait University, Faculty of Science, Department of Mathematics, Safat 13060, Kuwait

2. 

UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia

* Corresponding author

Received  March 2021 Revised  June 2021 Published  May 2022 Early access  August 2021

This article deals with the dynamic stability of a flexible cable attached at its top end to a cart and a load mass at its bottom end. The model is governed by a system of one partial differential equation coupled with two ordinary differential equations. Assuming that a time-dependent delay occurs in one boundary, the main concern of this paper is to stabilize the dynamics of the cable as well as the dynamical terms related to the cart and the load mass. To do so, we first prove that the problem is well-posed in the sense of semigroups theory provided that some conditions on the delay are satisfied. Thereafter, an appropriate Lyapunov function is put forward, which leads to the exponential decay of the energy as well as an estimate of the decay rate.

Citation: Boumedièene Chentouf, Sabeur Mansouri. Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1127-1141. doi: 10.3934/dcdss.2021090
References:
[1]

E. M. Abdel-RahmanA. H. Nayfeh and Z. N. Masoud, Dynamics and control of cranes: A review, J. Vibration Control, 9 (2003), 863-908.  doi: 10.1177/1077546303009007007.

[2]

F. Al-MusallamK. Ammari and B. Chentouf, Asymptotic behavior of a 2D overhead crane with input delays in the boundary control, Zeitschrift fur Angewandte Mathematik und Mechanik, 98 (2018), 1103-1122.  doi: 10.1002/zamm.201700208.

[3]

K. Ammari and B. Chentouf, Further results on the long-time behavior of a 2D overhead crane with a boundary delay: Exponential convergence, Applied Math. and Computation, 365 (2020), 124698, 17 pp. doi: 10.1016/j.amc.2019.124698.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitex, Springer, 2011.

[5]

F. Boustany, Commande non Linéaire Adaptive de Systèmes Mécaniques de Type Pont Roulant. Stabilisation Frontière d'EDP,, PhD thesis, 1992.

[6]

B. Chentouf, Compensation of the interior delay effect for a rotating disk-beam system, IMA Journal of Math. Control and Information, 33 (2016), 963–978. doi: 10.1093/imamci/dnv018.

[7]

B. Chentouf, Effect compensation of the presence of a time-dependent interior delay on the stabilization of the rotating disk-beam system, Nonlinear Dynamics, 84 (2016), 977–990. doi: 10.1007/s11071-015-2543-x.

[8]

B. Chentouf and Z.-J. Han, On the stabilization of an overhead crane system with dynamic and delayed boundary conditions, IEEE Transactions on Automatic Control, 65 (2020), 4273-4280.  doi: 10.1109/TAC.2019.2953782.

[9]

B. Chentouf and S. Mansouri, Exponential decay rate for the energy of a flexible structure with dynamic delayed boundary conditions and a local interior damping, Applied Math. Letters, 103 (2020), no. 106185. doi: 10.1016/j.aml.2019.106185.

[10]

B. Chentouf and S. Mansouri, On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only, Journal of the Franklin Institute, 358 (2021), 934-962.  doi: 10.1016/j.jfranklin.2020.10.027.

[11]

C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, vol. 70. Providence, (1999), RI: American Mathematical Society. doi: 10.1090/surv/070.

[12]

F. ConradG. O'Dowd and F.-Z. Saouri, Asymptotic behavior for a model of flexible cable with tip masses, Asymptot. Anal., 30 (2002), 313-330. 

[13]

B. d'Andréa-NovelF. BoustanyF. Conrad and B. R. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane, Math. Control. Signals Systems., 7 (1994), 1-22.  doi: 10.1007/BF01211483.

[14]

B. d'Andréa-Novel and J. M. Coron, Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach, Automatica, 36 (2000), 587-593.  doi: 10.1016/S0005-1098(99)00182-X.

[15]

B. d'Andréa-Novel and J. M. Coron, Stabilization of an overhead crane with a variable length flexible cable, Computational and Applied Mathematics, 21 (2002), 101-134. 

[16]

B. d'Andréa-NovelI. Moyano and L. Rosier, Finite-time stabilization of an overhead crane with a flexible cable, Math. Control. Signals Systems, 31 (2019), 1-19.  doi: 10.1007/s00498-019-0235-7.

[17]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time–delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.

[18]

R. Datko, Two examples of ill-posedness with respect to time–delays revisited, IEEE Trans. Automatic Control, 42 (1997), 511-515.  doi: 10.1109/9.566660.

[19]

R. DatkoJ. 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), 152-156.  doi: 10.1137/0324007.

[20]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[21]

A. Elharfi, Control design of an overhead crane system from the perspective of stabilizing undesired oscillations, IMA J. Math. Control Info., 28 (2011), 267-278.  doi: 10.1093/imamci/dnr007.

[22]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[23]

T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, C.I.E.M., 72, Springer, Heidelberg, 2011,125–191. doi: 10.1007/978-3-642-11105-1_4.

[24]

M. Kirane, B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Commun. Pure Appl. Anal., 10 (2011), 667–686. doi: 10.3934/cpaa.2011.10.667.

[25]

G. Kuralay and H. Özbay, Design of first order controllers for a flexible robot arm with time delay, Appl. Comput. Math., 16 (2017), 48-58. 

[26]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999.

[27]

Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0419-3.

[28]

A. Mifdal, Stabilisation uniforme d'un système hybride, C. R. Acad. Sci. Paris. Série I Math., 324 (1997), 37-42.  doi: 10.1016/S0764-4442(97)80100-0.

[29]

O. Morgul, On the stabilization and stability robustness against small delays of some damped wave equations, IEEE Trans. Automat. Control, 40 (1995), 1626-1630.  doi: 10.1109/9.412634.

[30]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[31]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

C. Prieur and E. Trélat, Feedback stabilization of a 1-D linear reaction-diffusion equation with delay boundary control, IEEE Trans. Automat. Control, 64 (2019), 1415-1425.  doi: 10.1109/TAC.2018.2849560.

[34]

B. Rao, Decay estimate of solution for hybrid system of flexible structures, Euro. J. Appl. Math., 4 (1993), 303-319.  doi: 10.1017/S0956792500001133.

[35]

H. Sano, Boundary stabilization of hyperbolic systems related to overhead cranes, IMA J. Math. Control Info., 25 (2008), 353-366.  doi: 10.1093/imamci/dnm031.

show all references

References:
[1]

E. M. Abdel-RahmanA. H. Nayfeh and Z. N. Masoud, Dynamics and control of cranes: A review, J. Vibration Control, 9 (2003), 863-908.  doi: 10.1177/1077546303009007007.

[2]

F. Al-MusallamK. Ammari and B. Chentouf, Asymptotic behavior of a 2D overhead crane with input delays in the boundary control, Zeitschrift fur Angewandte Mathematik und Mechanik, 98 (2018), 1103-1122.  doi: 10.1002/zamm.201700208.

[3]

K. Ammari and B. Chentouf, Further results on the long-time behavior of a 2D overhead crane with a boundary delay: Exponential convergence, Applied Math. and Computation, 365 (2020), 124698, 17 pp. doi: 10.1016/j.amc.2019.124698.

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitex, Springer, 2011.

[5]

F. Boustany, Commande non Linéaire Adaptive de Systèmes Mécaniques de Type Pont Roulant. Stabilisation Frontière d'EDP,, PhD thesis, 1992.

[6]

B. Chentouf, Compensation of the interior delay effect for a rotating disk-beam system, IMA Journal of Math. Control and Information, 33 (2016), 963–978. doi: 10.1093/imamci/dnv018.

[7]

B. Chentouf, Effect compensation of the presence of a time-dependent interior delay on the stabilization of the rotating disk-beam system, Nonlinear Dynamics, 84 (2016), 977–990. doi: 10.1007/s11071-015-2543-x.

[8]

B. Chentouf and Z.-J. Han, On the stabilization of an overhead crane system with dynamic and delayed boundary conditions, IEEE Transactions on Automatic Control, 65 (2020), 4273-4280.  doi: 10.1109/TAC.2019.2953782.

[9]

B. Chentouf and S. Mansouri, Exponential decay rate for the energy of a flexible structure with dynamic delayed boundary conditions and a local interior damping, Applied Math. Letters, 103 (2020), no. 106185. doi: 10.1016/j.aml.2019.106185.

[10]

B. Chentouf and S. Mansouri, On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only, Journal of the Franklin Institute, 358 (2021), 934-962.  doi: 10.1016/j.jfranklin.2020.10.027.

[11]

C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, vol. 70. Providence, (1999), RI: American Mathematical Society. doi: 10.1090/surv/070.

[12]

F. ConradG. O'Dowd and F.-Z. Saouri, Asymptotic behavior for a model of flexible cable with tip masses, Asymptot. Anal., 30 (2002), 313-330. 

[13]

B. d'Andréa-NovelF. BoustanyF. Conrad and B. R. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane, Math. Control. Signals Systems., 7 (1994), 1-22.  doi: 10.1007/BF01211483.

[14]

B. d'Andréa-Novel and J. M. Coron, Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach, Automatica, 36 (2000), 587-593.  doi: 10.1016/S0005-1098(99)00182-X.

[15]

B. d'Andréa-Novel and J. M. Coron, Stabilization of an overhead crane with a variable length flexible cable, Computational and Applied Mathematics, 21 (2002), 101-134. 

[16]

B. d'Andréa-NovelI. Moyano and L. Rosier, Finite-time stabilization of an overhead crane with a flexible cable, Math. Control. Signals Systems, 31 (2019), 1-19.  doi: 10.1007/s00498-019-0235-7.

[17]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time–delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.

[18]

R. Datko, Two examples of ill-posedness with respect to time–delays revisited, IEEE Trans. Automatic Control, 42 (1997), 511-515.  doi: 10.1109/9.566660.

[19]

R. DatkoJ. 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), 152-156.  doi: 10.1137/0324007.

[20]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[21]

A. Elharfi, Control design of an overhead crane system from the perspective of stabilizing undesired oscillations, IMA J. Math. Control Info., 28 (2011), 267-278.  doi: 10.1093/imamci/dnr007.

[22]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[23]

T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, C.I.E.M., 72, Springer, Heidelberg, 2011,125–191. doi: 10.1007/978-3-642-11105-1_4.

[24]

M. Kirane, B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Commun. Pure Appl. Anal., 10 (2011), 667–686. doi: 10.3934/cpaa.2011.10.667.

[25]

G. Kuralay and H. Özbay, Design of first order controllers for a flexible robot arm with time delay, Appl. Comput. Math., 16 (2017), 48-58. 

[26]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, 1999.

[27]

Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0419-3.

[28]

A. Mifdal, Stabilisation uniforme d'un système hybride, C. R. Acad. Sci. Paris. Série I Math., 324 (1997), 37-42.  doi: 10.1016/S0764-4442(97)80100-0.

[29]

O. Morgul, On the stabilization and stability robustness against small delays of some damped wave equations, IEEE Trans. Automat. Control, 40 (1995), 1626-1630.  doi: 10.1109/9.412634.

[30]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[31]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

C. Prieur and E. Trélat, Feedback stabilization of a 1-D linear reaction-diffusion equation with delay boundary control, IEEE Trans. Automat. Control, 64 (2019), 1415-1425.  doi: 10.1109/TAC.2018.2849560.

[34]

B. Rao, Decay estimate of solution for hybrid system of flexible structures, Euro. J. Appl. Math., 4 (1993), 303-319.  doi: 10.1017/S0956792500001133.

[35]

H. Sano, Boundary stabilization of hyperbolic systems related to overhead cranes, IMA J. Math. Control Info., 25 (2008), 353-366.  doi: 10.1093/imamci/dnm031.

Figure 1.  The overhead crane model
Figure 2.  The platform
Figure 3.  Payload
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