American Institute of Mathematical Sciences

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June  2016, 6(2): 271-292. doi: 10.3934/mcrf.2016004

Exponential stabilization of Timoshenko beam with input and output delays

Xiu-Fang Liu 1, and  Gen-Qi Xu 2,

1. 

School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
2. 

Department of Mathematics, Tianjin University, Tianjin 300072

Received  October 2014 Revised  October 2015 Published  April 2016

In this paper, we consider the exponential stabilization issue of Timoshenko beam with input and output delays. By using the Luenberger observer and Smith predictor we obtain an estimate of the state of the system, and by the partial state predictor we transform the delayed system into a without delay system, and then by the collocated feedback of the without delay system to obtain the control signal. We prove that under the control signal, the Timoshenko beam with output and input delays can be stabilized exponentially.
Keywords: input delay, Timoshenko beam, output delay., boundary control, exponential stabilization.
Mathematics Subject Classification: 93C20, 93B52, 93D15, 93D2.
Citation: Xiu-Fang Liu, Gen-Qi Xu. Exponential stabilization of Timoshenko beam with input and output delays. Mathematical Control and Related Fields, 2016, 6 (2) : 271-292. doi: 10.3934/mcrf.2016004
References:
[1]

T. Faria, On a planar system modelling a neuron network with memory, J. Differential Equations, 168 (2000), 129-149. doi: 10.1006/jdeq.2000.3881.

[2]

T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks, J. Differential Equations, 244 (2008), 1049-1079. doi: 10.1016/j.jde.2007.12.005.

[3]

Guo, Y. Chen and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, J. Differential Equations, 244 (2008), 444-486. doi: 10.1016/j.jde.2007.09.008.

[4]

B. Z. Guo, C. Z. Xu and H. Hammouri, Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation, ESAIM:Control, Optimization and Calculus of Variations, 18 (2012), 22-35. doi: 10.1051/cocv/2010044.

[5]

Z. J. Han and G. Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM:Control, Optimization and Calculus of Variations, 17 (2010), 552-574. doi: 10.1051/cocv/2010009.

[6]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM. J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[7]

X. F. Liu and G. Q. Xu, Exponenntial stabilization for Timoshenko beam with distributed delay in the boundary control, Abstract and Applied Analysis, (2013), Art. ID 726794, 15 pp. doi: 10.1155/2013/726794.

[8]

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

[9]

S. Nicaise and J. Valein, Stabilitization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.

[10]

S. Nicaise and C. Pignotti, Stabilitization of the wave equation with boundary or internal distributed delay, Differential and Integral Equation, 21 (2008), 935-958.

[11]

G. Stepan, Retarded dynamical system: stability and characteristic functions, Longman Scientific and Technical, John Wiley and Sons, Inc., New York, (1989), 136-147.

[12]

Y. F. Shang and G. Q. Xu, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Systems and Control letters, 61 (2012), 1069-1078. doi: 10.1016/j.sysconle.2012.07.012.

[13]

Y. F. Shang, G. Q. Xu and Y. L. Chen, Stability analysis of Euler-Bernoulli beam with input delay in the boundary control, Asian Journal of Control, 14 (2012), 186-196. doi: 10.1002/asjc.279.

[14]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Basel Hoston Berlin: Birkhaüser, 2009. doi: 10.1007/978-3-7643-8994-9.

[15]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control,Optimisation and Calculus of Variations, 12 (2006), 70-785. doi: 10.1051/cocv:2006021.

[16]

G. Q. Xu and H. X. Wang, Stabilization of Timoshenko beam system with delay in the boundary control, INT. J. Control, 86 (2013), 1165-1178. doi: 10.1080/00207179.2013.787494.

[17]

R. Yafia, Danamics and numerical simulations in a production and development of red blood cells model with one delay, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 582-592. doi: 10.1016/j.cnsns.2007.08.012.

show all references

References:
[1]

T. Faria, On a planar system modelling a neuron network with memory, J. Differential Equations, 168 (2000), 129-149. doi: 10.1006/jdeq.2000.3881.

[2]

T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks, J. Differential Equations, 244 (2008), 1049-1079. doi: 10.1016/j.jde.2007.12.005.

[3]

Guo, Y. Chen and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, J. Differential Equations, 244 (2008), 444-486. doi: 10.1016/j.jde.2007.09.008.

[4]

B. Z. Guo, C. Z. Xu and H. Hammouri, Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation, ESAIM:Control, Optimization and Calculus of Variations, 18 (2012), 22-35. doi: 10.1051/cocv/2010044.

[5]

Z. J. Han and G. Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM:Control, Optimization and Calculus of Variations, 17 (2010), 552-574. doi: 10.1051/cocv/2010009.

[6]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM. J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[7]

X. F. Liu and G. Q. Xu, Exponenntial stabilization for Timoshenko beam with distributed delay in the boundary control, Abstract and Applied Analysis, (2013), Art. ID 726794, 15 pp. doi: 10.1155/2013/726794.

[8]

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

[9]

S. Nicaise and J. Valein, Stabilitization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.

[10]

S. Nicaise and C. Pignotti, Stabilitization of the wave equation with boundary or internal distributed delay, Differential and Integral Equation, 21 (2008), 935-958.

[11]

G. Stepan, Retarded dynamical system: stability and characteristic functions, Longman Scientific and Technical, John Wiley and Sons, Inc., New York, (1989), 136-147.

[12]

Y. F. Shang and G. Q. Xu, Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Systems and Control letters, 61 (2012), 1069-1078. doi: 10.1016/j.sysconle.2012.07.012.

[13]

Y. F. Shang, G. Q. Xu and Y. L. Chen, Stability analysis of Euler-Bernoulli beam with input delay in the boundary control, Asian Journal of Control, 14 (2012), 186-196. doi: 10.1002/asjc.279.

[14]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Basel Hoston Berlin: Birkhaüser, 2009. doi: 10.1007/978-3-7643-8994-9.

[15]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control,Optimisation and Calculus of Variations, 12 (2006), 70-785. doi: 10.1051/cocv:2006021.

[16]

G. Q. Xu and H. X. Wang, Stabilization of Timoshenko beam system with delay in the boundary control, INT. J. Control, 86 (2013), 1165-1178. doi: 10.1080/00207179.2013.787494.

[17]

R. Yafia, Danamics and numerical simulations in a production and development of red blood cells model with one delay, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 582-592. doi: 10.1016/j.cnsns.2007.08.012.

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