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

Exponential stabilization of Timoshenko beam with input and output delays

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.
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.

[1]

Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303

[2]

Imene Aicha Djebour, Takéo Takahashi, Julie Valein. Feedback stabilization of parabolic systems with input delay. Mathematical Control and Related Fields, 2022, 12 (2) : 405-420. doi: 10.3934/mcrf.2021027

[3]

K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050

[4]

Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057

[5]

S. Hadd, F.Z. Lahbiri. A semigroup approach to stochastic systems with input delay at the boundary. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022004

[6]

Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011

[7]

Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028

[8]

Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks and Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297

[9]

Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 569-581. doi: 10.3934/dcdsb.2021055

[10]

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

[11]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693

[12]

Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004

[13]

James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445

[14]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial and Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061

[15]

Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021052

[16]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219

[17]

Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2959-2978. doi: 10.3934/dcdsb.2021168

[18]

M. Grasselli, Vittorino Pata, Giovanni Prouse. Longtime behavior of a viscoelastic Timoshenko beam. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 337-348. doi: 10.3934/dcds.2004.10.337

[19]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[20]

Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]