October  2016, 21(8): 2491-2507. doi: 10.3934/dcdsb.2016057

Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary

1. 

College of Science, Civil Aviation University of China, Tianjin 300300, China

2. 

Department of Mathematics, Tianjin University, Tianjin 300072, China, China

Received  October 2015 Revised  April 2016 Published  September 2016

This paper considers the stabilization of a wave equation with interior input delay: $\mu_1u(x,t)+\mu_2u(x,t-\tau)$, where $u(x,t)$ is the control input. A new dynamic feedback control law is obtained to stabilize the closed-loop system exponentially for any time delay $\tau>0$ provided that $|\mu_1|\neq|\mu_2|$. Moreover, some sufficient conditions are given for discriminating the asymptotic stability and instability of the closed-loop system.
Citation: Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057
References:
[1]

K. Ammaria, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay,, Systems and Control Letters, 59 (2010), 623.  doi: 10.1016/j.sysconle.2010.07.007.  Google Scholar

[2]

T. A. Apalara, Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks,, Applicable Analysis, 95 (2016), 187.  doi: 10.1080/00036811.2014.1000314.  Google Scholar

[3]

E. M. A. Benhassi, K. Ammari, S. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay,, Journal of Evolution Equations, 9 (2009), 103.  doi: 10.1007/s00028-009-0004-z.  Google Scholar

[4]

R. Datko, Not all feedback stabilized huperbolic systems are robust with respect to small time delays in their feedbacks,, SIAM Journal on Control and Optimimization, 26 (1988), 697.  doi: 10.1137/0326040.  Google Scholar

[5]

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

[6]

N. Dunford and J. T. Schwartz, Linear Operators, Part Iii, Spectral Operators,, Wiley-Interscience, (1971).   Google Scholar

[7]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping,, Journal of Differential Equations, 132 (1996), 338.  doi: 10.1006/jdeq.1996.0183.  Google Scholar

[8]

B. Z. Guo and Y. H. Luo, Controllability and stability of a second-order hyperbolic system with collocated sensor/ actuator,, Systems and Control Letters, 46 (2002), 45.  doi: 10.1016/S0167-6911(01)00201-8.  Google Scholar

[9]

B. S. Houari and A. Soufyane, Stability result of the Timoshenko system with delay and boundary feedback,, IMA Journal of Mathematical Control and Information, 29 (2012), 383.  doi: 10.1093/imamci/dnr043.  Google Scholar

[10]

Z. J. Han and G. Q. Xu, Output-based stabilization of Euler-Bernoulli beam with time delay in boundary control,, IMA journal of Mathematical Control and Information, 31 (2014), 533.  doi: 10.1093/imamci/dnt030.  Google Scholar

[11]

X. F. Liu and G. Q. Xu, Exponential stabilization for Timoshenko beam with distributed delay in the boundary control,, Abstract and Applied Analysis, (2013), 1.   Google Scholar

[12]

Yu. I. Byubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Bnanach spaces,, Studia Mathmatic, 88 (1988), 37.   Google Scholar

[13]

Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equation,, IEEE Trans. Automat. Control, 40 (1995), 1626.  doi: 10.1109/9.412634.  Google Scholar

[14]

S. Nacaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM Journal on Control and Optimimization, 45 (2006), 1561.  doi: 10.1137/060648891.  Google Scholar

[15]

S. Nacaise and J. Valein, Stabilization of the wave equations on 1-d networks with a delay term in the nodal feedbacks,, Networks and Heterogrneous Media, 2 (2007), 425.  doi: 10.3934/nhm.2007.2.425.  Google Scholar

[16]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay,, Electronic Journal of Differential Equations, 41 (2011), 1.   Google Scholar

[17]

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

[18]

Y. F. Shang and G. Q. Xu, Stabilization of Euler-Bernouli beam with input delay in the boundary control,, System Control Letters, 61 (2012), 1069.  doi: 10.1016/j.sysconle.2012.07.012.  Google Scholar

[19]

Y. F. Shang and G. Q. Xu, Dynamic feedback control and exponential stabilization of a compound system,, Journal of Mathematical Analysis and Applications, 422 (2015), 858.  doi: 10.1016/j.jmaa.2014.09.013.  Google Scholar

[20]

M. Slemrod, A Note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM Journal of Control and Optimation, 12 (1974), 500.  doi: 10.1137/0312038.  Google Scholar

[21]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Verlag, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[22]

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

[23]

G. Q. Xu and H. X. Wang, Stabilisation of Timoshenko beam system with delay in the boundary control,, International Journal of Control, 86 (2013), 1165.  doi: 10.1080/00207179.2013.787494.  Google Scholar

[24]

K. Y. Yang and C. Z. Yao, Stabilization of one-dimention schrödinger equation by boundary observation with time delay,, Asian Journal of Control, 15 (2013), 1531.   Google Scholar

show all references

References:
[1]

K. Ammaria, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay,, Systems and Control Letters, 59 (2010), 623.  doi: 10.1016/j.sysconle.2010.07.007.  Google Scholar

[2]

T. A. Apalara, Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks,, Applicable Analysis, 95 (2016), 187.  doi: 10.1080/00036811.2014.1000314.  Google Scholar

[3]

E. M. A. Benhassi, K. Ammari, S. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay,, Journal of Evolution Equations, 9 (2009), 103.  doi: 10.1007/s00028-009-0004-z.  Google Scholar

[4]

R. Datko, Not all feedback stabilized huperbolic systems are robust with respect to small time delays in their feedbacks,, SIAM Journal on Control and Optimimization, 26 (1988), 697.  doi: 10.1137/0326040.  Google Scholar

[5]

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

[6]

N. Dunford and J. T. Schwartz, Linear Operators, Part Iii, Spectral Operators,, Wiley-Interscience, (1971).   Google Scholar

[7]

P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping,, Journal of Differential Equations, 132 (1996), 338.  doi: 10.1006/jdeq.1996.0183.  Google Scholar

[8]

B. Z. Guo and Y. H. Luo, Controllability and stability of a second-order hyperbolic system with collocated sensor/ actuator,, Systems and Control Letters, 46 (2002), 45.  doi: 10.1016/S0167-6911(01)00201-8.  Google Scholar

[9]

B. S. Houari and A. Soufyane, Stability result of the Timoshenko system with delay and boundary feedback,, IMA Journal of Mathematical Control and Information, 29 (2012), 383.  doi: 10.1093/imamci/dnr043.  Google Scholar

[10]

Z. J. Han and G. Q. Xu, Output-based stabilization of Euler-Bernoulli beam with time delay in boundary control,, IMA journal of Mathematical Control and Information, 31 (2014), 533.  doi: 10.1093/imamci/dnt030.  Google Scholar

[11]

X. F. Liu and G. Q. Xu, Exponential stabilization for Timoshenko beam with distributed delay in the boundary control,, Abstract and Applied Analysis, (2013), 1.   Google Scholar

[12]

Yu. I. Byubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Bnanach spaces,, Studia Mathmatic, 88 (1988), 37.   Google Scholar

[13]

Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equation,, IEEE Trans. Automat. Control, 40 (1995), 1626.  doi: 10.1109/9.412634.  Google Scholar

[14]

S. Nacaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,, SIAM Journal on Control and Optimimization, 45 (2006), 1561.  doi: 10.1137/060648891.  Google Scholar

[15]

S. Nacaise and J. Valein, Stabilization of the wave equations on 1-d networks with a delay term in the nodal feedbacks,, Networks and Heterogrneous Media, 2 (2007), 425.  doi: 10.3934/nhm.2007.2.425.  Google Scholar

[16]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay,, Electronic Journal of Differential Equations, 41 (2011), 1.   Google Scholar

[17]

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

[18]

Y. F. Shang and G. Q. Xu, Stabilization of Euler-Bernouli beam with input delay in the boundary control,, System Control Letters, 61 (2012), 1069.  doi: 10.1016/j.sysconle.2012.07.012.  Google Scholar

[19]

Y. F. Shang and G. Q. Xu, Dynamic feedback control and exponential stabilization of a compound system,, Journal of Mathematical Analysis and Applications, 422 (2015), 858.  doi: 10.1016/j.jmaa.2014.09.013.  Google Scholar

[20]

M. Slemrod, A Note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM Journal of Control and Optimation, 12 (1974), 500.  doi: 10.1137/0312038.  Google Scholar

[21]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Verlag, (2009).  doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[22]

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

[23]

G. Q. Xu and H. X. Wang, Stabilisation of Timoshenko beam system with delay in the boundary control,, International Journal of Control, 86 (2013), 1165.  doi: 10.1080/00207179.2013.787494.  Google Scholar

[24]

K. Y. Yang and C. Z. Yao, Stabilization of one-dimention schrödinger equation by boundary observation with time delay,, Asian Journal of Control, 15 (2013), 1531.   Google Scholar

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