American Institute of Mathematical Sciences

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 and 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), 623-628. doi: 10.1016/j.sysconle.2010.07.007. [2] T. A. Apalara, Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks, Applicable Analysis, 95 (2016), 187-202. doi: 10.1080/00036811.2014.1000314. [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), 103-121. doi: 10.1007/s00028-009-0004-z. [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), 697-713. doi: 10.1137/0326040. [5] R. Datko, Two examples of ill-posedness with respect of time delays revised, IEEE Trans. Automatic Control, 42 (1997), 511-515. doi: 10.1109/9.566660. [6] N. Dunford and J. T. Schwartz, Linear Operators, Part Iii, Spectral Operators, Wiley-Interscience, New York, 1971. [7] P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, Journal of Differential Equations, 132 (1996), 338-352. doi: 10.1006/jdeq.1996.0183. [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), 45-65. doi: 10.1016/S0167-6911(01)00201-8. [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), 383-398. doi: 10.1093/imamci/dnr043. [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), 533-550. doi: 10.1093/imamci/dnt030. [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), 1-15. [12] Yu. I. Byubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Bnanach spaces, Studia Mathmatic, 88 (1988), 37-42. [13] Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equation, IEEE Trans. Automat. Control, 40 (1995), 1626-1630. doi: 10.1109/9.412634. [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), 1561-1585. doi: 10.1137/060648891. [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), 425-479. doi: 10.3934/nhm.2007.2.425. [16] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 41 (2011), 1-20. [17] 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. [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), 1069-1078. doi: 10.1016/j.sysconle.2012.07.012. [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), 858-879. doi: 10.1016/j.jmaa.2014.09.013. [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), 500-508. doi: 10.1137/0312038. [21] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [22] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM, Control Optim. Calc. Var., 12 (2006), 770-785. doi: 10.1051/cocv:2006021. [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), 1165-1178. doi: 10.1080/00207179.2013.787494. [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), 1531-1537.

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References:
 [1] K. Ammaria, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems and Control Letters, 59 (2010), 623-628. doi: 10.1016/j.sysconle.2010.07.007. [2] T. A. Apalara, Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks, Applicable Analysis, 95 (2016), 187-202. doi: 10.1080/00036811.2014.1000314. [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), 103-121. doi: 10.1007/s00028-009-0004-z. [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), 697-713. doi: 10.1137/0326040. [5] R. Datko, Two examples of ill-posedness with respect of time delays revised, IEEE Trans. Automatic Control, 42 (1997), 511-515. doi: 10.1109/9.566660. [6] N. Dunford and J. T. Schwartz, Linear Operators, Part Iii, Spectral Operators, Wiley-Interscience, New York, 1971. [7] P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, Journal of Differential Equations, 132 (1996), 338-352. doi: 10.1006/jdeq.1996.0183. [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), 45-65. doi: 10.1016/S0167-6911(01)00201-8. [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), 383-398. doi: 10.1093/imamci/dnr043. [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), 533-550. doi: 10.1093/imamci/dnt030. [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), 1-15. [12] Yu. I. Byubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Bnanach spaces, Studia Mathmatic, 88 (1988), 37-42. [13] Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equation, IEEE Trans. Automat. Control, 40 (1995), 1626-1630. doi: 10.1109/9.412634. [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), 1561-1585. doi: 10.1137/060648891. [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), 425-479. doi: 10.3934/nhm.2007.2.425. [16] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 41 (2011), 1-20. [17] 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. [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), 1069-1078. doi: 10.1016/j.sysconle.2012.07.012. [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), 858-879. doi: 10.1016/j.jmaa.2014.09.013. [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), 500-508. doi: 10.1137/0312038. [21] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [22] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM, Control Optim. Calc. Var., 12 (2006), 770-785. doi: 10.1051/cocv:2006021. [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), 1165-1178. doi: 10.1080/00207179.2013.787494. [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), 1531-1537.
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