具有边界输入时滞的一维波方程的稳定性
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摘要
研究了边界具有输入时滞的一维波方程的指数稳定性.首先,采用Smith预估器方法设计了状态反馈控制律,进而得到了相应的闭环系统;其次,利用算子的可容许理论证明了闭环系统的适定性;最后,使用李雅普诺夫方法证明了系统的指数稳定性.
In this paper,the exponential stability of one-dimensional wave equations with input delays on the boundary is studied.Firstly,the state feedback control law is designed by using the Smith predictor method,and then the corresponding closed-loop system is obtained.Secondly,the appropriateness of the closed-loop system is proved by using the admissibility theory of operators.Finally,the exponential stability of the system is proved by the Lyapunov method.
In this paper,the exponential stability of one-dimensional wave equations with input delays on the boundary is studied.Firstly,the state feedback control law is designed by using the Smith predictor method,and then the corresponding closed-loop system is obtained.Secondly,the appropriateness of the closed-loop system is proved by using the admissibility theory of operators.Finally,the exponential stability of the system is proved by the Lyapunov method.
引文
[1]SRIVIDHYA J,GOPINATHAN M S.A simple time delay model for eukaryotic cell cycle[J].Journal of Theoretical Biology,2006(241):617-627.
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[3]DATKO R.Two questions concerning the boundary control of certain elastic system[J].Journal of Differential Equations,1991(92):27-44.
[4]DATKO R.Two examples of ill-posedness with respect to time delays revisited[J].IEEE Transactions on Automatic Control,1997(42):511-515.
[5]XU G Q,YUNG S P,LI L K.Stabilization of wave systems with input delay in the boundary control[J].ESAIM:Control,Optimization and Calculus of Variations,2006(12):770-785.
[6]NICAISE S,VALEIN J.Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks[J].Networks and Hererogeneous Media,2007,2(3):425-478.
[7]SHANG Y F,XU G Q,CHEN Y L.Stability analysis of Euler-Bernoulli beam with input delay in the boundary control[J].Asian Journal of Control,2012(14):186-196.
[8]SHANG Y F,XU G Q.Stabilization of an Euler-Bernoulli beam with input delay in the boundary control[J].Systems and Control Letters,2012(61):1069-1078.
[9]COX S,ZUAZUZ E.The rate at which energy decays on a damped string[J].Comm Partial Differential Equations,1994(19):213-143.
[10]NAKAO M.Decay of solutions of the wave equation with a local nonlinears dissipation[J].Math Ann,1996(305):403-417.
[11]WANG H,XU G Q.Exponential stabilization of 1-d wave equation with input delay[J].WSEAS Transctions on Mathematics,2013,10(12):1001-1013.
[2]ZAVAREI M M,JARNSHIDI M.Time-delay systems:analysis,optimization and applications[J].North-Holland Systems and Control Series,1987(9):554.
[3]DATKO R.Two questions concerning the boundary control of certain elastic system[J].Journal of Differential Equations,1991(92):27-44.
[4]DATKO R.Two examples of ill-posedness with respect to time delays revisited[J].IEEE Transactions on Automatic Control,1997(42):511-515.
[5]XU G Q,YUNG S P,LI L K.Stabilization of wave systems with input delay in the boundary control[J].ESAIM:Control,Optimization and Calculus of Variations,2006(12):770-785.
[6]NICAISE S,VALEIN J.Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks[J].Networks and Hererogeneous Media,2007,2(3):425-478.
[7]SHANG Y F,XU G Q,CHEN Y L.Stability analysis of Euler-Bernoulli beam with input delay in the boundary control[J].Asian Journal of Control,2012(14):186-196.
[8]SHANG Y F,XU G Q.Stabilization of an Euler-Bernoulli beam with input delay in the boundary control[J].Systems and Control Letters,2012(61):1069-1078.
[9]COX S,ZUAZUZ E.The rate at which energy decays on a damped string[J].Comm Partial Differential Equations,1994(19):213-143.
[10]NAKAO M.Decay of solutions of the wave equation with a local nonlinears dissipation[J].Math Ann,1996(305):403-417.
[11]WANG H,XU G Q.Exponential stabilization of 1-d wave equation with input delay[J].WSEAS Transctions on Mathematics,2013,10(12):1001-1013.