Stabilisation for a wave equation with distributed time delay in the internal input
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- 英文论文题名：Stabilisation for a wave equation with distributed time delay in the internal input
- 论文作者：Yanfang Li ; Genqi Xu ; Yanni Guo
- 英文论文作者：Yanfang Li ; Genqi Xu ; Yanni Guo ; Department of Mathematics ; Tianjin University ; College of Science ; Civil Aviation University of China
- 年：2017
- 作者机构：Department of Mathematics,Tianjin University;College of Science,Civil Aviation University of China;
- 英文论文关键词：wave equation ; input time delay ; exact observability ; exponential stabilization
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（A）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
- 语种：英文
- 分类号：O175
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：215k
- 原文格式：D
- 会议级别：全国
摘要
This paper analyzes the stabilization problem of a wave equation subject to distributed time delay in the internal input.Based on a partial state predictor, we design a feedback control signal which is shown to stabilize an undelayed system by the spectral analysis and the Hilbert uniqueness method. The stability of the original system under the feedback control is obtained via the relation between the original system and the undelayed system in terms of stability.
This paper analyzes the stabilization problem of a wave equation subject to distributed time delay in the internal input.Based on a partial state predictor, we design a feedback control signal which is shown to stabilize an undelayed system by the spectral analysis and the Hilbert uniqueness method. The stability of the original system under the feedback control is obtained via the relation between the original system and the undelayed system in terms of stability.
This paper analyzes the stabilization problem of a wave equation subject to distributed time delay in the internal input.Based on a partial state predictor, we design a feedback control signal which is shown to stabilize an undelayed system by the spectral analysis and the Hilbert uniqueness method. The stability of the original system under the feedback control is obtained via the relation between the original system and the undelayed system in terms of stability.
引文
[1]K.Ammari,S.Nicaise,and C.Pignotti,Feedback boundary stabilization of wave equations with interior delay,Systems&Control Letters,59(10):623–628,2010.
[2]S.Gerbi,B.Said-Houari,Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term,Applied Mathematics&Computation,218(24):11900–11910,2012.
[3]J.Y.Park,Y.H.Kang,J.A.Kim,Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term,Acta Appl.Math.,104:287-301,2008.
[4]X.F.Liu,G.Q.Xu,Exponnential stabilization of Timoshenko beam with input and output delays,Mathematical Control and Related Fields,6(2):271–292 2016.
[5]R.Datko,J.Lagnese,and M.P.Polis,An example on the effect of time delays in boundary feedback stabilization of wave equations,SIAM J.Control Optim.,24:152–156,1986.
[6]R.Datko,Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,SIAM J.Control Optim.,26:697–713,1988.
[7]R.Datko,Two examples of ill-posedness with respect to time delays revisited,IEEE Trans.Autom Control,42:511–515,1997.
[8]G.Q.Xu,S.P.Yung,and L.K.Li,Stability of wave system with input delay in the boundary,ESAIM:Control Optimisation and Calculus of Variations,12:770–785,2006.
[9]S.Nicaise,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 Optimization,45(5):1561–1585,2006.
[10]S.Nicaise and C.Pignotti,Exponential stability of secondorder evolution equations with structural damping and dynamic boundary delay feedback,IMA J.Math.Control Inform.,28:417–446,2011.
[11]B.Z.Guo,Z.C.Shao,Regularity of an Euler-Bernoulli equation with Neumann control and collocated observation,Journal of Dynamic and Control System,12:405–418,2006.
[12]P.C.Han,Y.F.Li,G.Q.Xu,and D.H.Liu,The Exponential stability result of an Euler-Bernoulli beam equation with interior delays and boundary damping,Journal of Difference Equations,http://dx.doi.org/10.1155/2016/3732176.
[13]S.Nicaise and C.Pignotti,Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback.Discrete and Continuous Dynamical Systems,9(3):791–813,2016.
[14]Y.F.Shang and G.Q.Xu,Stabilization of an Euler-Bernoulli beam with input delay in the boundary control,Systems Control Letters,61(11):1069–1078,2012.
[15]H.Wang,G.Q.Xu,Exponential stabilization of 1-d wave equation with input delay.Wseas Transactions on Mathematics,12:(10),1001–1013,2013.
[16]G.Q.Xu,H.X.Wang,Stabilization of Timoshenko beam system with delays in the boundary control,International Journal of Control,86(6),1165–1178,2013.
[17]Z.J.Han and G.Q.Xu,Output-based stabilization of EulerBernoulli beam with time-delay in boundary input,IMA Journal of Mathematical Control and Information,34(4):533–550,2013.
[18]X.F.Liu,G.Q.Xu,Exponential Stabilization for Timoshenko beam with distributed delay in the boundary control,Abstract&Applied Analysis,4:1–15,2013.
[19]Y.F.Shang,G.Q.Xu,Dynamic feedback control and exponential stabilization of a compound system,J.Math.Anal.Appl.422:858–879,2015.
[20]Y.F.Shang,G.Q.Xu,Output-based stabilization for a onedimensional wave equation with distributed input delay in the boundary control,IMA Journal of Mathematical Control&Information,31(4):533–550,2014.
[21]X.F.Liu,G.Q.Xu,Output-Based Stabilization of Timoshenko Beam with the boundary control and input distributed delay,J.Dyn.Control.Syst.,22(2):347–367,2016.
[22]A.Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations,Berlin:Springer-Verlag,1983.
[2]S.Gerbi,B.Said-Houari,Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term,Applied Mathematics&Computation,218(24):11900–11910,2012.
[3]J.Y.Park,Y.H.Kang,J.A.Kim,Existence and exponential stability for a Euler-Bernoulli beam equation with memory and boundary output feedback control term,Acta Appl.Math.,104:287-301,2008.
[4]X.F.Liu,G.Q.Xu,Exponnential stabilization of Timoshenko beam with input and output delays,Mathematical Control and Related Fields,6(2):271–292 2016.
[5]R.Datko,J.Lagnese,and M.P.Polis,An example on the effect of time delays in boundary feedback stabilization of wave equations,SIAM J.Control Optim.,24:152–156,1986.
[6]R.Datko,Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,SIAM J.Control Optim.,26:697–713,1988.
[7]R.Datko,Two examples of ill-posedness with respect to time delays revisited,IEEE Trans.Autom Control,42:511–515,1997.
[8]G.Q.Xu,S.P.Yung,and L.K.Li,Stability of wave system with input delay in the boundary,ESAIM:Control Optimisation and Calculus of Variations,12:770–785,2006.
[9]S.Nicaise,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 Optimization,45(5):1561–1585,2006.
[10]S.Nicaise and C.Pignotti,Exponential stability of secondorder evolution equations with structural damping and dynamic boundary delay feedback,IMA J.Math.Control Inform.,28:417–446,2011.
[11]B.Z.Guo,Z.C.Shao,Regularity of an Euler-Bernoulli equation with Neumann control and collocated observation,Journal of Dynamic and Control System,12:405–418,2006.
[12]P.C.Han,Y.F.Li,G.Q.Xu,and D.H.Liu,The Exponential stability result of an Euler-Bernoulli beam equation with interior delays and boundary damping,Journal of Difference Equations,http://dx.doi.org/10.1155/2016/3732176.
[13]S.Nicaise and C.Pignotti,Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback.Discrete and Continuous Dynamical Systems,9(3):791–813,2016.
[14]Y.F.Shang and G.Q.Xu,Stabilization of an Euler-Bernoulli beam with input delay in the boundary control,Systems Control Letters,61(11):1069–1078,2012.
[15]H.Wang,G.Q.Xu,Exponential stabilization of 1-d wave equation with input delay.Wseas Transactions on Mathematics,12:(10),1001–1013,2013.
[16]G.Q.Xu,H.X.Wang,Stabilization of Timoshenko beam system with delays in the boundary control,International Journal of Control,86(6),1165–1178,2013.
[17]Z.J.Han and G.Q.Xu,Output-based stabilization of EulerBernoulli beam with time-delay in boundary input,IMA Journal of Mathematical Control and Information,34(4):533–550,2013.
[18]X.F.Liu,G.Q.Xu,Exponential Stabilization for Timoshenko beam with distributed delay in the boundary control,Abstract&Applied Analysis,4:1–15,2013.
[19]Y.F.Shang,G.Q.Xu,Dynamic feedback control and exponential stabilization of a compound system,J.Math.Anal.Appl.422:858–879,2015.
[20]Y.F.Shang,G.Q.Xu,Output-based stabilization for a onedimensional wave equation with distributed input delay in the boundary control,IMA Journal of Mathematical Control&Information,31(4):533–550,2014.
[21]X.F.Liu,G.Q.Xu,Output-Based Stabilization of Timoshenko Beam with the boundary control and input distributed delay,J.Dyn.Control.Syst.,22(2):347–367,2016.
[22]A.Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations,Berlin:Springer-Verlag,1983.