Controller design for a Schr?dinger Equation with an input delay via backstepping method
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- 英文论文题名：Controller design for a Schr?dinger Equation with an input delay via backstepping method
- 论文作者：Dongyi Liu ; Rumeng Han ; Genqi Xu
- 英文论文作者：Dongyi Liu ; Rumeng Han ; Genqi Xu ; School of Mathematics ; Tianjin University
- 年：2017
- 作者机构：School of Mathematics,Tianjin University;
- 英文论文关键词：Schr?dinger Equation ; Backstepping Method ; State-Feedback Control ; Stability
- 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（B）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（B）
- 语种：英文
- 分类号：O231
- 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：254k
- 原文格式：D
- 会议级别：全国
摘要
This paper is concerned with the controller design for a one dimensional Schr?dinger equation with an input delay on the boundary. A state-feedback control law is presented via the backstepping method to stabilize the delayed system. The numerical simulation verifies the feasibility of the suggested control law.
This paper is concerned with the controller design for a one dimensional Schr?dinger equation with an input delay on the boundary. A state-feedback control law is presented via the backstepping method to stabilize the delayed system. The numerical simulation verifies the feasibility of the suggested control law.
This paper is concerned with the controller design for a one dimensional Schr?dinger equation with an input delay on the boundary. A state-feedback control law is presented via the backstepping method to stabilize the delayed system. The numerical simulation verifies the feasibility of the suggested control law.
引文
[1]R.Datko,J.Lagnese,and M.Polis,An example on the effect of time delays in boundary feedback stabilization of wave equations,SIAM Journal on Control and Optimization,24(1):152-156,1986.
[2]R.Datko,Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,SIAM Journal on Control and Optimization,26(3):697-713,1988.
[3]O.M?rgul,On the stabilization and stability robustness against small delays of some damped wave equation,IEEE Trans.on Automatic Control,40(9):1626-1630,1995.
[4]J.Liang,Y.Chen,and B.Guo,A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors,in Proceedings of the 42nd IEEEConference on Decision and Control,2003:809-814.
[5]G.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(4):770-785,2006.
[6]Y.Guo,Y.Chen,G.Xu,and Y.Zhang,Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks.Journal of Mathematical Analysis and Applications,395(2):727-746,2012.
[7]M.Krstic,and A.Smyshlyaev,Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays,Systems and Control Letters,57(9):750-758,2008.
[8]F.Mazenc,and M.Malisoff,Extensions of Razumikhin’s theorem and Lyapunov Krasovskii functional constructions for time-varying systems with delay,Automatica,78:1-13,2017.
[9]E.Machtyngier,Exact controllability for the Schr?dinger equation,SIAM Journal on Control and Optimization,32(1):24-34,1994.
[10]E.Machtyngier,and E.Zuazua,Stabilization of the Schr?dinger equation,Portugaliae Mathematica,51(2):243-256,1994.
[11]J.Liu,and J.Wang,Output-feedback stabilization of an antistable Schr?dinger equation by boundary feedback with only displacement observation,Journal of Dynamical and Control Systems,19(4):471-482,2013.
[12]C.Bortot,M.Cavalcanti,W.Correa,and V.Cavalcanti,Uniform decay rateestimates for Schr?dinger and plate equations with nonlinear locally distributed damping,Journal of Differential Equations,254(9):3729-3764,2013.
[13]K.Yang,and C.Yao,Stabilization of one-dimensional Schr?dinger equation with variable coefficient under delayed boundary output feedback,Asian Journal of Control,15(5):375-380,2013.
[14]B.Guo,and K.Yang,Output feedback stabilization of a one-dimensional Schr?dinger equation by boundary observation with time delay,IEEE Transactions on Automatic Control,55(5):1226-1232,2010.
[15]M.Krstic,and B.Guo,Boundary controllers and observers for the linearized schr?dinger equation,SIAM Journal on Control and Optimization,49(4):1479-1497,2011.
[16]H.Cui,Z.Han,and G.Xu,Stabilization for Schr?dinger equation with a time delay in the boundary input,Applicable Analysis,95(5):963-977,2016.
[17]H.Cui,D.Liu,and G.Xu,Asymptotic behavior of a Schr?dinger equation under a constrained boundary feedback,under review.
[18]A.Pazy,Semigroups of linear operators and applications to partial differential equations.Berlin:Springer-Verlag,1983.
[19]M.Tucsnak,and G.Weiss,Observation and control for operator semigroups.Basel:Birkh¨auser,2009.
[2]R.Datko,Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,SIAM Journal on Control and Optimization,26(3):697-713,1988.
[3]O.M?rgul,On the stabilization and stability robustness against small delays of some damped wave equation,IEEE Trans.on Automatic Control,40(9):1626-1630,1995.
[4]J.Liang,Y.Chen,and B.Guo,A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors,in Proceedings of the 42nd IEEEConference on Decision and Control,2003:809-814.
[5]G.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(4):770-785,2006.
[6]Y.Guo,Y.Chen,G.Xu,and Y.Zhang,Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks.Journal of Mathematical Analysis and Applications,395(2):727-746,2012.
[7]M.Krstic,and A.Smyshlyaev,Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays,Systems and Control Letters,57(9):750-758,2008.
[8]F.Mazenc,and M.Malisoff,Extensions of Razumikhin’s theorem and Lyapunov Krasovskii functional constructions for time-varying systems with delay,Automatica,78:1-13,2017.
[9]E.Machtyngier,Exact controllability for the Schr?dinger equation,SIAM Journal on Control and Optimization,32(1):24-34,1994.
[10]E.Machtyngier,and E.Zuazua,Stabilization of the Schr?dinger equation,Portugaliae Mathematica,51(2):243-256,1994.
[11]J.Liu,and J.Wang,Output-feedback stabilization of an antistable Schr?dinger equation by boundary feedback with only displacement observation,Journal of Dynamical and Control Systems,19(4):471-482,2013.
[12]C.Bortot,M.Cavalcanti,W.Correa,and V.Cavalcanti,Uniform decay rateestimates for Schr?dinger and plate equations with nonlinear locally distributed damping,Journal of Differential Equations,254(9):3729-3764,2013.
[13]K.Yang,and C.Yao,Stabilization of one-dimensional Schr?dinger equation with variable coefficient under delayed boundary output feedback,Asian Journal of Control,15(5):375-380,2013.
[14]B.Guo,and K.Yang,Output feedback stabilization of a one-dimensional Schr?dinger equation by boundary observation with time delay,IEEE Transactions on Automatic Control,55(5):1226-1232,2010.
[15]M.Krstic,and B.Guo,Boundary controllers and observers for the linearized schr?dinger equation,SIAM Journal on Control and Optimization,49(4):1479-1497,2011.
[16]H.Cui,Z.Han,and G.Xu,Stabilization for Schr?dinger equation with a time delay in the boundary input,Applicable Analysis,95(5):963-977,2016.
[17]H.Cui,D.Liu,and G.Xu,Asymptotic behavior of a Schr?dinger equation under a constrained boundary feedback,under review.
[18]A.Pazy,Semigroups of linear operators and applications to partial differential equations.Berlin:Springer-Verlag,1983.
[19]M.Tucsnak,and G.Weiss,Observation and control for operator semigroups.Basel:Birkh¨auser,2009.