Improved Razumikhin and Krasovskii stability criteria for time-varying stochastic time-delay systems

The problem of p-th moment stability for time-varying stochastic time-delay systems with Markovian switching is investigated in this paper. Some novel stability criteria are obtained by applying the generalized Razumikhin and Krasovskii stability theorems. Both p-th moment asymptotic stability and (integral) input-to-state stability are considered based on the notion and properties of uniformly stable functions and the improved comparison principles. The established results show that time-derivatives of the constructed Razumikhin functions and Krasovskii functionals are allowed to be indefinite, which improve the existing results on this topic. By applying the obtained results for stochastic systems, we also analyze briefly the stability of time-varying deterministic time-delay systems. Finally, examples are provided to illustrate the effectiveness of the proposed results.

[1]  V. Kolmanovskii,et al.  Stability of Functional Differential Equations , 1986 .

[2]  P. Pepe,et al.  A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems , 2006, Syst. Control. Lett..

[3]  Wenbing Zhang,et al.  pth Moment stability of impulsive stochastic delay differential systems with Markovian switching , 2013, Commun. Nonlinear Sci. Numer. Simul..

[4]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[5]  Shengyuan Xu,et al.  Razumikhin method and exponential stability of hybrid stochastic delay interval systems , 2006 .

[6]  Min Wu,et al.  Indefinite derivative Lyapunov-Krasovskii functional method for input to state stability of nonlinear systems with time-delay , 2015, Appl. Math. Comput..

[7]  Shengyuan Xu,et al.  Robust H∞ control for uncertain discrete stochastic time-delay systems , 2004, Syst. Control. Lett..

[8]  David Angeli,et al.  A characterization of integral input-to-state stability , 2000, IEEE Trans. Autom. Control..

[9]  Xuerong Mao,et al.  On Input-to-State Stability of Stochastic Retarded Systems With Markovian Switching , 2009, IEEE Transactions on Automatic Control.

[10]  Rama Cont,et al.  A functional extension of the Ito formula , 2010 .

[11]  Bin Zhou,et al.  On Asymptotic Stability of Discrete-Time Linear Time-Varying Systems , 2017, IEEE Transactions on Automatic Control.

[12]  Jitao Sun,et al.  p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching , 2006, Autom..

[13]  Yang Tang,et al.  Input-to-state stability of impulsive stochastic delayed systems under linear assumptions , 2016, Autom..

[14]  D. Elworthy.  ASYMPTOTIC METHODS IN THE THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS , 1992 .

[15]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[16]  Bin Zhou,et al.  On asymptotic stability of linear time-varying systems , 2016, Autom..

[17]  Yong He,et al.  Improved Razumikhin-Type Theorem for Input-To-State Stability of Nonlinear Time-Delay Systems , 2014, IEEE Transactions on Automatic Control.

[18]  Jin-Hua She,et al.  Input-to-state stability of nonlinear systems based on an indefinite Lyapunov function , 2012, Syst. Control. Lett..

[19]  P. Olver.  Nonlinear Systems , 2013 .

[20]  Bin Zhou,et al.  Pseudo-predictor feedback stabilization of linear systems with time-varying input delays , 2014, Proceedings of the 33rd Chinese Control Conference.

[21]  Peng Shi,et al.  Stochastic Synchronization of Markovian Jump Neural Networks With Time-Varying Delay Using Sampled Data , 2013, IEEE Transactions on Cybernetics.

[22]  Shiguo Peng,et al.  Some criteria on pth moment stability of impulsive stochastic functional differential equations , 2010 .

[23]  X. Mao.  Stochastic Functional Differential Equations with Markovian Switching , 2004 .

[24]  Dong Yue,et al.  Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching , 2005, IEEE Transactions on Automatic Control.

[25]  Wei Xing Zheng,et al.  Delay-independent stabilization of a class of time-delay systems via periodically intermittent control , 2016, Autom..

[26]  Yun Zhang,et al.  Some New Criteria on $p$th Moment Stability of Stochastic Functional Differential Equations With Markovian Switching , 2010, IEEE Transactions on Automatic Control.

[27]  Xuerong Mao,et al.  Exponential stability of stochastic delay interval systems with Markovian switching , 2002, IEEE Trans. Autom. Control..

[28]  Bin Zhou.  Stability Analysis of Nonlinear Time-Varying Systems by Lyapunov Functions with Indefinite Derivatives , 2015, 1512.02302.

[29]  Guo-Ping Liu,et al.  Stability Analysis of A Class of Hybrid Stochastic Retarded Systems Under Asynchronous Switching , 2014, IEEE Transactions on Automatic Control.

[30]  Shengyuan Xu,et al.  Delay-Dependent $H_{\infty }$ Control and Filtering for Uncertain Markovian Jump Systems With Time-Varying Delays , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[31]  Zongli Lin,et al.  An analysis of the exponential stability of linear stochastic neutral delay systems , 2015 .

[32]  Zhengguang Wu,et al.  New results on delay-dependent stability analysis for neutral stochastic delay systems , 2013, J. Frankl. Inst..

[33]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[34]  Emilia Fridman,et al.  On input-to-state stability of systems with time-delay: A matrix inequalities approach , 2008, Autom..

[35]  Bin Zhou,et al.  Razumikhin and Krasovskii stability theorems for time-varying time-delay systems , 2016, Autom..

[36]  Feiqi Deng,et al.  Razumikhin-type theorems on stability of stochastic retarded systems , 2009, Int. J. Syst. Sci..

[37]  Feiqi Deng,et al.  Stability of Hybrid Stochastic Retarded Systems , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[38]  Michael Malisoff,et al.  Stability analysis for systems with time-varying delay: Trajectory based approach , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[39]  Shen Cong,et al.  On almost sure stability conditions of linear switching stochastic differential systems , 2016 .

[40]  T. M. Flett.  Differential Analysis: Appendix , 1980 .

[41]  Lei Liu,et al.  The asymptotic stability and exponential stability of nonlinear stochastic differential systems with Markovian switching and with polynomial growth , 2012 .

[42]  Hassan K. Khalil,et al.  Nonlinear Systems Third Edition , 2008 .

[43]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.