Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities

In this paper, stochastic stability for a class of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities is investigated. Some new results are given based on stochastic Lyapunov functionals. It is shown that an unstable discrete-time Markovian jump delay system can be stochastically stable under certain stabilizing impulses. It is also shown that, when the nearest impulsive time interval is appropriately large, a stable discrete-time Markovian jump delay system can retain its stochastic stability property even with destabilizing impulses. Numerical examples together with their simulations are provided to demonstrate the effectiveness of the derived results.

[1]  D. Sworder,et al.  Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria , 1975 .

[2]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[3]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[4]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[5]  Lihua Xie,et al.  Robust H/sub infinity / control for linear systems with norm-bounded time-varying uncertainty , 1992 .

[6]  M. Fragoso,et al.  Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters , 1993 .

[7]  M. Frigon,et al.  Impulsive differential equations with variable times , 1996 .

[8]  J. Lam,et al.  Stochastic stabilizability and H∞ control for discrete-time jump linear systems with time delay ☆ , 1999 .

[9]  Peng Shi,et al.  Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay , 1999, IEEE Trans. Autom. Control..

[10]  M. Mahmoud,et al.  Robust control for Markovian jump linear discrete-time systems with unknown nonlinearities , 2002 .

[11]  Wassim M. Haddad,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control , 2006 .

[12]  Zidong Wang,et al.  Exponential stability of delayed recurrent neural networks with Markovian jumping parameters , 2006 .

[13]  X. Lou,et al.  Delay-dependent stochastic stability of delayed Hopfield neural networks with Markovian jump parameters , 2007 .

[14]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

[15]  Jitao Sun,et al.  Stabilization of discrete-time Markovian jump linear systems via time-delayed and impulsive controllers , 2008, Autom..

[16]  Yu Zhang,et al.  Stability of impulsive functional differential equations , 2008 .

[17]  Lixian Zhang,et al.  Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities , 2009, Autom..

[18]  Yu Zhang,et al.  Impulsive Control of Discrete Systems With Time Delay , 2009, IEEE Transactions on Automatic Control.

[19]  Kondalsamy Gopalsamy,et al.  Exponential stability preservation in semi-discretisations of BAM networks with nonlinear impulses , 2009 .

[20]  Xiaodi Li,et al.  New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays , 2010 .

[21]  Xinzhi Liu,et al.  Observer-based impulsive chaotic synchronization of discrete-time switched systems , 2010 .

[22]  Jinde Cao,et al.  Robust Exponential Stability of Markovian Jump Impulsive Stochastic Cohen-Grossberg Neural Networks With Mixed Time Delays , 2010, IEEE Transactions on Neural Networks.

[23]  David J. Hill,et al.  Uniform stability and ISS of discrete-time impulsive hybrid systems , 2010 .

[24]  Kok Lay Teo,et al.  Global exponential stability of impulsive discrete-time neural networks with time-varying delays , 2010, Appl. Math. Comput..

[25]  Jie Zhang,et al.  Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices , 2011, Autom..

[26]  Jinde Cao,et al.  Exponential Stability of Stochastic Neural Networks With Both Markovian Jump Parameters and Mixed Time Delays , 2011, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[27]  Junping Shi,et al.  Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks , 2011 .

[28]  Alim P. C. Gonçalves,et al.  Filtering of discrete‐time Markov jump linear systems with uncertain transition probabilities , 2011 .

[29]  Yu Zhang.  Robust exponential stability of uncertain impulsive delay difference equations with continuous time , 2011, J. Frankl. Inst..

[30]  Eva Kaslik,et al.  Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis , 2011, Neural Networks.

[31]  Jitao Sun,et al.  Adaptive-impulsive synchronization of chaotic systems , 2011, Math. Comput. Simul..

[32]  Jinde Cao,et al.  Stability of Markovian jump neural networks with impulse control and time varying delays , 2012 .

[33]  Corentin Briat,et al.  A looped-functional approach for robust stability analysis of linear impulsive systems , 2012, Syst. Control. Lett..

[34]  Ju H. Park,et al.  State estimation of neural networks with time-varying delays and Markovian jumping parameter based on passivity theory , 2012 .

[35]  Xiaodi Li,et al.  Robust Exponential Stability of Stochastically Nonlinear Jump Systems with Mixed Time Delays , 2012, J. Optim. Theory Appl..

[36]  Jinde Cao,et al.  Stability Analysis of Markovian Jump Stochastic BAM Neural Networks With Impulse Control and Mixed Time Delays , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[37]  Jinde Cao,et al.  pth moment exponential synchronization for stochastic delayed Cohen–Grossberg neural networks with Markovian switching , 2011, Nonlinear Dynamics.

[38]  Yu Zhang,et al.  Stability of impulsive piecewise linear systems , 2013, Int. J. Syst. Sci..