This paper develops some criteria for a kind of hybrid stochastic systems with time-delay, which improve existing results on hybrid systems without considering noises. The improved results show that the presence of noise is quite involved in the stability analysis of hybrid systems. New results can be used to analyze the stability of a kind of stochastic hybrid impulsive and switching neural networks (SHISNN). Therefore, stability analysis of SHISNN can be turned into solving a linear matrix inequality (LMI). 1. Introduction With the development of social production, many practical systems cannot be modeled by linear time-invariant systems. In this case, hybrid systems are employed to model many practical systems. A hybrid system is a dynamical system with continuous dynamics, discrete dynamics, and the interaction between them (see, e.g., [1–3]). Hybrid systems are important from both the practical and theoretical point of view. In fact, hybrid systems naturally represent a wide class of practical systems which are subject to known or unknown abrupt parameter variations and which undergo sudden change of system structures due to the failure of a component. Another practical motivation for studying hybrid dynamical systems originates from the fact that the hybrid control scheme provides an effective approach for controlling highly nonlinear complex dynamical systems and systems with uncertain and/or unknown parameters. From theoretical viewpoints, the interactions between low-level continuous dynamics and high-level discrete logics, mainly governed by the switching and impulsive mechanisms, bring new challenges beyond the conventional system theory. An area of particular interest has been the analysis of stability of hybrid dynamical systems (see, e.g., [4–9]). Impulsive stabilization of dynamical systems has attracted increasing interests in fields such as population dynamics, automatic control, drug administration, and communication networks (see [4, 5, 8]) and references therein). In many cases, impulsive stabilization may give better performance than continuous stabilization since the former is implemented only at impulsive instants (see [10]) while the latter does so at every moment (see [5]). Even in some cases, only impulsive stabilization can be used for control purpose. For instance, a central bank cannot change its interest rate every day in order to regulate the money supply in a financial market. The fundamental theoretic and systemic method of impulsive dynamical systems have been established in the recent years (see [1, 5]). On the other
References
[1]
D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, Mass, USA, 2003.
[2]
X. Liao and K. W. Wong, “Global exponential stability of hybrid bidirectional associative memory neural networks with discrete delay,” Physical Review E, vol. 67, no. 41, Article ID 042901, 2003.
[3]
K. Yuan, J. Cao, and H. X. Li, “Robust stability of switched Cohen-Grossberg neural networks with mixed time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 36, no. 6, pp. 1356–1363, 2006.
[4]
H. Huang, Y. Qu, and H.-X. Li, “Robust stability analysis of switched Hopfield neural networks with time-varying delay under uncertainty,” Physics Letters A, vol. 345, no. 4–6, pp. 345–354, 2005.
[5]
Z. Sun and S. Ge, Switched Linear Systems Control and Assign, Springer, London, UK, 2005.
[6]
M. S. Branicky, V. S. Borkar, and S. K. Mitter, “A unified framework for hybrid control: model and optimal control theory,” IEEE Transactions on Automatic Control, vol. 43, no. 1, pp. 31–45, 1998.
[7]
X. Li, Y. Soh, and C. Wen, Switched and Impulsive Systems: Analysis,Design and Applications, Springer, Berlin, Germany, 2005.
[8]
Z.-H. Guan, D. J. Hill, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,” IEEE Transactions on Automatic Control, vol. 50, no. 7, pp. 1058–1062, 2005.
[9]
C. Li, G. Feng, and T. Huang, “On hybrid impulsive and switching neural networks,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 38, no. 6, pp. 1549–1560, 2008.
[10]
H. Liu, L. Zhao, Z. Zhang, and Y. Ou, “Stochastic stability of Markovian jumping Hopfield neural networks with constant and distributed delays,” Neurocomputing, vol. 72, no. 16–18, pp. 3669–3674, 2009.
[11]
C. Huang and J. Cao, “On pth moment exponential stability of stochastic Cohen-Grossberg neural networks with time-varying delays,” Neurocomputing, vol. 73, no. 4–6, pp. 986–990, 2010.
[12]
Y. Sun and J. Cao, “Stabilization of stochastic delayed neural networks with markovian switching,” Asian Journal of Control, vol. 10, no. 3, pp. 327–340, 2008.
[13]
L. Huang and X. Mao, “On almost sure stability of hybrid stochastic systems with mode-dependent interval delays,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1946–1952, 2010.
[14]
D. Li, X. Wang, and D. Xu, “Existence and global p-exponential stability of periodic solution for impulsive stochastic neural networks with delays,” Nonlinear Analysis: Hybrid Systems, vol. 6, no. 3, pp. 847–858, 2012.
[15]
L. Pan and J. Cao, “Robust stability for uncertain stochastic neural network with delay and impuses,” Neurocomputing, vol. 94, pp. 102–110, 2012.
[16]
L. Xu, D. He, and Q. Ma, “Impulsive stabilitization of stochastic differential equations with time delays,” Mathematical and Computer Modelling, vol. 57, pp. 997–1004, 2013.
[17]
C. Li, J. Shi, and J. Sun, “Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 10, pp. 3099–3111, 2011.
[18]
X. Wu, W. Zhang, and Y. Tang, “p-Moment stability of impulsive stochastic delay differential systems with Markovian switching,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 1870–1879, 2013.
