MEAN SQUARE EXPONENTIAL DISSIPATIVITY OF SINGULARLY PERTURBED STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

Title & Authors
MEAN SQUARE EXPONENTIAL DISSIPATIVITY OF SINGULARLY PERTURBED STOCHASTIC DELAY DIFFERENTIAL EQUATIONS
Xu, Liguang; Ma, Zhixia; Hu, Hongxiao;

Abstract
This paper investigates mean square exponential dissipativity of singularly perturbed stochastic delay differential equations. The L-operator delay differential inequality and stochastic analysis technique are used to establish sufficient conditions ensuring the mean square exponential dissipativity of singularly perturbed stochastic delay differential equations for sufficiently small $\small{{\varepsilon}}$ > 0. An example is presented to illustrate the efficiency of the obtained results.
Keywords
delay;stochastic;singularly perturbed;mean square exponential dissipativity;L-operator delay differential inequality;
Language
English
Cited by
1.
Lagrange $$p$$ p -Stability and Exponential $$p$$ p -Convergence for Stochastic Cohen–Grossberg Neural Networks with Time-Varying Delays, Neural Processing Letters, 2016, 43, 3, 611
References
1.
H. C. Chang and M. Aluko, Multi-scale analysis of exotic dynamics in surface catalyzed reactions-I, Chemical Engineering Science 39 (1984), 37-50.

2.
J. H. Chow, Time Scale Modelling of Dynamic Networks, Springer-Verlag, New York, 1982.

3.
P. D. Christofides and P. Dsoutidis, Feedback control of two-time-scale nonlinear systems, Internat. J. Control 63 (1996), no. 5, 965-994.

4.
J. H. Cruz and P. Z. Taboas, Periodic solutions and stability for a singularly perturbed linear delay differential equation, Nonlinear Anal. 67 (2007), 1657-1667.

5.
D. Da and M. Corless, Exponential stability of a class of nonlinear singularly perturbed systems with marginally sable boundary layer systems, In Proceedings of the American Control Conference, 3101-3106, San Francisco, CA, 1993.

6.
M. El-Ansary, Stochastic feedback design for a class of nonlinear singularly perturbed systems, Internat. J. Systems Sci. 22 (1991), no. 10, 2013-2023.

7.
M. El-Ansary and H. K. Khalil, On the interplay of singular perturbations and wide-band stochastic fluctuations, SIAM J. Control Optim. 24 (1986), no. 1, 83-94.

8.
E. Fridman, Effects of small delays on stability of singularly perturbed systems, Automatica J. IFAC 38 (2002), no. 5, 897-902.

9.
V. B. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992.

10.
X. Z. Liu, X. M. Shen, and Y. Zhang, Exponential stability of singularly perturbed systems with time delay, Appl. Anal. 82 (2003), no. 2, 117-130.

11.
X. R. Mao, Razumikhin-type theorems on exponential stability of stochastic functional-differential equations, Stochastic Process. Appl. 65 (1996), no. 2, 233-250.

12.
X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publication, Chichester, 1997.

13.
X. R. Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations 153 (1999), no. 1, 175-195.

14.
S.-E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1986.

15.
J. J. Monge and C. Georgakis, The effect of operating variables on the dynamics of catalytic cracking processes, Chemical Engineering Communications 60 (1987), 1-15.

16.
N. Prljaca and Z. Gajic, A method for optimal control and filtering of multitime-scale linear singularly-perturbed stochastic systems, Automatica J. IFAC 44 (2008), no. 8, 2149-2156.

17.
L. Socha, Exponential stability of singularly perturbed stochastic systems, IEEE Trans. Automat. Contr. 45 (2000), no. 3, 576-580.

18.
H. J. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl. 270 (2002), no. 1, 143-149.

19.
H. J. Tian, Dissipativity and exponential stability of ${\theta}$-method for singularly perturbed delay differential equations with a bounded lag, J. Comput. Math. 21 (2003), no. 6, 715-726.

20.
H. J. Tian, Numerical and analytic dissipativity of the ${\theta}$-method for delay differential equations with a bounded variable lag, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), no. 5, 1839-1845.

21.
L. G. Xu, Exponential p-stability of singularly perturbed impulsive stochastic delay differential systems, Int. J. Control. Autom. 9 (2011), no. 5, 966-972.

22.
D. Y. Xu, Z. G. Yang, and Y. M. Huang, Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Differential Equations 245 (2008), no. 6, 1681-1703.