Article

# Stability of impulsive functional differential equations

Department of Mathematics, Tongji University, 200092, PR China

Journal of Mathematical Analysis and Applications (Impact Factor: 1.05). 01/2005; DOI: 10.1016/j.na.2007.04.009 - [Show abstract] [Hide abstract]

**ABSTRACT:**Strict stability is the kind of stability that can give us some information about the rate of decay of the solution. There are some results about strict stability of functional differential equations. In this paper, we shall extend strict stability to Impulsive functional differential equations in which the state variables on the impulses are related to time delay. By using Lyapunov functions and Razumikhin technique, some criteria for strict stability for functional differential equations, in which the state variables on the impulses are related to the time delay are provided, and we can see that impulses do contribute to the system's strict stability behavior.Lecture Notes in Engineering and Computer. 08/2012; 2197(1):3. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, global exponential synchronization of Takagi–Sugeno (TS) fuzzy complex dynamical networks with multiple time-varying delays and stochastic perturbations is studied via delayed impulsive distributed control. Both impulsive and continuous parts of the fuzzy model have different multiple time-varying delays, which make the considered model general and practical. A novel lemma on exponential stability of impulsive delayed differential equations was established, in which the impulsive functions exhibit multiple time-varying delays. By utilizing the proposed lemma, Lyapunov functions, the stochastic analysis techniques, and the Kronecker product, general criteria ensuring global exponential synchronization in mean square of the addressed TS fuzzy complex networks are obtained. Moreover, the theory of function minimum value is utilized to reduce the conservativeness of the obtained synchronization criteria, which also makes them simple and easy to be verified in practical applications. Results of this paper improve and extend some existing ones. Numerical simulations including small-world network coupled with time-delayed Lorenz system are given to show the effectiveness of the theoretical results.Fuzzy Sets and Systems 01/2014; 235:25–43. · 1.75 Impact Factor -
##### Article: Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations

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**ABSTRACT:**Strict stability is the kind of stability that can give us some information about the rate of decay of the solution. There are some results about strict stability of functional differential equations. On other hand, in the study of stability, an interesting set of problems deal with bringing sets close to a certain state, rather than the equilibrium state. The desired state of a system may be mathematically unstable and yet the system may oscillate sufficiently near this state that its performance is acceptable. Many problems fall into this category. Such considerations led to the notion of practical stability which is neither weaker nor stronger than stability. In this paper, strict practical stability of Impulsive functional differential equations in which the state variables on the impulses are related to time delay is considered. By using Lyapunov functions and Razumikhin technique, some criteria for strict practical stability for functional differential equations, in which the state variables on the impulses are related to the time delay, are provided.Global Journal of Science Frontier Research Mathematics and Decision Sciences. 02/2013; 13(1):1-8.

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