tailieunhanh - Global exponential stability for nonautonomous cellular neural networks with unbounded delays

In this article, we study cellular neural networks (CNNs) with timevarying coefficients, bounded and unbounded delays. By introducing a new Liapunov functional to approach unbounded delays and using the continuation theorem of coincidence degree, we obtain some sufficient conditions to ensure the existence periodic solutions and global exponential stability of CNNs. | Journal of Science of Hanoi National University of Education Natural sciences Volume 52 Number 4 2007 pp. 38- 46 GLOBAL EXPONENTIAL STABILITY FOR NONAUTONOMOUS CELLULAR NEURAL NETWORKS WITH UNBOUNDED DELAYS Tran Thi Loan and Duong Anh Tuan Department of Mathematics Ha Noi National University of Education Abstract. In this article we study cellular neural networks CNNs with time- varying coefficients bounded and unbounded delays. By introducing a new Lia- punov functional to approach unbounded delays and using the continuation theorem of coincidence degree we obtain some sufficient conditions to ensure the existence periodic solutions and global exponential stability of CNNs. Many of the existing results in previous literature are extended and improved in this paper. 1 Introduction It is well known that cellular neural networks CNNs proposed by and in 1988 have been extensively studied both in theory and applications such as 1 2 3 4 in refs. They have been successfully applied in signal processing pattern recognition and associative memories and especially in static image treatment. Such applications rely on the qualitative properties of the neural networks. Usually the Liapunov functional method is used to study qualitative properties of CNNs. Such a method is performed in three steps. In step 1 we construct a Liapunov functions V t . In step 2 we use suitable technique to estimate V t . In step 3 we put some condi- tions on CNNs such that the function V t satisfies necessary properties. Thus we obtain sufficient criteria to check the qualitative properties of CNNs. In our knowledge results about the neural networks with variable unbounded delays and time varying coefficients have not been widely studied. Moreover we easy see that other authors have not used scale Liapunov functions in their studies refs 3 4 . In this paper we use scale Liapunov functions and the continuation theorem of coincidence degree to establish conditions. 2 Definitions and .