Search

Global asymptotic stability is an important issue for wide applications of recurrent neural networks with time-varying delays. The Lyapunov–Krasovskii functional method is a powerful tool to check the global asymptotic stability of a delayed recurrent neural network. When the Lyapunov–Krasovskii functional method is employed, three steps are necessary in order to derive a global asymptotic stability criterion: (i) constructing a Lyapunov–Krasovskii functional, (ii) estimating the derivative of the Lyapunov–Krasovskii functional, and (iii) formulating a global asymptotic stability criterion. This paper provides an overview of recent developments in each step with insightful understanding. In the first step, some existing Lyapunov–Krasovskii functionals for stability of delayed recurrent neural networks are anatomized. In the second step, a free-weighting matrix approach, an integral inequality approach and its recent developments, reciprocally convex inequalities and S -procedure are analyzed in detail. In the third step, linear convex and quadratic convex approaches, together with the refinement of allowable delay sets are reviewed. Finally, some challenging issues are presented to guide the future research. [ABSTRACT FROM AUTHOR]

This paper is concerned with stability of a linear system with a time-varying delay. First, an improved reciprocally convex inequality including some existing ones as its special cases is derived. Compared with an extended reciprocally convex inequality recently reported, the improved reciprocally convex inequality can provide a maximum lower bound with less slack matrix variables for some reciprocally convex combinations. Second, an augmented Lyapunov–Krasovskii functional is tailored for the use of a second-order Bessel–Legendre inequality. Third, a stability criterion is derived by employing the proposed reciprocally convex inequality and the augmented Lyapunov–Krasovskii functional. Finally, two well-studied numerical examples are given to show that the obtained stability criterion can produce a larger upper bound of the time-varying delay than some existing criteria. [ABSTRACT FROM AUTHOR]