Stability Analysis For Delayed Systems Based On Delay Decomposition Approach

Dynamical system is a mathematical model which describes thesystem state variables change over time. Because of the important theoretical valueand practical significance of dynamical system, it has received wide attention ofdomestic and international scholars. The stability is considered as the main controlperformance index in theoretical research and practical application, so the stability ofcontrol system and neural network has very important significance. The time delay iswidely existed in engineering system and frequently causes oscillation or instability indynamical system. Therefore, increasing attention of the scholars has been devoted todynamical systems with time delay in recent years. Based on the Lyapunov stabilitytheory, This paper considers the the robustly asymptotic stability of continuoussystems with interval time-varying delay, global asymptotic stability and globalexponential stability for delayed neural networks. More specifically, the maincontributions are as follows:1. Robustly asymptotic stability analysis for the continuous system with intervaltime-varying delayBy defining a Lyapunov–Krasovskii functional, which conclude the informationboth the bounds of the delay and their points, and using the reciprocally convexapproach, a new delay-range-dependent criterion is derived in terms of the linearmatrix inequalities (LMIs). Finally, some numerical examples are included toillustrate the effectiveness and the less conservativeness of the proposed method.2. Global asymptotic stability analysis for neural networks with time-varying delaysBy dividing the delay interval into three unequal subintervals, a newLyapunov-Krasovskii functional is defined. The reciprocally convex combinationtechnique and integral inequality are utilized to estimate the derivative of theLyapunov-Krasovskii functional. Some global asymptotic stability conditions arederived in term of the linear matrix inequalities (LMIs). These conclusions haveextended the results in the given references.3. Exponential stability analysis for neural networks with distributed time-varyingdelaysBy dividing the delay interval into two unequal subintervals, and defining aLyapunov–Krasovskii functional, an improved stability condition is derived in termsof the linear matrix inequalities (LMIs) by using the reciprocally convex approach.Finally, some numerical examples are included to illustrate the less conservativenessof the proposed method.