Stability Analysis of Differential Equations with Time-Dependent Delay

In recent years, increasing attention has been focused on the study of dynamical systems, often with time delay, which model in population dynamics [Chen & Blackmore, 2002; Gopalsamy, 1992; Yang, 1993; Matin & Ruan, 2001; May, 1973], physics and mechanics [Cahlon et al., 1997], remote control [Ryabov, 1960], radiation damping [Chicone et al., 2001], and so on and so forth. Such systems attract much attention and many theoretical results are obtained [Baer et al., 1989; Cahlon et al., 1998; Chicone et al., 2003; Favini et al., 2003; Hua & Lu, 2000, 2001; Tang, 2004]. However, the stability on finite interval of differential equations with time-dependent delay, has not been discussed yet. In this Letter, we attempt to analyze such stability in detail. We first introduce the concept of stability on finite time interval, called “interval stability” in brief. Then we consider interval stability and Liapunov stability of the following general system with time-dependent delay,