Stability analysis of some delay differential inequalities with small time delays and its applications

In this paper, we discuss the asymptotic stability of the trajectories governed by the scalar delay differential inequalities: D⁺x(t)≤-a(t)x(t)+b(t)sup(0≤s≤τ)x(t-s). Here, the requirements on a(t) and b(t) are more relaxed than those in previous works. For example, a(t), b(t), and a(t)-b(t) are not necessarily nonnegative. We prove that when τ is small, the asymptotic stability of x(t) can be obtained if the time average of a(t)-b(t) on some fixed length T is lower bounded by some positive δ. And we explicitly give the upper bound of τ. We also give two applications of the theoretical results. First, we consider self synchronization in Hopfield networks with time varying connections. Then we investigate consensus in networks with time varying topologies and arbitrary coupling weights. In both applications, we extend some of our previous works where time delays are not considered. At last, two numerical examples with simulations are provided to illustrate the effectiveness of the theoretical results.