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Extensions of Razumikhin-type stability theorems for nonlinear time-delay systems on time scales

This paper focuses on the stability analysis of nonlinear time-delay systems on time scales, which are of generality as they can include not only the traditional continuous and discrete ones, but also some other cases, such as systems on general uniform or non-uniform time domains. One existing classical method for analyzing stability of such systems is the Razumikhin-type theorem with requiring the time derivative of relating Lyapunov function to be non-positive for uniform stability and negative for uniform asymptotic or exponential stability, which are difficult to be satisfied. To relax these restrictions, by introducing the time-scale type uniformly stable function and uniformly asymptotically stable function, this paper presents several less conservative stability criteria, in which the time-scale time derivative of Lyapunov function can be positive or non-negative. To demonstrate the effectiveness of the theoretic results, a numerical example about non-continuous and non-discrete time-delay systems is given.