The purpose of this study is to obtain sufficient conditions for output asymptotic stability of nonlinear nonautonomous time-delay systems described by ordinary differential equations. Both uniform and non-uniform output asymptotic stability are considered separately — unlike classical Lyapunov asymptotic stability, these two properties are not equivalent in general, even for autonomous systems.

Methods. This work investigates the capabilities of Lyapunov’s direct method in formulating sufficient conditions for output asymptotic stability of nonlinear time-delay systems. Using the well-studied problem of partial stability as an example, known results on
output asymptotic stability for time-delay systems are analyzed in terms of Lyapunov functions and LyapunovKrasovskii functionals. The differences in requirements for auxiliary constructions compared to sufficient conditions for classical asymptotic stability are discussed, as well as conditions ensuring uniform convergence.

Results. New results on output asymptotic stability and uniform output asymptotic stability for nonautonomous timedelay systems are presented in terms of a Lyapunov–Razumikhin function, which is not required to be signdefinite with respect to the output.

Conclusion. New sufficient conditions for output asymptotic stability of nonlinear nonautonomous time-delay systems are formulated. Conditions for both non-uniform and uniform output asymptotic stability are obtained in terms of Lyapunov–Razumikhin functions. The requirements imposed on these functions as well as on the right-hand side of the system are less restrictive than those in known similar results, thus expanding the applicability of the method to the analysis of specific systems.