Sufficient conditions for the stability of systems of ordinary differential time-dependent delay equations. Part I. Linear equations

Background. The paper is devoted to the analysis of stability in the sense of Lyapunov steady-state solutions of systems of linear differential equations with time-dependent coefficients and with time-dependent delays. Cases of continuous and pulsed perturbations are considered. Materials and methods. The study of stability was based on the application of the method of freezing of time-dependent coefficients and the subsequent analysis of the stability of the solution of the system in the vicinity of the freezing point. When analyzing systems of differential equations thus transformed, the properties of the logarithmic norms were used. Results. An algorithm is proposed that allows one to obtain sufficient stability criteria for solutions of finite systems of linear differential equations with coefficients and time-dependent delays. The algorithms are effective in both continuous and impulsive perturbations. Conclusions. The proposed method can be used in the study of nonstationary dynamic systems described by systems of ordinary linear differential equations with time-dependent delays.