On exponential stability of nonlinear time-varying differential equations

Weak Exponential Stability for Time-Periodic Differential Inclusions via First Approximation Averaging

“…Later, in [7,11], under some homogeneity conditions, it was shown that the asymptotic stability properties of the original equation can be derived from the respective asymptotic stability properties of the averaged equation. In [1,6,9] similar results were established without any homogeneity conditions.…”

Models of n-th order linear time – varying systems

“…The parametric functions are strictly positive due to the assumed variability of the coefficients. It is significant for the stability of parametric systems [1,9]. It ought to be noticed that for considered parametric function classes (4), (5) the solutions to the Equations (2), (3) exist in a closed form [6].…”

“…For parametric systems, the relation between input x(t) and output y(t) signal [4,9] (1) where: h(t, J) -impulse response of the system. The impulse responses of the first and second order LTV systems have been determined in work [6].…”

Stability analysis via averaging for nonlinear systems with delays

“…Solutions of (1) are absolutely continuous functions that satisfies the equation almost everywhere. With the assumptions imposed on f , the usual regularity conditions such as existence, uniqueness, and maximum continuation of solutions hold for (1).…”

“…A powerful tool in this context is the averaging method, see, for instance, [1], [3], [5], [6], [7], [10], [11], and references therein. The key idea of the averaging method is to convert the stability analysis for a time varying system to the analysis of a time invariant system that can be approximated in certain manners by the original system as the small parameter approaches 0 (see Definitions 3.1 and 3.4).…”