Uniform exponential stability of linear time-varying systems: revisited

Stability analysis of PE systems via Steklov's averaging technique

“…As it is well known, the system (3) is exponentially stable; see, for example, [1][2][3][4]. Up to now, this fact was established via Lyapunov-like arguments similar to [11] exemplified by, for example, Lemma 5 in [4]; see also Lemma 1 in [12].…”

“…Up to now, this fact was established via Lyapunov-like arguments similar to [11] exemplified by, for example, Lemma 5 in [4]; see also Lemma 1 in [12]. However to the best of our knowledge, no Lyapunov function or functional is available so far that proves exponential stability of this system, though a design of such a functional might be of self-interest when treating, for example, system identification, adaptive control, or robustness issue.…”

“…This stability has been established a long time ago; see, for example, [1][2][3][4]. The theory of converse Lyapunov functions guarantees existence of a Lyapunov function in many situations where stability does hold; however, this theory often offers only an implicit, hardly tractable, and abstract function of such a kind.…”

Fast Sampling of Evolving Systems with Periodic Trajectories

“…Recall (see e.g. [17]) that assumptions of the theorem imply existence of positive numbers ρ, D > 0:…”

“…This is followed by presentation of main results in Section 3. The results are based on the periodicity assumption we impose on the data and also on known facts from the theory of adaptive observes [19], [17]. In Section 4 we illustrate the approach with two examples as well as provide an assessment of scalability of the approach in the problems considered.…”