For nonautonomous linear differential equations v′ = A(t)v in a Banach space, we consider general exponential dichotomies that extend the notion of (uniform) exponential dichotomy in various ways. Namely, the new notion allows: stable and unstable behavior with respect to growth rates e for an arbitrary function ρ(t); nonuniform exponential behavior, causing that any stability or conditional st...

The aim of this paper is to present a general overview concerning the Rolewicz-Zabzczyk type techniques in the stability theory of dynamical systems. We discuss the main methods based on trajectories that may be used in order to characterize the uniform exponential stability of variational discrete systems and their applications to the case of skew-product flows. Beside our techniques used in t...

We prove that a general system of variational difference equations is uniformly exponentially stable if and only if certain associated sets are of the second category. We also deduce necessary and sufficient conditions for uniform exponential stability of systems with uniformly bounded coefficients. We apply our results for the study of exponential stability of linear skew-product flows, genera...

In this paper we give a characterization for the uniform exponential stability of evolution families {Φ(t, t0)}t≥t0 on R+ that do not have an exponential growth, using the hypothesis that the pairs of function spaces (L1(X), L∞(X)) and (Lp(X), Lq(X)), (p, q) 6= (1,∞), are admissible to the evolution families.

We present several equivalent characterizations of uniform global exponential stability (UGES) and uniform global asymptotic stability (UGAS) of arbitrary closed (not necessarily compact) sets for nonlinear difference inclusions. In particular, we provide several characterizations of these stability properties via summability criteria that do not require the knowledge of a Lyapunov function. We...

In this paper we show that for linear switching systems of the form xn+1 = Anxn + un+1, where the matrices An are chosen arbitrarily from a given set of matrices, bounded-input-bounded-output stability implies uniform exponential stability.

This paper deals with the global uniform exponential stability independent of delay of timedelay linear and time-invariant systems subject to point and distributed delays for the initial conditions being continuous real functions except possibly on a set of zero measure of bounded discontinuities. It is assumed that the delay-free system as well as an auxiliary one are globally uniformly expone...

We consider an integral variational control system on a Banach space X and we study the connections between its uniform exponential stability and the I R , X , O R , X stability, where I andO are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input sp...