On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow

Abstract

Let π = (Φ, σ) be an exponentially bounded, strongly continuous cocycle over a continuous semiflow σ. We prove that π = (Φ, σ) is uniformly exponentially stable if and only if there exist T > 0 and c ∈ (0, 1), such that for each θ ∈ Θ and x ∈ X there exists τ θ,x ∈ (0, T ] with the property thatAs a consequence of the above result we obtain generalizations, in both continuous-time and discrete-time, of the the well-known theorems of Datko-Pazy, Rolewicz and Zabczyk for an exponentially bounded, strongly continuous cocycle over a semiflow σ. A version of the above theorems for the case of the exponential instability is also obtained.

Datko-Pazy conditions for nonuniform exponential stability

“…As a by-product of our results described in previous paragraph, we also extend (for a certain class of cocycles) results in [25] that deal with the uniform exponential stability of cocycles. More precisely, the assumptions formulated in [25] that imply that the cocycle exhibits uniform exponential stability require that certain conditions hold for every point that belongs to the base space M of our dynamics.…”

“…Finally, we note that Theorem 3 generalizes (for continuous cocyles) the following result established in [25].…”

“…In addition, in [25] and [23] the authors have obtained Datko-Pazy type of theorems for linear cocycles. The importance of those results stems from the fact that the notion of a linear cocycle arises naturally in the study of the nonautonomous dynamical systems.…”

“…Afterwards, we show that negativity of Lyapunov exponents corresponds to the nonuniform exponential stability even in the Banach-space setting (see Theorem 2). We conclude by providing conditions for the uniform exponential stability of cocycles that extend those in [25] (see Theorem 3). In Section 3, we obtain the corresponding results for cocycles over semi-flows.…”

“…More precisely, the assumptions formulated in [25] that imply that the cocycle exhibits uniform exponential stability require that certain conditions hold for every point that belongs to the base space M of our dynamics. Here we show that assumptions in [25] can be considerably relaxed by assuming that those conditions hold for every point that belongs to the subset E of M , which is as large as M from the measure-theoretic point of view but can be considerably smaller from let's say dimension theory point of view (see the discussion after Theorem 3 for more details). This is achieved by using useful results of Morris [18] on the so-called subadditive ergodic optimisation.…”