On Uniform Exponential Stability of Exponentially Bounded Evolution Families

Abstract

A result of Barbashin ([1], [15]) states that an exponentially bounded evolution family $$\{U(t, s)\}_{t \geq s \geq 0}$$ defined on a Banach space and satisfying some measurability conditions is uniformly exponentially stable if and only if for some 1 ≤ p < ∞, we have that: $$\mathop {\sup}\limits_{t \geq 0} \int^{t}_{0} ||U(t, s)||^{p} ds < \infty .$$ Actually the Barbashin result was formulated for non-autonomous differential equations in the framework of finite dimensional spaces. Here we replace the above ”uniform” condition be a ”strong” one. Among others we shall prove that the evolution family $$\{U(t, s)\}_{t \geq s \geq 0}$$ is uniformly exponentially stable if there exists a non-decreasing function $$\phi : {\bf R}_+ \rightarrow {\bf R}_+$$ with $$\phi(r) > 0$$ for all r > 0 such that for each $$x^{*} \in X^{*}$$ , one has: $$\mathop {\sup}\limits_{t \geq 0} \int^{t}_{0} \phi (||U(t, s)^{*}x^{*}||) ds < \infty .$$ In particular, the family U is uniformly exponentially stable if and only if for some 0 < p < ∞ and each $$x^{*} \in X^{*}$$ , the inequality $$\mathop {\sup}\limits_{t \geq 0} \int^t_0 ||U(t, s)^{*}x^{*}||^p ds < \infty$$ is fulfilled. The latter result extends a similar one from the recent paper [4]. Related results for periodic evolution families are also obtained.