Integral Characterizations For Exponential Stability Of Semigroups And Evolution Families On Banach Spaces

Abstract

A proof of a sufficient condition for a strongly continuous semigroup $\{ T(t)\}_{t\ge 0}$ on a Banach space $X$ to be uniformly exponentially stable is given. This result is a simplification of an earlier theorem by van Neerven, and concludes that a semigroup is uniformly exponentially stable provided $\sup\nolimits_{||x||\le 1}J(||T(\cdot)x||)<\infty$ ; here $J$ is a certain nonlinear functional with certain natural properties. A non-autonomous version of this theorem for evolution families is also given. This implies the well-known Datko-Pazy and Rolewicz Theorems. This result is connected to the uniform asymptotic stability of the well-posed linear and non-autonomous abstract Cauchy problem \begin{equation*} \left\{ \begin{array}{rcl} \dot{u}(t)& = & A(t)u(t),\quad t\geq s\geq 0, \\ u(s) & = & x\in X. \end{array} \right. \end{equation*}