Integral Characterizations for Exponential Stability of Semigroups and Evolution Families on Banach Spaces

Let X be a real or complex Banach space and U = {U(t, s)}t≥s≥0 be a strongly continuous and exponentially bounded evolution family on X. Let J be a non-negative functional on the positive cone of the space of all realvalued locally bounded functions on R+ := [0,∞). We suppose that J satisfies some extra-assumptions. Then the family U is uniformly exponentially stable provided that for every x ∈ X we have: sup s≥0 J(||U(s + ·, s)x||) < ∞. This result is connected to the uniform asymptotic stability of the well-posed linear and non-autonomous abstract Cauchy problem { u̇(t) = A(t)u(t), t ≥ s ≥ 0, u(s) = x x ∈ X. In the autonomous case, i.e. when U(t, s) = T (t − s) for some strongly continuous semigroup {T (t)}t≥0 we obtain the well-known theorems of Datko, Littman, Neerven, Pazy and Rolewicz.