On uniform exponential stability of evolution families (Q2777990)

For the evolution family \(\Phi= \{\Phi(t,s)\}_{t\geq s\geq 0}\) of bounded linear operators in a Banach space, the uniform exponential stability is analyzed: \(\Phi\) is said to be uniformly exponentially stable iff there are numbers \(N\), \(\nu>0\) such that for all \(t\geq s\geq 0\) NEWLINE\[NEWLINE\|\Phi(t, s)\|\leq Ne^{-\nu(t- s)}.NEWLINE\]NEWLINE Here, especially a generalization of Neerven's theorem [\textit{J. van Neerven}, The asymptotic behaviour of semigroups of linear operators. Operator Theory: Advances and Applications 88 (Birkhäuser, Basel) (1996; Zbl 0905.47001)] is proved. Neerven's theorem provides a sufficient condition for a \(C_0\)-semigroup being uniformly exponentially stable. The generalization given here provides necessary and sufficient conditions for an evolution family being uniformly exponentially stable. The authors also consider periodic evolution families.