Exponential growth bound and exponential stability

Summary

In Section 5.1, we characterize the exponential growth bound of the propagators of (ACP _n ) in a Hilbert space in terms of the behavior of $$ on vertical lines in a half complex plane. As a consequence we show that the propagators are exponentially stable if P _λ is boundedly invertible in {λ ∈ C; Reλ ≥ 0} with $$ uniformly bounded there.

Section 5.2 investigates the condition ensuring stability of every single solution of (ACP _n ) in Banach spaces. It turns out to be a concise condition only requiring the uniform boundedness of R _λ in {λ ∈ C; Reλ > −δ} for some δ > 0.