zbMATH — the first resource for mathematics

Some remarks about the Perron condition for \(C_0\)-semigroups. (English) Zbl 1003.93043
Summary: We prove the following. Let \(\mathbf{T}=\{T(t)\}_{t\geq 0}\) be a bounded \(C_0\)-semigroup on a Banach space \(X\) and \(A\) its infinitesimal generator. Then Re\(\sigma(A)<0\) if and only if \[ \sup_{t>0} \Biggl\|\int_0^t e^{i\mu\xi}T(\xi)x d\xi\Biggr\|<\infty,\quad\forall\;\mu\in{\mathbb R}, \forall\;x\in X. \] In particular we obtain that a strongly continuous and exponentially bounded family of bounded linear operators \({\mathcal U}\) on \(X\) is uniformly exponentially stable if and only if the spectrum of the infinitesimal generator of the evolution semigroup associated to \({\mathcal U}\) lies in \({\mathbb C}_{-}:=\{z\in{\mathbb C}:\text{Re}(z)<0\}\).

93D20 Asymptotic stability in control theory
93C25 Control/observation systems in abstract spaces
93B28 Operator-theoretic methods