# 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 $${\mathbb C}_{-}:=\{z\in{\mathbb C}:\text{Re}(z)<0\}$$.

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