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Exponential stability for periodic evolution families of bounded linear operators. (English) Zbl 1179.47038
Summary: We prove that a $$q$$-periodic evolution family $\mathcal U=\{U(t,s): t\geq s\geq 0\}$ of bounded linear operators is uniformly exponentially stable if and only if $\sup_{t>0}\|\int^t_0 e^{-i\mu\xi}U(t,\xi)f(\xi)d\xi\|=M(\mu,f)<\infty$ for all $$\mu\in\mathbb R$$ and $$f\in P_q(\mathbb R_+,X)$$ (that is, $$f$$ is a $$q$$-periodic and continuous function on $$\mathbb R_+$$).
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34D05 Asymptotic properties of solutions to ordinary differential equations
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