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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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