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Exponential stability of linear and almost periodic systems on Banach spaces. (English) Zbl 1043.35022
Summary: Let $$v_f(\cdot, 0)$$ the mild solution of the well-posed inhomogeneous Cauchy problem $\dot v(t)=A(t)v(t)+f(t), \quad v(0)=0\quad t\geq 0$ on a complex Banach space $$X$$, where $$A(\cdot)$$ is an almost periodic (possible unbounded) operator-valued function. We prove that $$v_f(\cdot, 0)$$ belongs to a suitable subspace of bounded and uniformly continuous functions if and only if for each $$x\in X$$ the solution of the homogeneous Cauchy problem $\dot u(t)=A(t)u(t), \quad u(0)=x\quad t\geq 0$ is uniformly exponentially stable. Our approach is based on the spectral theory of evolution semigroups.

##### MSC:
 35B15 Almost and pseudo-almost periodic solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 47A10 Spectrum, resolvent 47D03 Groups and semigroups of linear operators 35B10 Periodic solutions to PDEs 47D06 One-parameter semigroups and linear evolution equations
##### Keywords:
uniform exponential stability; evolution semigroups
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