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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
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