Exponential stability of linear and almost periodic systems on Banach spaces
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Let vƒ(·, 0) the mild solution of the well-posed inhomogeneous Cauchy problem
v̇(t) = A(t)v(t) + ƒ(t), v(0) = 0 t ≥ 0
on a complex Banach space X, where A(·) is an almost periodic (possible unbounded) operator-valued function. We prove that vƒ(·, 0) belongs to a suitable subspace of bounded and uniformly continuous functions if and only if for each x ∈ X the solution of the homogeneous Cauchy problem
u̇(t) = A(t)u(t), u(0) = x t ≥ 0
is uniformly exponentially stable. Our approach is based on the spectral theory of evolution semigroups.
CitationBuse, C., & Lupulescu, V. (2003). Exponential stability of linear and almost periodic systems on Banach spaces. Electronic Journal of Differential Equations, 2003(125), pp. 1-7.
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