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On uniform exponential stability of exponentially bounded evolution families. (English) Zbl 1165.47027
The authors deduce new characterizations for the uniform exponential stability of evolution families in terms of rearrangement Banach function spaces. By applying their methods, the authors obtain stability conditions both for semigroups and for periodic evolution families.
Let $$X$$ be a Banach space, let $$J \in \{{\mathbb R}, {\mathbb R}_+\}$$ and $$\Delta_J := \{(t, s) \in J \times J : t\geq s\}$$. One of the main results of the paper states that if $$U=\{U(t, s) : (t, s) \in \Delta_J\}$$ is an exponentially bounded evolution family on $$X$$ and $$E(J)$$ is a rearrangement invariant Banach function space over $$J$$ such that $$\lim_{r \to \infty} \|\chi_{[0, r]}(\cdot)\|_{E^\prime(J)} = \infty,$$ then the following statements are equivalent: 7mm
(i)
the family $$U$$ is uniformly exponentially stable;
(ii)
there exists a positive constant $$M^\prime(J)$$ such that, for each $$(s, t) \in \Delta_J$$ and each $$x^* \in X^*$$, the map $$\chi_{[s, t]}(\cdot)\;\|U(t, \cdot)^*x^*\|$$ defines an element of the space $$E^\prime(J)$$ and
$\sup_{(t, s) \in \Delta_J}\|\chi_{[s, t]}(\cdot)\;\|U(t, \cdot)^*x^*\|\;\|_{E^\prime(J)} \leq M^\prime(J)\;\|x^*\|.$
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 35B35 Stability in context of PDEs 34D20 Stability of solutions to ordinary differential equations 35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
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