On the admissibility of observation operators for evolution families

Abstract

This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix τ > 0 and let (A(t)) t∈[0,τ ] ⊂ L(D, X), where D and X are two Banach spaces such that D is continuously and densely embedded into X. We assume that the operator A(t) has maximal regularity for all t ∈ [0, τ ] and that A(•) : [0, τ ] → L(D, X) satisfies a regularity condition (viz. relative p-Dini for some p ∈ (1, ∞)). At first sight, we show that there exists an evolution family on X associated to the problemThen we prove that an observation operator is admissible for A(•) if and only if it is admissible for each A(t) for all t ∈ [0, τ ).