On the admissibility of observation operators for evolution families

Abstract

This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix $\tau > 0$ and let $\left( A(t)\right) _{t \in [0,\tau ]} \subset \mathcal {L}(D,X)$, 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 $t \in [0,\tau ]$ and that $A(\cdot ) : [0,\tau ] \rightarrow \mathcal {L}(D,X)$ satisfies a regularity condition (viz. relative p-Dini for some $p \in (1,\infty )$). At first sight, we show that there exists an evolution family on X associated to the problem

Then we prove that an observation operator is admissible for $A(\cdot )$ if and only if it is admissible for each A(t) for all $t \in [0,\tau )$.