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 $A(\cdot ):[0,\tau ]\to \mathcal{L}(D,X)$ 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 $$\dot{u}(t)+A(t)u(t)=0t\text{ a.e. on }[0,\tau ],u(0)=x\in X.$$ 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 $t\in [0,\tau )$.

13 pages