Yassine Kharou | Searcharxiv

arXiv2022

On the admissibility of observation operators in the context of maximal regularity

We study admissible observation operators for perturbed evolution equations using the concept of maximal regularity. We first show the invariance of the maximal $L^p$-regularity under non-autonomous Miyadera-Voigt perturbations. Second, we establish the invariance of admissibility of observation operators under such a class of perturbations. Finally, we illustrate our result with two examples, one on a non-autonomous parabolic system, and the other on an evolution equation with mixed boundary conditions and a non-local perturbation.

math.AP↗

arXiv2021

On the admissibility of observation operators for evolution families

This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix $τ> 0$ and let $\left(A(t)\right)_{t \in [0,τ]} \subset \mathcal{L}(D,X)$, 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 $t \in [0,τ]$ and that $ A(\cdot) : [0,τ] \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 $$ \dot{u}(t) + A(t) u(t) = 0 \quad t\text{ a.e. on } [0,τ], \qquad u(0) = x \in X. $$ 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 $t \in [0,τ)$.

math.OC↗