关于发展族观测算子的可容许性
On the admissibility of observation operators for evolution families
摘要：
研究了非自治发展方程的无界观测算子。固定$τ> 0$，设$\left(A(t)\right)_{t \in [0,τ]} \subset \mathcal{L}(D,X)$，其中$D$和$X$是两个Banach空间，使得$D$连续稠密地嵌入到$X$中。我们假设算子$A(t)$对所有的$t \in [0,τ]$具有最大正则性，并且$ A(\cdot) : [0,τ] \to \mathcal{L}(D,X) $满足正则性条件（即对某些$p \in (1,\infty)$，相对$p$-Dini）.首先证明了$X$上存在一个与问题$$ \dot{u}(t) + A(t) u(t) = 0 \quad t\text{ a.e. on } [0,τ], \qquad u(0) = x \in X. $$相关的演化族，然后证明了观测算子对$A(\cdot)$是容许的当且仅当它对每个$A(t)$对所有$t \in [0,τ)$都是容许的.
英文摘要：
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 $t \in [0,τ)$.
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