论文标题
非自主系统的可观察性
Observability for Non-autonomous Systems
论文作者
论文摘要
我们研究非自主观察系统\ BEGIN {ALIGN*} \ dot {x}（t）= a（t）x（t），\ quad y（t）= c（t）= c（t）x（t），\ quad x（t）= x（0）= x_0 \ in x，\ end \ end \ end {align*}，其中$（a（a（t））$是封闭的opertors opertor opertor opertor opertor opertor opertor oper opertor in and op ober $ x $（$ x $）（$ x $ $ x $（c） $ x $ to Banach Space $ y $。基于抽象的不确定性原理和耗散估计，我们证明该观察系统满足了可测量子集$ e \ subseteq [0，t]，t> 0 $的最终状态可观察性估计。我们将上述结果的申请介绍给家庭$（a（t））$均匀椭圆形的差异操作员以及$ \ Mathrm {l}^p（\ mathbb {r}^d）$ p（l}^d）$ p（l}^d）$ p（t）$ p（t）$ p（t）$ p（t）$ p（t）$ p（r}^d）$ cobservation p（r}^d）$ p（r}^d）$ p（t）在非自主性强烈椭圆运算符的情况下，我们在集合$（ω（t））$的家族中得出了必要且充分的几何条件，以使相应的观察系统满足最终的可观察性估计。
We study non-autonomous observation systems \begin{align*} \dot{x}(t) = A(t) x(t),\quad y(t) = C(t) x(t),\quad x(0) = x_0\in X, \end{align*} where $(A(t))$ is a strongly measurable family of closed operators on a Banach space $X$ and $(C(t))$ is a family of bounded observation operators from $X$ to a Banach space $Y$. Based on an abstract uncertainty principle and a dissipation estimate, we prove that the observation system satisfies a final-state observability estimate in $\mathrm{L}^r(E; Y)$ for measurable subsets $E \subseteq [0,T], T > 0$. We present applications of the above result to families $(A(t))$ of uniformly strongly elliptic differential operators as well as non-autonomous Ornstein-Uhlenbeck operators $P(t)$ on $\mathrm{L}^p(\mathbb{R}^d)$ with observation operators $C(t)u = u|_{Ω(t)}$. In the setting of non-autonomous strongly elliptic operators, we derive necessary and sufficient geometric conditions on the family of sets $(Ω(t))$ such that the corresponding observation system satisfies a final-state observability estimate.
