论文标题
Hautus-山摩托标准，用于线性差延迟方程的近似和确切可控性
Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations
论文作者
论文摘要
该论文介绍了有限维线性差延迟方程的可控性，即，在给定时间$ t $的状态作为在时间$ t $评估的控件的线性组合以及在有限的许多以前的瞬时评估的状态的状态$ t-λ_1，\λ_1，\ dots，t-λ_n$中获得的状态。基于Y.Yamamoto为一般无限维动力系统开发的实现理论，我们获得了在频域中表达的必要和充分条件，以在$ l^q $空间中的有限时间内，$ q \ in [1， +\ infty）$。我们还提供了$ l^1 $精确可控性的必要条件，可以看作是$ l^1 $近似可控性标准的关闭。此外，我们在最小和确切可控性的最小时间上提供了一个明确的上限，由$ d \ max \ {λ_1，\ dots，λ_n\} $给出，其中$ d $是状态空间的维度。
The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-Λ_1,\dots,t-Λ_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q \in [1, +\infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $d\max\{Λ_1,\dots,Λ_N\}$, where $d$ is the dimension of the state space.
