AI 中文总结
本文针对可数无穷多的离散线性集成系统，给出使其闭环系统满足极点在闭左半平面且无穷小生成元可对角化的静态线性反馈控制律存在的充要条件。
AI 中文摘要
我们考虑可数无穷多个标量线性控制系统组成的集合，其中每个系统由对$(a_n,b_n)$表示，$a_n>0$（$n\in \mathbb{N}$），所有系统均由同一个标量控制输入驱动，我们将这类系统集合称为离散线性集成系统。若存在公共反馈控制输入可同时渐近镇定所有系统，则该集成系统是反馈可镇定的。与有限维线性系统不同，无穷维线性系统的极点位于开左半平面时不保证稳定，但当满足以下两个条件时可保证稳定：（i）其极点包含于闭左半平面；（ii）其无穷小生成元可对角化。本文给出静态线性反馈控制律存在的充要条件，该控制律可确保闭环集成系统满足上述（i）和（ii）。特别地，我们证明序列$(|b_n|)_{n\in \mathbb{N}}$和$(a_n/|b_n|)_{n\in \mathbb{N}}$的指数衰减是必要条件，且在期望极点的假设下是充分条件。
英文摘要
We consider a countably infinite collection of linear, scalar control systems, where each system is represented by the pair $(a_n,b_n)$ with $a_n>0$ for $n\in \mathbb{N}$, and all systems are forced by a common scalar control input. We refer to this collection of systems as a discrete linear ensemble system. The ensemble system is feedback stabilizable if there exists a common feedback control input that asymptotically stabilizes every system simultaneously. Unlike finite-dimensional linear systems, an infinite-dimensional linear system is not guaranteed to be stable if its poles lie in the open left half-plane. Stability is guaranteed, however, if (i) its poles are contained in the closed left half-plane and (ii) its infinitesimal generator is diagonalizable. In this paper, we provide necessary and sufficient conditions for the existence of a static, linear feedback control law that ensures the closed-loop ensemble system satisfies (i) and (ii). In particular, we show that the exponential decay of the sequences $(|b_n|)_{n\in \mathbb{N}}$ and $(a_n/|b_n|)_{n\in \mathbb{N}}$ is necessary and, under an assumption on the desired poles, sufficient.
