AI 中文总结
本文针对离散时间后结构随机线性系统，建立了精确可控性、精确零可控性及L2-镇定性的Hautus型准则，证明精确零可控性蕴含L2-镇定性，为系统相关特性提供代数与谱检验方法。
AI 中文摘要
本文研究离散时间后结构随机线性系统的精确可控性与L2-镇定性。针对每个给定的有限时域，基于可控性Gramian、Kalman型秩条件及可达子空间，建立了精确可控性的等价刻画；随后通过有限维秩条件刻画了无给定时域的精确零可控性，并利用正算子的半正定特征矩阵推导了Hautus型准则。对于L2-镇定性，本文建立了镇定分解并得到对应的Hautus型谱准则，作为副产品证明了精确零可控性蕴含L2-镇定性。这些结果为所研究系统的可控性与镇定性提供了代数与谱检验方法。
英文摘要
This paper studies exact controllability and L2-stabilizability of discrete-time backward-structured stochastic linear systems. For each prescribed finite horizon, equivalent characterizations of exact controllability are established in terms of the controllability Gramian, a Kalman-type rank condition, and the reachable subspace. Exact null controllability without a prescribed horizon is then characterized by a finite-dimensional rank condition, and a Hautus-type criterion is derived using positive semidefinite eigenmatrices of a positive operator. For L2-stabilizability, we establish a stabilizability decomposition and obtain a corresponding Hautus-type spectral criterion. As a byproduct, we show that exact null controllability implies L2-stabilizability. These results provide algebraic and spectral tests for the controllability and stabilizability of the considered systems.
