网络上演化系统的迫零、线性和量子能控性
Zero forcing, linear and quantum controllability for systems evolving on networks
摘要：
我们从线性代数的角度研究网络系统的动力学。控制理论的可控性概念描述了这些系统可以达到的状态集。在适当的条件下，能控性的量子（李理论）性质和线性系统（卡尔曼）能控性条件之间存在联系。我们研究了迫零集的图论概念如何影响可控性。特别是，我们证明了，如果一组顶点是迫零集，相关的动力系统是可控的。结果开辟了进一步利用网络，线性控制系统理论和量子系统李代数理论之间的类比的可能性。这项研究的动机是目前正在研究的几个量子系统，包括模拟传输现象的连续量子行走。此外，它提出了零强迫作为一个新的概念，在复杂网络的分析。
英文摘要：
We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks.
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