This paper investigates the controllability of probabilistic Boolean control networks (PBCNs) with time-variant delays in states. By cutting the time sequence, we split the network into at most countably infinitely many subnetworks with no delays, where any one of the longest subnetworks is called a controllability constructed path (CCP). When the CCP is of infinite length, we prove that the network is controllable iff any CCP is controllable, and give an equivalent condition for the controllability of the network. When it is of finite length, we give a necessary condition and a sufficient condition for the controllability of the network, and show that the controllability of the network is not equivalent to the controllability of a CCP.
本文研究时变时延概率布尔控制网络的能控性。 研究思路为: 通过切割时间序列, 该网络被分割成至多可数不带时延的子网络, 其中时间序列最长的子网络被称为能控性结构路径。 当能控性结构路径长度为无穷时, 证明该网络能控当且仅当任意一个能控性结构路径能控, 并且给出该网络能控的充分必要条件。 当能控性结构路径长度有限时, 给出该网络能控的一个充分条件和一个必要条件, 并且说明该网络的能控性并不一定等价于能控性结构路径的能控性。
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Zhang, K., Zhang, L. Controllability of probabilistic Boolean control networks with time-variant delays in states. Sci. China Inf. Sci. 59, 92204 (2016). https://doi.org/10.1007/s11432-015-5423-6
- probabilistic Boolean control network
- time delay
- controllability constructed path
- semi-tensor product of matrices