Set Stability and Set Stabilization of Boolean Control Networks Avoiding Undesirable Set

Abstract

The traditional set stability of Boolean networks (BNs) refers to whether all the states can converge to a given state subset. Different from the existing results, the set stability investigated in this paper is whether all states in a given initial set can converge to a given destination set. This paper studies the set stability and set stabilization avoiding undesirable sets of BNs and Boolean control networks (BCNs), respectively. First, by virtue of the semi-tensor product (STP) of matrices, the dynamics of BNs avoiding a given undesirable set are established. Then, the set reachability and set stability of BNs from the initial set to destination set avoiding an undesirable set are investigated, respectively. Furthermore, the set stabilization of BCNs from the initial set to destination set avoiding a given undesirable set are investigated. Finally, a design method for finding the time optimal set stabilizer is proposed, and an example is provided to illustrate the effectiveness of the results.

Further results for synchronization of two coupled Boolean networks

“…Based on this, the definition of set stability of BNs and set stabilization of BCNs are given in Guo et al [24]. In Liu et al [25], based on external interference, all states of a given initial state set can reach a given target state set, in order to avoid the undesirable state set reaching the target state set.…”

Algebraic Representation of Topologies on a Finite Set

“…Since the STP can convert any finite-valued functions into algebraic forms, Prof. Cheng and his collaborators developed an algebraic state space representation method for the analysis of BNs and BCNs and obtained many essential results [12,13]. In the following years, the STP made many achievements in the study of Boolean networks and gradually formed a complete framework of Boolean network control theory [14][15][16]. Moreover, the STP method also shows strong vitality in many other fields, such as information security [17], finite automata [18,19], vehicle control, and networked evolutionary games [20,21].…”