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.
Highlights
A Boolean network (BN) was first proposed by Kauffman, which laid the foundation for the study of gene regulation networks [1]
This paper investigates the set stability and set stabilization of Boolean control networks (BCNs) avoiding a given undesirable set
This paper investigates the set stability and set stabilization of BCNs with state constraints in the framework of semi-tensor products (STP)
Summary
A Boolean network (BN) was first proposed by Kauffman, which laid the foundation for the study of gene regulation networks [1]. It can be seen that STP is becoming an effective tool for studying BNs. This paper investigates the set stability and set stabilization of Boolean control networks (BCNs) avoiding a given undesirable set. Different from the existing results, the set stability and set stabilization problem studied in this paper are from the initial state set to the destination state set, which is the generation of traditional set stability and set stabilization Another concern that frequently occurs in biological systems is to prevent certain states of genes from being in an unfavorable or dangerous environment [31]. (ii) The set reachability and set stability of BNs and BCNs from the initial set to destination set are provided avoiding a given undesirable set, respectively.
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