In this article, the stabilization issues for probabilistic Boolean Networks (PBNs) with time delays are discussed. This article's objective is designing an efficient algorithm to choose suitable nodes to be pinning controlled for PBNs with time delays. By using the semi-tensor product (STP) of matrices, a PBN with time delays can be converted into a discrete-time linear system, and the transition matrix also can be obtained. Then, the necessary and sufficient conditions in the form of algebraic expression are given for the existence and solvability of the pinning feedback controllers with minimum pinning nodes for PBNs with time delays. Besides, three algorithms are proposed for designing and solving minimum pinning controllers.INDEX TERMS Probabilistic Boolean networks, pinning control, time delays, semi-tensor product. I. INTRODUCTIONBoolean Networks (BNs), which were first proposed by Kauffman in 1969 [1], are a kind of logical dynamical models to describe gene regulatory networks (GRNs) [2]. As we all know, in a gene regulatory network, each gene can be expressed (1) or not expressed (0), which corresponds to binary state variables. A BN is a deterministic model to simulate the evolution of binary state variables. What's more, Boolean Networks have been widely studied in state estimation [3], logical networks [4], neural networks [5], etc. Recently, the STP of matrices was introduced by Cheng's team. With the help of STP, a BN can be transformed into a discrete-time linear system. Moreover, a logic function can be represented by an algebraic form with STP. This new matrix product was introduced to the study of BNs in many fields, such as the controllability, event-triggered control, ect., which have been studied in [6]-[11].To better handle of biological system uncertainty, Shmulevich etc. in [12] generalized the concept of BNs for application to probabilistic Boolean Networks (PBNs). In general, the PBNs can be seen as a kind of randomly switched BNs in given sets of BNs. Every BN is chosen with an definite probability. Many interesting results have been obtained for The associate editor coordinating the review of this manuscript and approving it for publication was Tingwen Huang.
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