# Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks
## 1 Introduction  
The Boolean network (BN) is a discrete dynamic model proposed by [PERSON] [1; 2] in 1969 to represent complicated gene regulatory networks in an accessible and effective manner. Compared with detailed kinetic models of biomolecules, BNs may be useful in understanding the dynamic key properties of regulatory processes when they are applied to biological systems [3; 4; 5], computer systems [6; 7], and power systems [8], and other fields. Recently, [PERSON] and co-authors proposed a new method that can multiply two matrices of arbitrary dimensions, called semi-tensor product (STP) [9]. The original purpose of this approach is to address the Morgan issue of nonlinear systems. The STP of matrices is an expansion of ordinary matrix multiplication, which breaks through the limitation of ordinary matrix multiplication on dimension. Based on this method, many achievements [10; 11; 12; 13] related to BNs have been obtained. The study of Boolean control networks (BCNs) also provides an effective method to simulate gene regulatory networks. Many related problems [14; 15; 16; 17; 18; 19] of BCNs were also well studied and developed in recent years. For example, the set stabilization issue of Markovian jump BCNs was addressed in [20]. Disturbance decoupling issue of BCNs was considered in [21]. In addition, switched Boolean control networks (SBCNs) are further proposed by adding deterministic switching signals to the BCNs, and it is suitable for modeling biological systems. The controllability and observability were investigated for a class of SBCNs in [22]. Ref. [23] addressed the observability of a SBCNs by the STP of matrices. To simulate more realistic biological systems, probabilistic Boolean control networks (PBCNs) [24; 25] were further studied. In [26], pinning control was carried out on the PBCNs to attain a certain reachability level. In [27], the finite-time controllability and set controllability of impulsive PBCNs were investigated. On this basis, a model-free reinforcement learning technique [28] was also proposed to solve the control problem in gene regulatory networks. Specifically, the feedback stabilization problem of PBCNs was studied based on the model-free reinforcement learning technique in [29]. In the same year, without knowing the underlying dynamics of the network, literature [30] applied the model-free control method to applications that are difficultto deduce dynamics. The cooperative design of the model-free trigger drive controller proposed in [31] can achieve feedback stabilization with minimal controller efforts. In [32], the flip sequences was found through the Q-learning algorithm, and the global stabilization was further realized. Moreover, the STP approach can be utilized to investigate mix-valued logical control networks [33; 34], logical equations [35], game theory [36; 37; 38] and so on. To sum up, many results have been obtained which are valuable in BNs systems, such as BCNs, SBCNs and PBCNs. From something like the systems biology [39] perspective, very few of these works considered reachability. For example, [PERSON]. [26; 34; 40] studied the problems associated with the reachability of a single-layer network. However, for the complex gene regulatory networks, the same gene can participate in different regulatory processes or metabolic processes, that is, the same gene can work simultaneously in many different signaling pathways. In this case, a simple single-layer network cannot satisfy the modeling of complex gene regulatory networks. Therefore, probabilistic Boolean multiplex control networks (PBMCs) are proposed to simulate the complex variability of biological systems in this paper. Then, inspired by [18; 41; 42], this paper further studies the finite-time set reachability of the PBMCs with a global state layer. The necessary and sufficient condition of the finite-time set reachability of PBMCs is obtained. Finally, a numerical example is used to demonstrate the validity of the proposed approach.