Stabilization and set stabilization of delayed Boolean control networks based on trajectory stabilization☆
Introduction
Time delay often occurs in the modelling and analysis of gene regulatory networks (GRNs) due to slow biochemical reactions such as gene transcription and translation, and protein diffusion between the cytosol and nucleus [6], [19], [28], [29], [30], [31], [32]. As was shown in [11], [32], the effect of time delay may prohibit the evolutionary behavior and the controllability of GRNs. On the other hand, as an important model of GRNs, Boolean networks with time delay have attracted many scholars’ interest from systems biology [6], physics [27] and systems science [12], [23] in the last few decades. For example, the coupled oscillations biochemical networks in the cell cycle are modeled by the following Boolean network with time delay: where A(t) and B(t) denote the state of cell A and cell B at time t, respectively, and time delay is caused by the translocation delay between cells [12].
Recently, an algebraic state space representation method has been proposed to the analysis and control Boolean networks [1], [2], [3], [4], [5], [7], [8], [10], [14], [15], [16], [17], [18], [21], [22], [24], [25], [26], [33], [34], [35], [36], [41], [43]. Especially, this method has been used to the controllability, observability and synchronization of delayed Boolean control networks (DBCNs) [9], [11], [13], [23], [37], [38], [40], [42]. Lu et al. [23] proposed some necessary and sufficient conditions for the trajectory and state controllability of DBCNs by constructing a proper controllability matrix. The synchronization of DBCNs was studied in [42], and some necessary and sufficient conditions were presented for the synchronization of delay-coupled Boolean networks.
As an important issue of GRNs, the stabilization problem has found wide applications in the gene therapy, where therapeutic interventions are designed to steer a GRN to a health state [18]. It is noted that although the controllability, observability and synchronization of DBCNs are well studied, the stabilization of DBCNs is still untilled. Hence, it is necessary for us to develop new techniques for the study of the stabilization of DBCNs.
In this paper, we investigate the stabilization and set stabilization of DBCNs by using the algebraic state space representation method. The main contributions of this paper are twofold. On one hand, the state feedback stabilization problem of DBCNs is proved to be equivalent to the trajectory stabilization of DBCNs, which facilitates the control design of DBCNs. On the other hand, based on the algebraic form and trajectory reachability of DBCNs, several criteria are proposed for the state feedback stabilization and set stabilization of DBCNs, which is easily verifiable via MATLAB. In addition, the proof of these criteria provides a constructive procedure for the design of state feedback stabilizers of DBCNs.
The rest of this paper is organized as follows. Section 2 gives some notations and recalls some preliminary results on the semi-tensor product of matrices. Section 3 investigates the stabilization and set stabilization of DBCNs, and presents the main results of the paper. In Section 4, two illustrative examples are worked out to support our new results, which is followed by a brief conclusion in Section 5.
Section snippets
Preliminaries
The following notations will be used in the sequel.
- (1)
and denote the set of real numbers, the set of natural numbers and the set of positive integers, respectively.
- (2)
and .
- (3)
where denotes the kth column of the n × n identity matrix In. When we briefly denote Δ ≔ Δ2.
- (4)
An n × t matrix M is called a logical matrix, if . We express M briefly as . Denote the set of n × t logical matrices by .
- (5)
Given a matrix
Main results
In this section, we study the state feedback stabilization and set stabilization of DBCNs, and present the main results of this paper.
Illustrative examples
In this section, we give two illustrative examples to show the effectiveness of the obtained new results.
Example 4.1 Consider the following DBCN with :
[23]
Our objective is to check whether or not there exists a state feedback control such that the DBCN (4.1) is globally stabilizable to with respect to state.
Using the canonical vector form of logical variables and setting and . According to [23], we
Conclusion
In this paper, we have investigated the state feedback stabilization and set stabilization of DBCNs. We have proved that the state feedback stabilization of DBCNs is equivalent to the trajectory stabilization. We have proposed a constructive procedure to design state feedback gain matrix for the stabilization of DBCNs. Moreover, we have presented several necessary and sufficient conditions for the set stabilization and partial stabilization of DBCNs based on the algebraic form and trajectory
Acknowledgments
The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments and suggestions which improved the quality of this paper.
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The research was supported by the National Natural Science Foundation of China under grants 61374065 and 61503225, the Major International (Regional) Joint Research Project of the National Natural Science Foundation of China under grant 61320106011, the Natural Science Fund for Distinguished Young Scholars of Shandong Province under grant JQ201613, and the Natural Science Foundation of Shandong Province under grant ZR2015FQ003.