Stochastic stability analysis of switched genetic regulatory networks without stable subsystems
Introduction
Genetic regulatory networks (GRNs), describing the regulatory mechanism of interactions among DNAs, mRNAs and proteins of biological systems at the molecular level, have been a significant research domain in the biological science due to their engineering applications in such as biotherapy, gene correction, gene replacement. To promote these application practices, it is of great significance to have a deep comprehension of the properties and functions of GRNs. To this end, appropriately mathematical models are useful and necessary to represent expression mechanisms and signal-transduction ways. Recently it has been shown that SUM regulatory logic is suitable to describe the feedback regulation models in GRNs [1], [2]. Subsequently, massive efforts have been devoted to the research of GRNs, e.g. stability and bifurcation analysis [3], [4], [5], [6], [7], [8], [9], [10], disturbance attenuation [2], [11], filter design [12], [13], [14], [15], finite-time stability [16], [17] and so on.
Switched systems, as one class of hybrid dynamic systems, have been a hot topic of modern control theory since many power or physical systems display switching features and can be modeled and analyzed by using switched systems. As far as the studies of analysis and synthesis of switched systems are concerned, there has been great progress in later years (see e.g. [18], [19], [20], [21], [22], [23], [24]). With the rapid development of switched system theory, the stability problem of switched GRNs has also received increasing attention recently and there have been a number of ways to deal with it [25], [26], [27], [28], [29]. Specifically, by a Lyapunov-Krasovskii functional approach as well as average dwell time, the investigation of exponential stability of switched time-delay GRNs with all stable subsystems was dealt with in [27]. When stable and unstable subsystems exist simultaneously in the switched GRNs, [29], [30] addressed the stability by means of the method proposed in [31]. Recently, [32] analyzed switched time-delay GRNs again and provided some new sufficient conditions by constructing a novel Lyapunov-Krasovskii functional.
It is worth stressing that the aforementioned works are all concerned on switched systems with at least one stable subsystem. When all the subsystems are not stable, it is more attractive to consider the stability problem due to its importance both in theory and application. Recent years have witnessed growing attention on this subject. For instance, work [33] investigated the stabilization problem via a discretized Lyapunov function approach. In [34], a new concept of average dwell time was introduced, upon which some sufficient conditions of stabilization for switched nonlinear systems with unstable subsystems was proposed. By exploiting the stabilization property of partial switching behaviors, [35] developed some new stability conditions. With the aid of the results in [33], the reduced-order impulsive stabilization of a class of nonlinear singular switched systems was studied in [36]. When it comes to SSGRNs with all the unstable subsystems, however, there has been no result focusing on the stability analysis to the authors’ knowledge. How to investigate the stability of such systems is still an open and challenging problem. Moreover, from the perspective of GRNs theory development, it is necessary to analyze the stability of SSGRNs with all the unstable subsystems. These findings motivate our research in this paper.
Our main interest in the paper is to study the globally asymptotic stability of SSGRNs without/with time delays composed of all the unstable subsystems. For SSGRNs without time delays, with the exploitation of the stabilization property of switching behaviors, we first derive a stability result in terms of multiple Lyapunov functions; in order to develop computable ways to represent stability of SSGRNs, the multiple discretized Lyapunov functions approach is utilized and a stability criterion is provided in terms of matrix inequalities. We then extend these results to the case of SSGRNs with time delays by using multiple discretized Lyapunov-Krasovskii functionals. Compared with the published work, our main contributions are listed as follows. First, the requirement that at least one subsystem is stable in the existing results [27], [29], [30], [32], [37], [38] is removed in this paper, which makes the system under consideration more general. Second, by considering the stabilization effect raised by switchings and restricting an upper bound of the dwell time, effective stability conditions for SSGRNs without/with time delays are developed. Third, to improve the computability of the obtained results, multiple discretized Lyapunov functions/Lyapunov-Krasovskii functionals are constructed, based on which easily verifiable sufficient conditions are given.
The rest structure of this paper is the following. Section 2 makes a description of SSGRN models without/with time delays and reviews two stability definitions for the analysis of SSGRNs. In Section 3, the stability criteria for SSGRNs are derived. An example is provided to illustrate our results in Section 4 and a conclusion of this paper as well as our future research interest is presented in Section 5.
Section snippets
Model formulation and preliminary results
As in [1], the dynamical model of a GRN with n nodes is described bywhere and for the ith node, the concentration of mRNA is described by and the concentration of protein by . are diagonal matrices with positive entries defined aswhere for the ith node, denote the decay and translation
GAS-M of SSGRN without time delays
In this section, our objective is to develop sufficient conditions ensuring GAS-M of SSGRN (10) when all the subsystems are not stable. We now present the first result of this paper. Theorem 1 Consider SSGRN (10) with Assumption 1. If there exist quadratic differential functions and c > 0, 0 < μ < 1 such that for any where denotes the switching
Simulation studies
We now provide a simulation example to illustrate the applicability and effectiveness of the theorems provided in this paper. Example 1 Consider SSGRN (7) composed by two subsystems withIn general, the Hill coefficient of the regulation function is chosen as 2, in this case . On the other hand, the noise density functions take the following forms
Concluding remarks
In the paper we have investigated GAS-M of SSGRNs. Two conditions have been developed in a general form to check GAS-M of SSGRNs without/with time delays. Moreover, we have also established two easily verifiable criteria by using the approach of multiple discretized Lyapunov functions and Lyapunov-Krasovskii functionals. Our results have been verified with a numerical example at last. One of our future research directions is to consider consensus of GRNs with Markov jumps [47], [48], [49]
Acknowledgments
This work is supported by National Natural Science Foundation of China (61703249), (61773235), (61673197), (61703132) and (51707110), Natural Science Foundation of Shandong ZR2017MF045, a Project of Shandong Province Higher Educational Science and Technology Program under Grant J17KA079, National Key R &D Program of China (2016YFB0900600) and Taishan Scholar Project of Shandong Province (TSQN20161033). The work of J.H. Park was supported by Basic Science Research Programs through the National
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