Abstract
Time delay arising in a genetic regulatory network may cause the instability. This paper is concerned with the stability analysis of genetic regulatory networks with interval time-varying delays. Firstly, a relaxed double integral inequality, named as Wirtinger-type double integral inequality (WTDII), is established to estimate the double integral term appearing in the derivative of Lyapunov-Krasovskii functional with a triple integral term. And it is proved theoretically that the proposed WTDII is tighter than the widely used Jensen-based double inequality and the recently developed Wiringter-based double inequality. Then, by applying the WTDII to the stability analysis of a delayed genetic regulatory network, together with the usage of useful information of regulatory functions, several delay-range- and delay-rate-dependent (or delay-rate-independent) criteria are derived in terms of linear matrix inequalities. Finally, an example is carried out to verify the effectiveness of the proposed method and also to show the advantages of the established stability criteria through the comparison with some literature.
Highlights
In the past few years, genetic regulatory networks (GRNs), which describe the interactions of many molecules (DNA, RNA, proteins, etc.), have been becoming a new research area of biological and biomedical sciences [1,2,3,4]
A Wirtinger-based double integral inequality (WBDII) was developed to general linear timedelay system and it was proved to be less conservative than the Jensen-based double integral inequality (JBDII) [72]
This paper aims to analyze the asymptotical stability of GRN (2) and to determine the delay bounds, named as maximal admissible delay bounds (MADBs), under which the GRN is asymptotically stable
Summary
In the past few years, genetic regulatory networks (GRNs), which describe the interactions of many molecules (DNA, RNA, proteins, etc.), have been becoming a new research area of biological and biomedical sciences [1,2,3,4]. A Wirtinger-based double integral inequality (WBDII) was developed to general linear timedelay system and it was proved to be less conservative than the JBDII [72] Such inequality has not been used to discuss the GRNs. the gap between term (1) and its estimated value obtained by the WBDII still leads to conservatism. This paper further investigates the delay-dependent stability of the GRNs by developing a more effective inequality to estimate the double integral term (1). For the GRNs with time-varying delays satisfying different conditions, two stability criteria are, respectively, established by applying the proposed WTDII to estimate the double integral terms appearing in the derivative of the LKFs. The rest of the paper is organized as follows. In the Notations, the list of notations and abbreviations used throughout this paper is shown
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