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The global asymptotical stability analysis for genetic regulatory networks with time delays is concerned. By using Lyapunov functional theorem, LMIs, and convex combination method, a new delay-dependent stability criterion has been presented in terms of LMIs to guarantee the delayed genetic regulatory networks to be asymptotically stable. The restriction that the derivatives of the time-varying delays are less than one is removed. Our result is applicable to both fast and slow time-varying delays. The stability criterion has less conservative and wider application range. Experimental result has been used to demonstrate the usefulness of the main results and less conservativeness of the proposed method.

Genetic regulatory networks (GRNs) play a key role in systems biology as they explain the interactions between genes (mRNA) and proteins. In a biological cell, genes may be expressed constantly (i.e., constitutive gene expression) or expressed based on molecular signals (i.e., regulated gene expression) [

Since 1960s, many notable researchers have proposed various kinds of mathematical models to describe GRN. So far, during the past few years, there are two basic models for genetic network models: the Boolean model and the differential equation model [

On the other hand, time delays which usually exist in transcription, translation, diffusion, and translocation processes especially in a eukaryotic cell are one of the key factors affecting the dynamics of genetic regulatory network. The delays could be time invariant or time variant. The study of stability is essential for designing or controlling genetic regulatory networks. Up to now, there are already some sufficient conditions that have been proposed to guarantee the asymptotic or robust stability for genetic regulatory networks [

Motivated by the above discussions, we aim to analyze the stability of genetic regulatory networks with SUM logic in the forms of differential equations. Besides the basic case, we will make contributions on the issues of asymptotical stability for genetic networks with time-varying delays. By choosing an appropriate new Lyapunov functional and employing convex combination method, new delay-derivative-dependent stability criterion is derived based on the consideration of ranges for the time-varying delays. The obtained criterion is given in terms of linear matrix inequalities (LMIs) and is applicable to both fast and slow time-varying delays. Finally, one numerical example is given to demonstrate the effectiveness and the merit of the proposed method.

In [

Furthermore, nonlinear function

In the following, we will always shift an intended equilibrium point

Given constant symmetric matrices

For given scalars

The Lyapunov functional of system (

From the Leibniz-Newton formula, the following equations are true for any matrices

In addition, from (

Notice that

Applying Lemma

In [

Consider the following genetic regulatory networks with time-varying delays, borrowed from [

Using LMI Control Toolbox, by our Theorem

It should be pointed out that Theorem 1 in [

This paper presents some new results of stability analysis for genetic regulatory networks with time-varying delays. An appropriate Lyapunov functional is proposed to investigate the delay-derivative-dependent stability problem. The present results improve the existing ones due to a method to estimate the upper bound of the derivative of Lyapunov functional without ignoring some useful terms and the introduction of convex combination method into the proposed Lyapunov functional, which takes into account the relationship between the time-varying delays and their lower and upper bounds. The supplementary requirements that the time derivatives of time-varying delays must be less than one are removed. As a result, the new stability criterion in terms of LMIs is applicable to both fast and slow time-varying delays. One numerical example shows that the proposed criterion is an improvement over some existing results in the literature. In the future, our work will include the problems of filter design and state estimation for genetic networks.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China (Grant no. 61103211), Postdoctoral Science Foundation of Chongqing (Grant no. XM201310), and Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1401403).