Stability Of Genetic Regulatory Networks Research Articles

Finite-Time Stability of Genetic Regulatory Networks With Nondifferential Delays

Studying the stability of genetic regulatory networks (GRNs) helps to explore the laws of automatic regulation in organisms. By designing proper controllers, the finite-time stability problem of GRNs with nondifferential discrete time-varying delays is discussed in this paper. Rather than constructing the Lyapunov functional or using the finite-time stability theorem, inequality technique and comparison method are firstly adopted in the analyzation. Then, several sufficient algebraic criteria are derived to ensure the finite-time stability of addressed delayed GRNs. The results are general and contain the exponential stability as a special case. Finally, numerical simulations are provided to illustrate the validness of theoretical results.

New Results on Global Exponential Stability of Genetic Regulatory Networks with Diffusion Effect and Time-Varying Hybrid Delays

New Results on Global Exponential Stability of Genetic Regulatory Networks with Diffusion Effect and Time-Varying Hybrid Delays

Finite-time adaptive stability of gene regulatory networks

This paper is concerned with the finite-time adaptive stability of genetic regulatory networks(GRNs). Traditionally, the finite-time stability criterion is often established based on the prior information on gene regulatory parameters. However, it is difficult to acquire the above prior information for GRNs. To overcome the above difficulty, we design controllers in non-switching and switching gene regulatory networks with adaptive technique, respectively. Then, we derive a sufficient criterion for system to achieve finite-time stability under the action of an adaptive controller and give an estimate of the steady-state time. Furthermore, we can also shorten the setting time required for the system to achieve finite-time stability under the designed adaptive controller. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design method.

A reduced-order approach to analyze stability of genetic regulatory networks with discrete time delays

Abstract This paper addresses the problem of establishing the asymptotic stability criteria for genetic regulatory networks with discrete time delays. First, the system model is simplified to a reduced-order system with bounded uncertain parameters and distributed delays by exploiting calculus’s properties and Lagrange’s mean–value theorem. Second, the relationship between the asymptotic stability of the primal system and the robust asymptotic stability of the reduced-order one is investigated. Third, a new reduced-order approach is proposed to derive a sufficient condition for the robust asymptotic stability of the reduced-order system (i.e., the asymptotic stability of the primal system). At last, a numerical example illustrates the effectiveness of the theoretical results obtained in this paper.

Finite-time stability of genetic regulatory networks with impulsive effects

We study the finite-time stability of genetic regulatory networks with impulsive effects. Using the method of Lyapunov function, sufficient conditions of the finite-stability, in terms of linear matrix inequalities, are established. A numerical example is provided to further illustrate the significance of our results.

Delay-dependent asymptotic stability criteria for genetic regulatory networks with impulsive perturbations

This paper considers the problem for asymptotic stability of genetic regulatory networks with impulse control using the delay partitioning approach. By using the direct Lyapunov method, a new Lyapunov–Krasovskii functional is introduced based on the decomposition approach. Time delays here are assumed to be time-varying and belong to the given intervals. When dealing with the time derivative of Lyapunov functional, a new tight integral inequality is adopted for bounding the cross terms. Then, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by the Matlab LMI toolbox. Finally, the proposed method is validated through the numerical simulation, which shows the effectiveness of the presented stability criteria.

Stability of genetic regulatory networks based on switched systems and mixed time-delays

In this paper, the switched genetic regulatory networks (GRNs) are modeled from a real biological system, based on switched systems, noise and mixed time-delays. Global asymptotical stability for the proposed switched GRNs are studied by the Lyapunov method and the matrix inequality techniques. Some new sufficient conditions are obtained to ensure the global asymptotical stability of the proposed switched GRNs. Furthermore, the proposed LMI results are computationally efficient as it can be solved numerically with standard commercial software. Finally, an example is provided to illustrate the usefulness of the results.

Stability of genetic regulatory networks with time-varying delay: Delta operator method

This paper investigates the stability problem for a class of uncertain genetic regulatory networks (GRNs) with time-varying delay via delta operator approach. Both the parameter uncertainty and the generalized activations are considered in the model under study. By constructing an appropriate Lyapunov-Krasovskii functional, the stability and robust stability conditions of GRNs are presented under the delta operator frame. These conditions can be expressed in terms of linear matrix inequalities (LMIs). Finally, a numerical example is employed to illustrate the effectiveness of the proposed results.

Effect of Leakage Delay on Stability of Neutral-Type Genetic Regulatory Networks

The stability of neutral-type genetic regulatory networks with leakage delays is considered. Firstly, we describe the model of genetic regulatory network with neutral delays and leakage delays. Then some sufficient conditions are derived to ensure the asymptotic stability of the genetic regulatory network by the Lyapunov functional method. Further, the effect of leakage delay on stability is discussed. Finally, a numerical example is given to show the effectiveness of the results.

Analysis on Global Asymptotical Stability of Genetic Regulatory Networks with Time-Varying Delays via Convex Combination Method

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.

Stability of Genetic Regulatory Networks with Interval Time-Varying Delays via Convex Combination Method

In this paper, asymptotical stability of genetic regulatory networks with interval time-varying delays is investigated. By choosing an appropriate new Lyapunov functional and employing convex combination method to estimate the derivative of the Lyapunov functional, some new delay-range-dependent and delay-derivative-dependent/independent stability criteria are presented in terms of linear matrix inequalities (LMIs). The important feature is that the obtained stability criteria are applicable to both fast and slow time-varying delays due to the ranges for the time-varying delays have been carefully considered. Three numerical examples are used to demonstrate the usefulness of the main results and less conservativeness of the presented results

Comments and Corrections on “Stability of Genetic Regulatory Networks With Time Delay” [May 02 602-608

This paper shows that the necessary and sufficient condition for the local stability of gene regulatory networks with time delay (Theorem 2) presented in the highly cited paper [L. Chen and K. Aihara, IEEE TCAS-I, vol. 49, No. 5, pp. 602-608, 2002] is erroneous. An illustrative counterexample is presented, and it is pointed out that there are essentially two errors in their derivation. A corrected theorem is then derived using the approach shown in the aforementioned paper. It turns out that the corrected theorem provides only a sufficient condition for the local stability. Finally, a necessary and sufficient condition that was recently presented by the authors' of this paper is presented. Both the sufficient and the necessary and sufficient conditions are demonstrated using the Repressilator, the oscillatory gene regulatory network, which was also used for the illustration of the Chen and Aihara's original result.

A Less Conservative Stability Criterion for Delayed Stochastic Genetic Regulatory Networks

This paper concerns the problem of stability analysis for delayed stochastic genetic regulatory networks. By introducing an appropriate Lyapunov-Krasovskii functional and employing delay-range partition approach, a new stability criterion is given to ensure the mean square stability of genetic regulatory networks with time-varying delays and stochastic disturbances. The stability criterion is given in the form of linear matrix inequalities, which can be easily tested by the LMI Toolbox of MATLAB. Moreover, it is theoretically shown that the obtained stability criterion is less conservative than the one in W. Zhang et al., 2012. Finally, a numerical example is presented to illustrate our theory.

Stochastic Stability of Genetic Regulatory Networks with Mixed Delays

In this paper, the asymptotical stability of stochastic genetic regulatory networks with time-varying delays and continuous distributed delays are investigated. The constraint that derivative of time-varying delays must be smaller than 1 is relaxed, and the activation functions are of more general descriptions, which generalize and improve the earlier methods. Based on the Lyapunov-Krasovskii functional approach and linear matrix inequality (LMI) techniques defined on convex set, stability criteria for stochastic delayed genetic regulatory networks are established in the form of LMIs, which can be easily verified. Numerical example is also given to demonstrate the effectiveness of the proposed criteria. Keywords: Distributed delays, Genetic regulatory network, Linear matrix inequality (LMI), Stochastic system, Time-varying delays.

Analysis on Exponential Stability of Genetic Regulatory Networks with Variable Delays

Analysis on Exponential Stability of Genetic Regulatory Networks with Variable Delays

M‐matrix‐based stability conditions for genetic regulatory networks with time‐varying delays and noise perturbations

Stability is essential for designing and controlling any dynamic systems. Recently, the stability of genetic regulatory networks has been widely studied by employing linear matrix inequality (LMI) approach, which results in checking the existence of feasible solutions to high-dimensional LMIs. In the previous study, the authors present several stability conditions for genetic regulatory networks with time-varying delays, based on M-matrix theory and using the non-smooth Lyapunov function, which results in determining whether a low-dimensional matrix is a non-singular M-matrix. However, the previous approach cannot be applied to analyse the stability of genetic regulatory networks with noise perturbations. Here, the authors design a smooth Lyapunov function quadratic in state variables and employ M-matrix theory to derive new stability conditions for genetic regulatory networks with time-varying delays. Theoretically, these conditions are less conservative than existing ones in some genetic regulatory networks. Then the results are extended to genetic regulatory networks with time-varying delays and noise perturbations. For genetic regulatory networks with n genes and n proteins, the derived conditions are to check if an n × n matrix is a non-singular M-matrix. To further present the new theories proposed in this study, three example regulatory networks are analysed.

Robust delay-probability-distribution-dependent stability of uncertain genetic regulatory networks with time-varying delays

This study is concerned with the problem of robust delay-probability-distribution-dependent stability of genetic regulatory networks with time-varying delays and parameter uncertainties. By dividing the time-varying delays into multiple nonuniformly subinterval and choosing an appropriate novel Lyapunov functional which contains some triple integral terms, new stability criteria are obtained and formulated in terms of linear matrix inequalities (LMIs). The uncertain parameters are modeled as having a structured linear fractional form. Besides, the derivatives of the time delays are considered having different bounds in various delay intervals. Moreover, less conservative conditions are obtained by using the lower bound lemma together with Jensen inequality lemma. Furthermore, the criteria can be applicable to both fast and slow time-varying delays. Finally, numerical examples are presented to illustrate the effectiveness of the theoretical results.

Robust stability analysis of delayed Takagi–Sugeno fuzzy genetic regulatory networks

In this paper, the nonlinear model of genetic regulatory networks is described by the Takagi–Sugeno fuzzy model representation with time-varying delays. Due to the highly complicated nonlinear stability and robust stability problems, a fuzzy approximation method is employed to interpolate several linear genetic regulatory networks at different operation points via fuzzy bases to approximate the nonlinear genetic regulatory network. In this context, the methods of the linear matrix inequality (LMI) technique could be employed to simplify the problems related to robust stability of genetic regulatory networks. Further, by involving triple integral terms in Lyapunov–Krasovskii functionals and LMI techniques, the stability criteria for the delayed fuzzy genetic regulatory networks are expressed as a convex combination of LMIs, which can be solved numerically by any LMI solvers. Several numerical examples are given to verify the effectiveness and applicability of the derived approach.

Robust and Global Delay-Dependent Stability for Genetic Regulatory Networks With Parameter Uncertainties

The study of stability is essential for designing or controlling genetic regulatory networks, which can be described by nonlinear differential equations with time delays. Much attention has been paid to the study of delay-independent stability of genetic regulatory networks and as a result, many sufficient conditions have been derived for delay-independent stability. Although it might be more interesting in practice, delay-dependent stability of genetic regulatory networks has been studied insufficiently. Based on the linear matrix inequality (LMI) approach, in this study we will present some delay-dependent stability conditions for genetic regulatory networks. Then we extend these results to genetic regulatory networks with parameter uncertainties. To illustrate the effectiveness of our theoretical results, gene repressilatory networks are analyzed .

Stochastic stability of genetic regulatory networks with a finite set delay characterization

In this paper, the delay-distribution-dependent stability is derived for the stochastic genetic regulatory networks (GRNs) with a finite set delay characterization and interval parameter uncertainties. One important feature of the obtained results here is that the time-varying delays are assumed to be random and the sum of the occurrence probabilities of the delays is assumed to be 1. By employing a new Lyapunov-Krasovskii functional dependent on auxiliary delay parameters which allow the time-varying delays to be not differentiable, less conservative mean-square stochastic stability criteria are obtained. Finally, two examples are given to illustrate the effectiveness and superiority of the derived results.