Stability Analysis For Delayed Genetic Regulatory Networks

Genetic regulatory networks(GRNs) have become an significant research subject and played an important role in the biological and biomedical sciences. How to understand the interaction between genes, the relationship between genes and proteins accurately, master the dynamic character of genetic regulatory networks, is an important research area. This article studies the stability of delayed GRNs. By employing Lyapunov-Krasovskii functional theory, some less conservative delay-range-dependent and delay-derivative-dependent stability criteria have been derived in terms of linear matrix inequal-ities (LMIs). The main works of this article is organized as follows:1. The asymptotic stability criteria for GRNs with time-varying delays is obtained. Due to the finite speed of the transcription and translation of DNA, mRNA, the diffu-sion to a certain place of a protein needs time. Time delay is inevitable in modeling gene regulation processes. One key point is that the decomposition of the matrix D into D=D1+D2 to obtain the new stability criteria. Moreover, the time-varying delays are divided into multiple nonuniformly subinterval and the derivatives of the time delays are estimated having different bounds in various delay intervals. Meanwhile, less con-servative condition are obtained by using the lower bound lemma together with Jensen inequality lemma. Numerical examples are presented to verify the effectiveness and ad-vantages of the theoretical results in this chapter.2. The robust stability analysis for uncertain GRNs with interval time-varying de-lays is investigated. The uncertainties such as external perturbations, parameter fluctu-ations, and data errors are inevitable by using approximate models, which means un-certain parameter is also an important factor to influence the stability of gene regula-tory network. The parameter uncertainties are modeled as having structured linear frac-tional form and norm-bounded form, respectively. By choosing an augmented novel Lyapunov-Krasovskii functional which contains some triple integral terms and using the Lower bound lemma to handle the integral terms, less conservative condition are obtained. Meanwhile, this article also studies the problem of exponential convergence of uncertain genetic regulatory networks with time-varying delays in the case of the unknown equilib-rium point. Finally, numerical examples are presented to illustrate the effectiveness and advantages of the stability criteria.3. The stability analysis of delayed GRNs with nonlinear disturbance and stochastic noise is studied. The gene regulation process is not in an ideal environment, but always subjects to intrinsic noise and extrinsic noise. By choosing an appropriate new Lyapunov-Krasovskii functional and reciprocally convex approach, new delay-dependent stability criteria are obtained and formulated in terms of LMIs. The important feature is that the obtained stability criteria are applicable to both fast and slow time-varying delays since the ranges for the time-varying delays have been carefully considered. Finally, numerical examples are presented to check the effectiveness and advantages of the stability condi-tions.4. The problem of robust filtering for stochastic GRNs with time-varying delays and parameter uncertainties is concerned. For the stability analysis of the GRNs, L2-L∞ filtering is a very effective method. By choosing an appropriate novel Lyapunov-Krasovskii functional and establishing a new integral inequality in the stochastic setting, less conservative conditions are obtained to ensure the error systems are mean-square robustly asymptotically stable. Then the filters are designed in terms of LMIs which can be checked efficiently via the LMI toolbox. Finally, numerical example is presented to illustrate the effectiveness and advantages of the theoretical results.