Asymptotic Stability Criteria for Genetic Regulatory Networks with Time-Varying Delays and Reaction–Diffusion Terms

Abstract

This paper is concerned with the state estimation problem for genetic regulatory networks with time-varying delays and reaction-diffusion terms under Dirichlet boundary conditions. It is assumed that the nonlinear regulation function is of the Hill form. The purpose of this paper is to design a state observer to estimate the concentrations of mRNA and protein through available measurement outputs. By introducing new integral terms in a novel Lyapunov-Krasovskii functional and employing Wirtinger-based integral inequality, Wirtinger's inequality, Green's identity, convex combination approach, and reciprocally convex combination approach, an asymptotic stability criterion of the error system is established in terms of linear matrix inequalities (LMIs). The obtained stability criterion depends on the upper bounds of the delays and their derivatives. It should be highlight that if the set of LMIs are feasible, the desired observer exists and can be determined. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed designed scheme. Index Terms-Genetic regulatory networks, Reaction-diffusion terms, State estimation, Wirtinger-based integral inequality.

Finite‐Time Robust Stability of Uncertain Genetic Regulatory Networks with Time‐Varying Delays and Reaction‐Diffusion Terms

“…A new inequality introduced in [29] solves this problem and makes the diffusion-dependent stability criteria in terms of LMIs. However, there are still a few papers [30][31][32][33][34] that have considered the dynamic properties of GRNs with reaction-diffusion terms. References [30,31] investigate the asymptotic stability problem for delayed GRNs with reactiondiffusion terms under Dirichlet boundary conditions, Neumann boundary conditions, and Robin boundary conditions.…”

“…However, there are still a few papers [30][31][32][33][34] that have considered the dynamic properties of GRNs with reaction-diffusion terms. References [30,31] investigate the asymptotic stability problem for delayed GRNs with reactiondiffusion terms under Dirichlet boundary conditions, Neumann boundary conditions, and Robin boundary conditions. From [28,30,31], we can find that the reaction-diffusion terms have no effect on the stability of GRNs under Neumann and Robin boundary conditions.…”

An r‐Order Finite‐Time State Observer for Reaction‐Diffusion Genetic Regulatory Networks with Time‐Varying Delays

“…x , and p i t, x stand for the concentrations of mRNAs and proteins, respectively; a i , c i , and b i are the rate constants; D k > 0 and D * k > 0 denote the diagonal diffusion rate matrices; W represents the coupling matrix with elements defined as in [15]; g j s → s H /1 + s H is the Hill function, q i is the sum of dimensionless transcriptional rates which repress gene i, κ t and ρ t are delays subject to…”

“…In the proof of Theorem 1, we employ the socalled Wirtinger's inequality to obtain the equalities (18,19), which claims the necessity of Dirichlet boundary conditions. However, by employing the technique used in [14,15], one can deal with the cases of Robin boundary conditions and Neumann boundary conditions, which will make the LMI conditions corresponding to ones in Theorems 1 and 2 more conservative.…”

“…To the best our knowledge, those works in [12][13][14][15][16][17] have researched the problem of stability analysis of delayed RDGRNs. The asymptotic stability analysis of delayed RDGRNs have been involved in [13][14][15] by constructing an appropriate LKF and applying some inequality techniques. In [16], a sufficient condition of F-T stability for delayed RDGRNs has been given by constructing an LKF including quad-slope integrations and applying the Gronwall inequality and Wirtinger-type integral inequality.…”