Stability Analysis And State Estimation Of Delayed Gene Regulating Network With Reaction Diffusion Term

In the last decades, due to the increasing progress in genome sequencing and gene recognition, genetic regulatory networks(GRNs) have become an significant area in biological and biomedical sciences. However, reaction-diffusion and time delays are unavoidably encountered in the process of gene regulation. Hence, It is necessary to study the dynamical behaviors for GRNs with time-varying delays and reaction-diffusion terms.In this thesis, asymptotic stability and state estimation problems for a class of GRNs with time-varying delays and reaction-diffusion terms are studied.Firstly, under Dirichlet boundary conditions, a delay-dependent and delayrate-dependent stability criterion for delayed GRNs with reaction-diffusion terms is established by constructing a novel Lyapunov–Krasovskii functional and employing Jensen’s inequality, Wirtinger’s inequality, Green’s second identity and reciprocally convex approach. The obtained stability criterion is in the form of liner matrix inequalities(LMIs) and does not require the restriction on the upper bounds of delays’ derivatives being less than 1. Thus they extend the range of applications of theoretical results. Then, theoretical proof illustrates that the obtained criterion is less conservative than the existing ones. Next, the asymptotic stability of delayed GRNs with reaction-diffusion terms is considered under Nuemman boundary conditions. By using a similar method, we obtain delay-dependent asymptotic stability criterion. It is worth emphasizing that the obtained criterion under Dirichlet boundary conditions remains the information of reaction-diffusion terms, while they do not exist in the criterion under Neumann boundary conditions. This will lead difference in conservatism. In the end, numerical examples are given to illustrate the effectiveness of the theoretical results.Secondly, under Dirichlet boundary conditions, the state estimation problem for delayed GRNs with reaction-diffusion terms is investigated. The purpose is to design a state observer to estimate the concentrations of m RNAs and proteins through available measurement outputs. By introducing new integral terms in a novel Lyapunov–Krasovskii functional and employing Wirtinger-based integral 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 LMIs. The obtained stability criterion depends on the upper bounds of the delays and their derivatives. It should be highlight that if the LMIs are feasible, the desired observer exists and can be determined. Finally, two numerical examples are presented to illustrate the effectiveness of observer designed here.