  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.
