Stability Analysis Of Gene Network With Mixed Time Delay

This paper ?rstly concerned with the problem of the existence for nonnegative equilibrium points of the genetic regulatory networks with mixed(discrete and distributed) delays. Then, the problems of stability and stabilization of nonnegative equilibrium points are studied for genetic regulatory networks with mixed delays.Firstly, the equilibrium points of the genetic regulatory networks represent the concentrations of proteins and mRNAs in a stable state. So it should be nonnegative.The nonnegativity is ignored in almost all of the literature which research on stability. This departs from the realistic background. For genetic regulatory networks with mixed delays, this paper proves the existence of the nonnegative equilibrium points by Brouwer’s ?xed point theorem.Secondly, on the basis of M-matrix theory and functional di?erential equation theory, the su?cient conditions are given for guaranteeing that genetic regulatory networks with mixed delays have only one nonnegative equilibrium point which is globally asymptotically stable. That is to test whether a constant matrix is a nonsingular M-matrix or not. There are more than 50 equivalent conditions for nonsingular M-matrix. One of them is that the real part of all eigenvalues are positive. Thus, the M-matrix-based stability criteria proposed in this paper could be judged whether hold or not through the function eig in MATLAB. Compared with the LMI-based stability criteria in most of the literature, the M-matrix-based stability criteria in this paper reduce the computational complexity greatly.Thirdly, based on nonsingular M-matrix theory and strictly diagonally dominant matrix theory, we design the state feedback stabilization controller for genetic regulatory networks with mixed delays. M-matrix-based and LMI-based su?cient conditions for the existence of the controller are given, respectively. The feedback gain matrices can be obtained by solving several simple LMIs at last.Furthermore, some numerical examples and their simulation results show the correctness and e?ectiveness for the stability and stabilization criteria established in this paper, and are less conservative than ones in the related literature.