Global existence and stability of solutions of matrix riccati equations

Abstract

We consider a matrix Riccati equation containing two parameters C and cu. The quantity c denotes the average total number of particles emerging from a collision, which is assumed to be conservative (i.e., 0 < c less than or equal to 1), and alpha (0 less than or equal to alpha < 1) is an angular shift. Let S = {(c, alpha):0 < c 1 and 0 <less than or equal to> alpha < 1}. Stability analysis for two steady-state solutions X-min and X-max are provided. In particular, we prove that X-min is locally asymptotically stable for S - {(1, 0)}, while X-max is unstable for S - {(1, 0)}. For c = 1 and alpha = 0, X-min = X-max is neutral stable. We also show that such equations have a global positive solution for (c, cu) E S, provided that the initial value is small and positive. (C) 2001 Academic Press.