Regular Article
Global Existence and Stability of Solutions of Matrix Riccati Equations

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Abstract

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

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Submitted by Horst, R. Thieme

1

Supported in part by NSC of R.O.C., Taiwan. E-mail: [email protected].