Global Existence and Stability of Solutions of Matrix Riccati Equations

Abstract

We consider a matrix Riccati equation containing two parameters c and ␣. The quantity c denotes the average total number of particles emerging from a collision, Ž . Ž . which is assumed to be conservative i.e., 0 -c F 1 , and ␣ 0 F ␣ -1 is an ÄŽ . 4 angular shift. Let S s c, ␣ : 0 -c F 1 and 0 F ␣ -1 . Stability analysis for two steady-state solutions X and X are provided. In particular, we prove that

Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

“…As shown in [4,Proposition 3.4], that nonsymmetric matrix RDE is a special case of (1). Moreover, the condition imposed on X 0 in this paper is much weaker than that in [8].…”

“…(ii) It follows from (i) that X (m) (t) converges pointwise to a function X(t) on [0, ∞), and X(t) S. Letting m → ∞ in the Picard iteration and applying the monotone convergence theorem for Lebesgue integrals, we conclude that the limit function X(t) satisfies (8). It then follows from the boundedness of X(t) that X(t) is also continuous, which in turn implies that X(t) is differential and is a solution to (1).…”

“…The initial value problem (7) can be written in its equivalent integral form; namely, premultiplying and postmultiplying the differential equation in (7) by the integrating factors e −(t−s)A 1 and e −(t−s)D 1 , respectively, and integrating the resulting equation with respect to s from 0 to t, we obtain as in [8] …”

“…For a particular nonsymmetric RDE, these problems have been studied in [8]. As shown in [4,Proposition 3.4], that nonsymmetric matrix RDE is a special case of (1).…”

The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond

“…Equations of this kind describe different models in the applications like fluid queues [76,79,98] and transport equations [61,62]. The solution of interest is the matrix S with nonnegative entries which, among all the nonnegative solutions, is the one with component-wise minimal entries.…”