Existence of algebraic matrix Riccati equations arising in transport theory

General Matrix Pencil Techniques for Solving Discrete-Time Nonsymmetric Algebraic Riccati Equations

“…This means that the players are committed to follow an a priori strategy. In a linear-quadratic game with infinite time horizon, the existence of a Nash equilibrium is closely related to the solvability of a set of coupled algebraic Riccati equations which can be reformulated as a single nonsymmetric algebraic Riccati equation [1,7,6,22,18]. We show further how these results can be recast by our pencil approach while removing the restrictive invertibility hypotheses present in the literature.…”

“…Besides the symmetric Riccati equation, the theory of nonsymmetric algebraic Riccati equation, both continuous and discrete-time, has received recently a renewed interest due to a large number of applications in various fields, including applied mathematics, engineering and economic science (see for instance [1,6,22,18]). The nonsymmetric algebraic Riccati equation extends the concept of standard Riccati equation by allowing for a not necessarily square matrix variable.…”

Explicit Solutions for a Riccati Equation from Transport Theory

“…, p n > 0. Such Riccati equations arise in Markov models [27] and in nuclear physics [6,16,20]. In the latter application, to study the transport of particles, one introduces integral equations of the form…”

“…For this the function T (x, y) is approximated via a matrix X = [x ij ], where x ij is an approximation of T (µ i , ν j ) with µ i , ν j being the ith and jth nodes of the quadrature formula on [−α, 1] and [α, 1], respectively, e.g. [16].…”

“…The computation of this minimal solution has been investigated in several publications. Various direct and iterative methods [1,10,11,12,13,14,15,17,16,24] have been proposed by either directly solving the Riccati equation or by computing specific invariant subspaces of the 2n × 2n matrix…”