On algebraic Riccati equations associated with M-Matrices
For researchers working on Riccati equations and M-matrices, this fills a gap by addressing the reducible singular case, though the extension is incremental.
The paper extends the existence of a minimal nonnegative solution to algebraic Riccati equations where the coefficient matrix is a reducible singular M-matrix, under a regularity assumption. It also characterizes properties of this solution and shows how existing methods can compute it.
We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.