On the algebraic Riccati inequality arising in cone-preserving time-delay systems

Abstract

This paper studies the problem of Riccati stability of a pair of matrices. For a matrix pair (A,B), it was recently shown that if its corresponding time-delay system is internally positive, meaning that A is Metzler and B is nonnegative, then the pair (A,B) is diagonally Riccati stable if and only if A+B is Hurwitz. We extend this to the case when the pair (A,B) corresponds to a time-delay system with a more general cone-preserving property. We show that if the time-delay system relating to the pair (A,B) is invariant on a symmetric cone, the corresponding algebraic Riccati inequality admits positive definite solutions, which can be constructed via the scaling transformation on the Euclidean Jordan algebra associated with the symmetric cone. For the special case when the symmetric cone is the positive semi-definite cone, an application to a class of stochastic systems is discussed.