Nonsymmetric Riccati equations: Partitioned algorithms

Abstract

Generalized partitioned solutions (GPS) of nonsymmetric matric Riccati equations are presented in terms of forward and backward time differential equations that are of theoretical interest and also are computationally powerful. The GPS are the natural framework for the effective change of initial conditions, and the transformation of backward Riccati equation to forward Riccati equation and vice versa. Based on the GPS, computationally effective algorithms are obtained for the numerical solution of Riccati equations. These partitioned numerical algorithms have a decomposed or “partitioned” structure. They are given exactly in terms of a set of elemental solutions which are completely decoupled, and as such computable in either a parallel or serial processing mode. The overall solution is given exactly in terms of a simple recursive operation on the elemental solutions. Except for a subinterval of the total computation interval, the partitioned numerical algorithms are integration-free for the Riccati equation with constant or periodic matrices. Most importantly based on the GPS, a computationally attractive numerical algorithm is obtained for the computation of the steady-state solution of time-invariant Riccati equations. By making use of the GPS and some simple iterative operations, the Riccati solution is obtained in an interval which is twice as long as the previous interval requiring integration only in the initial subinterval.