# Preprint No. MPIMD/11-06

#### Abstract:

The differential Riccati equation (DRE) arises in several fields like optimal control, optimal filtering, $H$ control of linear-time varying systems, differential games, etc. In the literature there is a large variety of approaches to compute its solution. Particularly for stiff DREs, matrix-valued versions of the standard multi-step methods for solving ordinary differential equations have given good results. In this paper we discuss a particular class of one-step methods. These are the linear-implicit Runge-Kutta methods, i.e, the so called Rosenbrock methods. We show that they offer a practical alternative for solving stiff DREs. They can be implemented with good stability properties and they allow a cheap way to control the step size. The matrix valued version of the Rosenbrock methods for DREs requires the solution of one Sylvester equation in each stage of the method. For the case in which the coefficient matrices of the Sylvester equation are dense, the Bartels-Stewart method can be efficiently applied for solving the equations. The computational cost (computing time and memory requirements) is smaller than for the multi step methods.

#### BibTeX:

@TECHREPORT{MPIMD11-06,
author = {Hermann Mena and Peter Benner},
title = {Rosenbrock methods for solving differential Riccati equations},
number = {MPIMD/11-06},
month = oct,
year = 2011,
institution = {Max Planck Institute Magdeburg},
type = {Preprint},
note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},
}