Rosenbrock Methods for Solving Riccati Differential Equations

The Riccati differential equation (RDE) arises in several fields like optimal control, optimal filtering, 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 RDEs, matrix-valued versions of the standard multi-step methods for solving ordinary differential equations have given good results. In this technical note we discuss a particular class of one-step methods. These are the linear-implicit Runge-Kutta methods or Rosenbrock methods. We show that they offer a practical alternative for solving stiff RDEs. They can be implemented with good stability properties and allow for a cheap step size control. The matrix valued version of the Rosenbrock methods for RDEs 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 multi-step methods.