Galerkin trial spaces and Davison-Maki methods for the numerical solution of differential Riccati equations

The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently, Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. Known solution formulas are reviewed and extended. Because of memory-efficient approximations, invariant subspaces for a possibly low-dimensional solution representation are identified. A Galerkin projection onto a trial space related to a low-rank approximation of the solution of the algebraic Riccati equation is proposed. The modified Davison-Maki method is used for time discretization. Known stability issues of the Davison-Maki method are discussed. Numerical experiments for large-scale autonomous differential Riccati equations and a comparison with high-order splitting schemes are presented.