Global extended Krylov subspace methods for large-scale differential Sylvester matrix equations

Abstract

In this paper, we present a new numerical methods for solving large-scale differential Sylvester matrix equations with low rank right hand sides. These differential matrix equations appear in many applications such as robust control problems, model reduction problems and others. We present two approaches based on extended global Arnoldi process. The first one is based on approximating exponential matrix in the exact solution using the global extended Krylov method. The second one is based on a low-rank approximation of the solution of the corresponding Sylvester equation using the extended global Arnoldi algorithm. We give some theoretical results and report some numerical experiments to show the effectiveness of the proposed methods compared with the extended block Krylov method given in Hached and Jbilou (Numer Linear Algebra Appl 255:e2187, 2018).