Projection methods for large Lyapunov matrix equations

Abstract

In the present paper, we propose Krylov subspace methods for solving large Lyapunov matrix equations of the form AX + XA T + BB T = 0 where A and B are real n × n and n × s matrices, respectively, with s ≪ n. Such problems appear in many areas of control theory such as the computation of Hankel singular values, model reduction algorithms and others. The proposed methods are based on the Arnoldi process. We show how to extract low rank approximate solutions to Lyapunov matrix equations and give some theoretical results. Finally, some numerical tests will be reported to illustrate the effectiveness of the proposed method.