Numerical Solution Of Generalized Lyapunov Matrix Equation

Generalized Lyapunov equations arise in controllability analysis and model reduction of bilinear systems, stability analysis and optimal or robust stabilization of stochastic linear control systems and so on. This thesis is concerned with the numerical solution of the generalized Lyapunov equations. Firstly, a Hermitian and Skew-Hermitian spiltting(HSS) iteration method is presented for solving the generalized Lyapunov equations. The convergence of the HSS iteration method is analyzed, and an upper bound on convergence rate is derived. In order to reduce the computational cost, an inexact variant of the HSS iteration method is established and its convergence property is discussed. Secondly, a global Arnoldi process for the generalized Lyapunov equations is developed, and global full orthogonalization method(FOM) and global generalized minimal residual(GMRES) method are proposed by projecting the generalized Lyapunov equations onto a linear operator Krylov subspace. Finally, in order to accelerate the convergence rate of the global FOM and global GMRES methods, preconditioned global FOM and preconditioned global GMRES methods for solving the generalized Lyapunov equations are presented based on the HSS preconditioner. Some numerical examples are given to show that the proposed methods are efficient.