Lyapunov equation is one of the most important topics in the field of numerical algebra and non-linear analysis,and it has been widely used in scientific and engineering computation such as system and control theory,transportation theory and signal processing.This article systematically studied theoretical and numerical methods for generalized Lyapunov equations.In Chapter 2,we studied the positive semi-definite solution of the following generalized Lyapunov equations AX + XAT +m? j=1 Nj XNjT +Q = 0.By using the properties of the generalized Lyapunov equation,the constraint conditions are well described and a new theory of solvability is established.The perturbation analysis of the generalized Lyapunov equation is performed and the exact expressions of the three conditions are obtained.A numerical method was constructed to solve the equation.Finally,the numerical results showed that the method is feasible and effective.In Chapter 3,we focused on the rank-p symmetric positive semi-definite solutions of generalized Lyapunov equations AX + XAT + m? j=1 MjXMjT +BBT=0.We firstly reformulate the original problem into a least squares problem,and introduce new variables to deal with.Then,the partial inexact alternating direction method was constructed to solve this problem,and use the SPG algorithm and LSQR algorithm to solve the subproblems.The convergence of the algorithm was given.Numerical experiments are presented to illustrate the feasibility and efficiency of the proposed algorithm.In Chapter 4,we considered the Hermitian positive definite solution of the mixed Lyapunov equations X-AXB*-BXA*+NXN*= Q.Based on the Bhaskar-Lakshmikantham fixed point theorem,the sufficient conditions for the Hermitian positive definite solution of the equation was given and the numerical solution method was proposed.Finally,numerical methods are presented to illustrate the feasibility of the proposed algorithm. |