Solution Bounds and Numerical Methods of the Unified Algebraic Lyapunov Equation

In this paper, applying some properties of matrix inequality and Schur complement, we give new upper and lower bounds of the solution for the unified algebraic Lyapunov equation that generalize the forms of discrete and continuous Lyapunov matrix equations. We show that its positive definite solution exists and is unique under certain conditions. Meanwhile, we present three numerical algorithms, including fixed point iterative method, the acceleration fixed point method and the alternating direction implicit method, to solve the unified algebraic Lyapunov equation. The convergence analysis of these algorithms is discussed. Finally, some numerical examples are presented to verify the feasibility of the derived upper and lower bounds, and numerical algorithms.

Automated summary

In this paper, new upper and lower bounds of the solution for the unified algebraic Lyapunov equation are derived using matrix inequality and Schur complement. It is shown that a positive definite solution exists and is unique under certain conditions. Three numerical algorithms are presented and analyzed for solving the equation. Numerical examples are provided to verify the feasibility of the derived bounds and algorithms.