Abstract
As is known well, Lyapunov matrix equation, together with Riccati equation, plays a fundamental role in various aspects of linear and nonlinear system theory. Due to its extreme importance, a lot of papers concerning numerical methods of solving it efficiently have been published, though it is merely a linear algebraic equation. Moreover, in recent years many simple inequalities have been proposed to evaluate the “size” of its solution at hand.
In this paper, some new explicit formulae for the solution of the Lyapunov matrix equation are presented and lower and upper bounds for its extremal characteristic roots are derived using these formulae. The obtained formulae may be theoretically interesting in its own right and may possess some points of advantages in solving the equation numerically. The bounds derived here are quite useful because they can be applied in an unified manner to the equation with any coefficient matrix whenever it has a positive definite solution. Simple numerical examples are illustrated to make the significance of the present approach clear.
