Modified Lyapunov matrix equations with positive definite solutions for descriptor systems

Descriptor systems (singular systems, differential algebraic equations) are a recent topic of research in numerical mathematics, mechanics and control theory as well. In this contribution the asymptotic stability of linear time-invariant descriptor systems is discussed relating the stability problem to symmetric positive (semi-) definite solutions of a modified Lyapunov matrix equation. This modified equation is obtained by changing the original matrix pencil in another one preserving stability and generating a regular Lyapunov matrix equation, which is much easier to solve.