Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems

We consider linear systems of differential equations $I \ddot{x}+B \dot{x}+C{x}={0}$ where $I$ is the identity matrix and $B$ and $C$ are general complex $n$ x $n$ matrices. Our main interest is to determine conditions for complete marginalstability of these systems. To this end we find solutions of the Lyapunov matrix equation and characterize the set of matrices $(B, C)$ which guarantees marginal stability. The theory is applied to gyroscopic systems, to indefinite damped systems, and to circulatory systems, showing how to choose certain parameter matrices to get sufficient conditions for marginal stability.Comparison is made with some known results for equations with real system matrices.Moreover more general cases are investigated and several examples are given.