On Structural Properties of the Lyapunov Matrix Equation for Optimal Diagonal Solutions

We revisit the classical problem of finding a positive diagonal solution P to the Lyapunov equation A^T P + PA < 0 that minimizes the Lyapunov exponent (the maximum eigenvalue of A^T P + PA) for A isin R^nXn, with the aim of identifying structural properties of the Lyapunov matrix equation at the optimum. Using eigenvalue sensitivity notions together with optimization machinery, we are able to obtain an explicit characterization of the minimum Lyapunov exponent that provides such structural insight.