Refined Eigenvalue Decay Bounds for Controllability Gramians of Sparsely-Actuated Symmetric LTI Systems

Refined bounds are obtained for the eigenvalues of the controllability Gramian for a linear system with a Hurwitz, symmetric state matrix. The new bounds are phrased in terms of partial condition numbers (ratios of intermediate eigenvalues) of the state matrix. The bounds are found to compare favorably with existing results for several examples, particularly in cases where the system has time-scale separations or multiple eigenvalues in narrow bands.