Analysis and Numerical Solution of Generalized Lyapunov Equations

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Stabilization of projection‐based reduced‐order models (2012, background): “Theorem 1 Let E be nonsingular and Q 2Rn n be a symmetric matrix such that vQv> 0 for all nonzero vectors v 2Rn, where the superscript T designates the transpose [18].”
Low rank iterative methods for projected generalized Lyapunov equations (2005, methodology): “Due to the practical importance of the numerical solution of large-scale projected generalized Lyapunov equations that occur in balanced truncation model reduction for descriptor systems [12, 13], development of iterative methods for such equations is a challenge problem.”
Balanced truncation model reduction for semidiscretized Stokes equation (2006, methodology): “To compute the full rank factors of the Gramians we can use the generalized Schur-Hammarling method as proposed in [35].”
Gramian-Based Reduction Method Applied to Large Sparse Power System Descriptor Models (2008, methodology): “The balanced truncation technique requires the computation of the system controllability and observability gramians [9], [12], [18].”
Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations (2009, methodology): “…strangeness index, Lyapunov exponent, Bohl exponent, Sacker-Sell spectrum, exponential dichotomy, spectral interval, smooth QR factorization, continuous QR algorithm, discrete QR algorithm, kinematic equivalence, Steklov function AMS(MOS) subject classification: 65L07, 65L80, 34D08, 34D09”