A common Lyapunov matrix for two Hurwitz matrices exists iff the intersection of strict Lyapunov matrices for one and non-strict for the other is nonempty; this yields exponential stability for the LPV representation of augmented primal-dual gradient flow under relaxed strong convexity.
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A Common Lyapunov Matrix Approach to the Exponential Stability of Augmented Primal-Dual Gradient Flow as LPV Systems
A common Lyapunov matrix for two Hurwitz matrices exists iff the intersection of strict Lyapunov matrices for one and non-strict for the other is nonempty; this yields exponential stability for the LPV representation of augmented primal-dual gradient flow under relaxed strong convexity.