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Symmetric matrix-valued dynamical systems are an important class of systemsthat can describe important processes such as covariance/second-order momentprocesses, or processes on manifolds and Lie Groups. We address here the caseof processes that leave the cone of positive semidefinite matrices invariant,thereby including covariance and second-order moment processes. Both thecontinuous-time and the discrete-time cases are first considered. In the LTVcase, the obtained stability and stabilization conditions are expressed asdifferential and difference Lyapunov conditions which are equivalent, in theLTI case, to some spectral conditions for the generators of the processes.Convex stabilization conditions are also obtained in both the continuous-timeand the discrete-time setting. It is proven that systems with constant delaysare stable provided that the systems with zero-delays are stable -- whichmirrors existing results for linear positive systems. The results are thenextended and unified into a impulsive formulation for which similar results areobtained. The proposed framework is very general and can recover and/or extendalmost all the results on the literature of linear systems related to(mean-square) exponential (uniform) stability. Several examples are discussedto illustrate this claim by deriving stability conditions for stochasticsystems driven by Brownian motion and Poissonian jumps, Markov jump systems,(stochastic) switched systems, (stochastic) impulsive systems, (stochastic)sampled-data systems, and all their possible combinations.

Stability Analysis and Stabilization of Linear Symmetric Matrix-Valued  Continuous, Discrete, and Impulsive Dynamical Systems -- A Unified Approach  for the Stability Analysis of Linear Systems