  Symmetric matrix-valued dynamical systems are an important class of systems
that can describe important processes such as covariance/second-order moment
processes, or processes on manifolds and Lie Groups. We address here the case
of processes that leave the cone of positive semidefinite matrices invariant,
thereby including covariance and second-order moment processes. Both the
continuous-time and the discrete-time cases are first considered. In the LTV
case, the obtained stability and stabilization conditions are expressed as
differential and difference Lyapunov conditions which are equivalent, in the
LTI case, to some spectral conditions for the generators of the processes.
Convex stabilization conditions are also obtained in both the continuous-time
and the discrete-time setting. It is proven that systems with constant delays
are stable provided that the systems with zero-delays are stable -- which
mirrors existing results for linear positive systems. The results are then
extended and unified into a impulsive formulation for which similar results are
obtained. The proposed framework is very general and can recover and/or extend
almost all the results on the literature of linear systems related to
(mean-square) exponential (uniform) stability. Several examples are discussed
to illustrate this claim by deriving stability conditions for stochastic
systems 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.
