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

TL;DR

This paper presents a unified framework for analyzing and stabilizing symmetric matrix-valued linear systems, covering continuous, discrete, and impulsive cases, with broad applicability to stochastic and hybrid systems.

Contribution

It introduces a comprehensive approach that unifies stability analysis and stabilization methods for symmetric matrix-valued systems across various types, extending existing results.

Findings

Derived Lyapunov-based stability conditions for continuous and discrete systems.

Established that systems with delays are stable if delay-free systems are stable.

Unified impulsive system analysis with similar stability results.

Abstract

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.