Invariance properties of time-varying linear systems

Abstract

This paper formulates and proves two types of necessary and sufficient conditions for the characterization of positively (flow) invariant sets with respect to the state-space trajectories of the time-varying (non-autonomous) linear systems in both continuous- and discrete-time case. These conditions are expressed in terms of inequalities involving the matrix function that defines the system dynamics and a constant matrix that defines the shape of the invariant set. The first type of results refers to contractive invariant sets which decrease exponentially, and the second one considers invariant sets that remain constant. Our approach to non-autonomous systems accommodates, as particular cases, the elements of the invariant set analysis already elaborated for autonomous systems.