Assessing Asymptotic Stability of Linear Continuous Time-Varying Systems by Computing the Envelope of all Trajectories

In this note, necessary and sufficient numerical conditions for asymptotic stability and for uniform asymptotic stability of linear continuous time-varying systems are derived. For a given set of initial conditions, a tube containing all the trajectories of the system is constructed in the state space. At each instant of time, there exists an initial condition inside the set such that the resulting trajectory attains the border of the tube. Based on the above formulation, necessary and sufficient conditions for asymptotic stability and for uniform asymptotic stability are expressed through the solution of a linear differential Lyapunov equation. The conditions can deal with the stability of periodic systems as well. One of the main characteristics of the proposed necessary and sufficient conditions is that the only assumption on the dynamical matrix of the time-varying system is continuity. Examples from the literature illustrate the superiority of the proposed conditions when compared to other methods.