Stability of a class of linear time-varying systems

This paper firstly considers the exponential stability of unforced linear systems of slowly time-varying dynamics. Possible switchings of the system structure to unstable dynamics during certain finite time intervals are admitted. The maintenance of global exponential stability does not necessarily require at most a finite number of switchings in the dynamics while infinitely many switches can also lead to stability. The mechanism to achieve stability under infinitely many switches in the dynamics is to maintain the system in the stable region during time intervals of sufficiently large length without switches provided that the system dynamics evolves at a sufficiently small rate with time. Special attention is paid to the robust tolerance to a class of state disturbances and to the case of time-varying matrix of dynamics that possess either piecewise constant or constant eigenvalues. The obtained results can be relevant for their use in stability issues for the cases of multimodel non-adaptive and adaptive control with improved transient performances.