On Lipschitz Continuity of the Top Lyapunov Exponent of Linear Parameter Varying and Linear Switching Systems

We study families of time-varying linear systems, where time-variations have to satisfy restrictions on the dwell time, that is, on the minimum distance between discontinuities, as well as on the derivative in between discontinuities. For this class of systems we study continuity properties of the growth rate as a function of the systems’ data. It is shown by example, that a straightforward topology on the space of systems does not yield the desired continuity result. A new natural metric is introduced and a continuity result is obtained. Furthermore, local Lipschitz continuity may be shown for the (generic) case of irreducible systems. The methods rely heavily on a recent converse Lyapunov theorem for the class under consideration.