A Converse Lyapunov Theorem for Linear Parameter-Varying and Linear Switching Systems

We study families of linear time-varying 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. Such classes of systems may be formulated as linear flows on vector bundles. The main objective of the paper is to construct parameter dependent Lyapunov functions, which characterize the exponential growth rate. This is possible in the generic irreducible case. As an application the Gelfand formula is generalized to the class of systems studied here. In other words, the maximal exponential growth rate may be approximated by only considering the periodic systems in the family of time-varying systems. An outlook regarding the question of continuous dependence of the exponential growth rate on the data is given.