Comprehensive Lyapunov functions for linear switching systems

For a linear dynamical switching system, we analyse maximal asymptotic growth of trajectories depending on the initial point. Both discrete and continuous time systems in R-d are considered. We prove the existence of a Lyapunov norm in Rd with the following property: for every invariant linear subspace L subset of R-d of the system, the restriction of the norm on L provides a tight upper bound for the growth of trajectories on L. For this, we introduce the concept of the spectral normal form of a family of matrices. Properties of the comprehensive Lyapunov norms are analysed and methods of their construction are discussed. (C) 2019 Published by Elsevier Ltd.