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 Rd are considered. We prove the existence of a Lyapunov norm in Rd with the following property: for every invariant linear subspace L⊂Rd 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.