On universal classes of Lyapunov functions for linear switched systems

In this paper we discuss the notion of universality for classes candidate common Lyapunov functions linear switched systems. On one hand, prove that a family absolutely homogeneous is universal as soon it approximates arbitrarily well every convex function C0 topology unit sphere. other several obstructions class to be universal, showing, in particular, families piecewise-polynomial continuous whose construction involves at most l polynomials degree m (for given positive integers l,m) cannot universal.