Uniformly regular triangulations for parameterizing lyapunov functions

The computation of Lyapunov functions to determine the basins of attraction of equilibria in dynamical systems can be achieved using linear programming. In particular, we consider a CPA (continuous piecewise affine) Lyapunov function, which can be fully described by its values at the vertices of a given triangulation. The method is guaranteed to find a CPA Lyapunov function, if a sequence of finer and finer triangulations with a bound on their degeneracy is considered. Hence, the notion of (h,d)-bounded triangulations was introduced, where h is a bound on the diameter of each simplex and d a bound on the degeneracy, expressed by the so-called shape-matrices of the simplices. However, the shape-matrix, and thus the degeneracy, depends on the ordering of the vertices in each simplex. In this paper, we first remove the rather unnatural dependency of the degeneracy on the ordering of the vertices and show that an (h,d)-bounded triangulation, of which the ordering of the vertices is changed, is still (h,d^∗)-bounded, where d^∗ is a function of d, h, and the dimension of the system. Furthermore, we express the degeneracy in terms of the condition number, which is a well-studied quantity.