Triangulations and Error Estimates for Interpolating Lyapunov Functions

Abstract

AbstractThe CPA method to compute Lyapunov functions depends on a triangulation of the relevant part of the state space. In more detail, a CPA (Continuous and Piecewise Affine) function is affine on each simplex of a given triangulation and is determined by the values at the vertices of the triangulation. Two important aspects in the proof that the CPA method is always able to generate a CPA Lyapunov function if the triangulation is sufficiently fine, are (a) the geometry of the simplices of the triangulation and (b) error estimates of CPA interpolations of functions. In this paper the aspect (a) is tackled by extending the notion of (h, d)-boundedness, which so far has depended on the order of the vertices in each simplex, and it is shown that it is essentially independent of the order and can be expressed in terms of the condition number of the shape matrix. Concerning (b), existing error estimates are generalised to other norms to increase the flexibility of the CPA method. In particular, when the CPA method is used to verify Lyapunov function candidates generated by other methods. Parts of the results were presented in Giesl and Hafstein (Uniformly regular triangulations for parameterizing Lyapunov functions. In: Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO), 549–557, 2021).