Peter Giesl-Accepted-11.12.20.pdf (1.13 MB)
Computation and verification of contraction metrics for exponentially stable equilibria
journal contribution
posted on 2023-06-09, 22:39 authored by Peter GieslPeter Giesl, Sigurdur Hafstein, Iman MehrabinezhadThe determination of exponentially stable equilibria and their basin of attraction for a dynamical system given by a general autonomous ordinary differential equation can be achieved by means of a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions decreases as time increases. The Riemannian metric can be expressed by a matrix-valued function on the phase space. The determination of a contraction metric can be achieved by approximately solving a matrix-valued partial differential equation by mesh-free collocation using Radial Basis Functions (RBF). However, so far no rigorous verification that the computed metric is indeed a contraction metric has been provided. In this paper, we combine the RBF method to compute a contraction metric with the CPA method to rigorously verify it. In particular, the computed contraction metric is interpolated by a continuous piecewise affine (CPA) metric at the vertices of a fixed triangulation, and by checking finitely many inequalities, we can verify that the interpolation is a contraction metric. Moreover, we show that, using sufficiently dense collocation points and a sufficiently fine triangulation, we always succeed with the construction and verification. We apply the method to two examples.
History
Publication status
- Published
File Version
- Accepted version
Journal
Journal of Computational and Applied MathematicsISSN
0377-0427Publisher
ElsevierExternal DOI
Page range
1-31Article number
a113332Pages
31.0Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2021-01-07First Open Access (FOA) Date
2022-01-03First Compliant Deposit (FCD) Date
2021-01-06Usage metrics
Categories
No categories selectedLicence
Exports
RefWorks
BibTeX
Ref. manager
Endnote
DataCite
NLM
DC