Computation and verification of contraction metrics for exponentially stable equilibria

Giesl, Peter, Hafstein, Sigurdur and Mehrabinezhad, Iman (2021) Computation and verification of contraction metrics for exponentially stable equilibria. Journal of Computational and Applied Mathematics. a113332 1-31. ISSN 0377-0427

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Abstract

The 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.

Item Type: Article
Keywords: Contraction Metric, Lyapunov Stability, Basin of Attraction, Numerical Method,
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
SWORD Depositor: Mx Elements Account
Depositing User: Mx Elements Account
Date Deposited: 07 Jan 2021 07:48
Last Modified: 07 Jan 2021 08:01
URI: http://sro.sussex.ac.uk/id/eprint/96357

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