## 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.

Original language | English |
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Article number | 113332 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 390 |

DOIs | |

Publication status | Published - Jul 2021 |

### Bibliographical note

Publisher Copyright:© 2020 Elsevier B.V.

## Other keywords

- Basin of attraction
- Continuous piecewise affine interpolation
- Contraction metric
- Lyapunov stability
- Radial basis functions
- Reproducing kernel Hilbert spaces