## Peter Giesl, University of Sussex, UK

Math colloquium

**Speakers: **Peter Giesl, University of Sussex, UK

**Title: **Existence and construction of a contraction metric as solution of a matrix-valued PDE**.**

Room: Via Zoom. Link to be sent.

Time: Friday 2^{nd }October, 10:00am

#### Abstract:

A contraction metric is a Riemannian metric, with respect to which the distance between adjacent solutions of an ordinary differential equation (ODE) decreases.

A contraction metric can be used to prove existence and uniqueness of an equilibrium of an autonomous ODE and determine a subset of its basin of attraction without requiring information about its location. Moreover, a contraction metric is robust to small perturbations of the system.

We will prove a converse theorem, showing the existence of a contraction metric for an equilibrium by characterising it as a matrix-valued solution of a certain linear partial differential equation (PDE). This leads to a construction method by numerically solving the matrix-valued PDE using mesh-free collocation. We use and present a recent extension of mesh-free collocation of scalar-valued functions, solving linear PDEs, to matrix-valued ones. Finally, we briefly discuss a method to verify that the computed metric satisfies the conditions of a contraction metric.

This is partly work with Holger Wendland, Bayreuth as well as Sigurdur Hafstein and Iman Mehrabinezhad, Iceland.