Peter Giesl (University of Sussex)
Construction of Lyapunov functions and Contraction Metrics to determine the Basin of Attraction
In this talk we consider two methods of determining the basin of attraction. In the first part, we discuss the construction of a Lyapunov function using Radial Basis Functions. The basin of attraction of equilibria or periodic orbits of an ODE can be determined through sublevel set of a Lyapunov function. To construct such a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the ODE, a linear PDE is solved approximately using Radial Basis Functions. Error estimates ensure that the approximation is a Lyapunov function. For the construction of a Lyapunov function it is necessary to know the position of the equilibrium or periodic orbit. A different method to analyse the basin of attraction of a periodic orbit without knowledge of its position uses a contraction metric and is discussed in the second part of the talk. A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. In this talk, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine function, which is affine on each simplex of a triangulation of the phase space.