Hausdorff Continuity of Region of Attraction Boundary Under Parameter Variation with Application to Disturbance Recovery

Abstract

Consider a parameter dependent vector field on either Euclidean space or a compact Riemannian manifold. Suppose that it possesses a parameter dependent initial condition and a parameter dependent stable hyperbolic equilibrium point. It is valuable to determine the set of parameter values, which we call the recovery set, whose corresponding initial conditions lie within the region of attraction of the corresponding stable equilibrium point. A boundary parameter value is a parameter value whose corresponding initial condition lies in the boundary of the region of attraction of the corresponding stable equilibrium point. Prior algorithms numerically estimated the recovery set by estimating its boundary via computation of boundary parameter values. The primary purpose of this work is to provide theoretical justification for those algorithms for a large class of parameter dependent vector fields. This includes proving that, for these vector fields, the boundary of the recovery set consists of boundary parameter values, and that the properties exploited by the algorithms to compute these desired boundary parameters will be satisfied. The main technical result which these proofs rely on is establishing that the region of attraction boundary varies continuously in an appropriate sense with respect to small variation in parameter value for this class of vector fields. Hence, the majority of this work is devoted to proving this result, which may be of independent interest. The proof of continuity proceeds by proving that, for this class of vector fields, the region of attraction permits a decomposition into a union of the stable manifolds of the equilibrium points and periodic orbits it contains, and this decomposition persists under small perturbations to the vector field.