A necessary condition for path-finding by the homotopy continuation method

Abstract

This paper is concerned with a path-finding method proposed by H. J. Sussmann, known as the homotopy continuation method (HCM). Given a control system initialized at some state, the HCM can be used to find an admissible open-loop control that performs a desired state transfer. This is accomplished by lifting curves in the state space of the system to curves in the space of regular admissible controls. The lifted curves are constructed as solutions of an ordinary differential equation called the path-lifting equation (PLE). The obstruction to applying the HCM, in general, is that solutions of the PLE may not be globally defined. We show that if this obstruction does not exist, then the endpoint map of the control system, restricted to the space of regular admissible controls, is necessarily a locally trivial fiber bundle. This result is valid for wide classes of control systems and controls.