Homotopy methods for zero finding from a learning/control Liapunov function viewpoint

Abstract

This paper revisits a class of recently proposed so-called invariant manifold methods for zero finding, showing that this class of homotopy methods can be designed in a natural manner from the control Liapunov function (CLF) approach proposed earlier by the authors. Moreover, the CLF approach clarifies the interplay between the homotopy parameter, which can be interpreted as a learning parameter and the choice of descent direction, which is the control vector and guides the choice of both. From this viewpoint, maintaining manifold invariance is equivalent to ensuring that the CLF satisfies a certain ordinary differential equation, involving the learning parameter, that allows an estimate of rate of convergence. In order to illustrate this approach, algorithms recently proposed using the invariant manifold approach, are rederived, via CLFs, in a unified manner.