Minimum Projection Method for nonsmooth control Lyapunov function design on general manifolds

Abstract

Asymptotic stabilization on noncontractible manifolds is known as a difficult control problem. On the other hand, an important fact is every control system that is globally asymptotically stabilizable at a desired equilibrium must have nonsmooth control Lyapunov functions. This paper considers the problem of construction of nonsmooth control Lyapunov functions on general manifolds, and we propose a nonsmooth control Lyapunov function design method called the ‘ Minimum Projection Method’. The proposed method considers a simple-structured smooth manifold associated with the original manifold by a surjective immersion, and then a control Lyapunov function defined on the simple-structured manifold is projected to the original manifold. A function on the original manifold is thus obtained. In this paper, we prove that the control system on another manifold associated with a surjective immersion is determined uniquely, and the resulting function by the proposed method is a nonsmooth control Lyapunov function on the original manifold. The effectiveness of the proposed method is confirmed by examples.