On local canonical form of boundary tangency manifolds for 2-dimensional gradient-like Morse-Smale controlled systems

摘要

Boundary tangency manifolds (BTMs) of a nonlinear state equation are for constructing the level surface of an unknown control Lyapunov function generalized to gradient-like Morse-Smale controlled systems. We differentiate a Morse-Smale function repeatedly along the 'state equation' by regarding the input as an independent variable. The BTMs are the nested sequence of submanifolds obtained from these Lie-derivatives. In this paper, we show that there is a certain kind of local canonical form of BTMs on the neighborhood of a noteworthy point. If the BTMs are in local canonical form then the trajectories of the closed-loop system under a constant input has only the points of external tangency on its boundary of the defining set of the BTMs.