Prolongational Controllability and Weak Attraction for Control Affine Systems

Abstract

This paper studies the concept of prolongational controllability for control affine system with piecewise constant controls. The purpose is to show the relationship between the controllability by prolongations and the notion of weak uniform attraction, presenting criteria for controllability and controllability by prolongations. The central result assures that a control affine system is controllable by prolongations if and only if each state point is a global weak uniform attractor. For systems with nonextensive semi-orbits, the controllability by prolongations coincides with the usual one; in this case, the system is controllable if and only if each state point is a weak attractor. In particular, for invariant control affine system on Lie group, a necessary and sufficient condition for the system to be controllable is the identity of the group to be a weak attractor.