Approximation of Curves in Phase Space by Solutions of Control System

In this article, we introduce the notion of a CO - controllable systems and study some relationships between controllability, local controllability, and CO - controllability. A control system in state-space representation is said to be CO-controllable if any oriented curve in the phase space can be CO - approximated by a solution of this system. We show that controllable (according to Kalman theory) linear systems with linear additive control are not CO - controllable and that linear or nonlinear systems with a “weakly nonlinear” additive control are generally CO - controllable.