On the relationship between global and local controllability

Abstract

Given a pair(X ⁰,D), whereX ⁰ is a vector field andD is a family of vector fields on a manifoldM, we can define a system affine in the control, i.e., the new family of all the vector fields of the formX ⁰ +uX, whereX ∈D andu is a real parameter. Such a system will be called globally controllable if each state is reachable from each other, whenever unbounded values foru are allowed. It is proved that a system affine in the control is globally controllable if and only if in any set of a suitable partition ofM there exist points locally controllable with bounded values foru. Further, it is proved that, under more restrictive assumptions, global controllability implies the existence of points locally controllable at a fixed time with bounded values foru. In the case of simply connected manifolds, a full equivalence among all the forms of controllability considered here is obtained.