Projective Controllability of Complete Lifted Control Systems

We introduce the complete lifted control system associated with a control system on a smooth manifold by replacing each vector field with its complete lift to the tangent bundle. We prove that complete lifted control systems are never controllable on the whole tangent bundle, due to the invariance of the zero section. Motivated by this obstruction and by the invariance of complete lifts under fiberwise dilations, we study the induced control system on the projectivized tangent bundle. We establish the relationship between the controllability properties of the lifted and projectivized systems, showing in particular that projective controllability implies controllability of the original system. Our main result provides a sufficient condition for controllability of the projectivized system in terms of a Lie rank condition modulo the Euler vector field.