Local controllability of control systems with feedback

We prove sufficient conditions for the instantaneous local controllability of nonlinear (nonsmooth) control systems with feedback. We introduce for that purpose the high-order variations of the reachable map, which is related strongly to its shape. It is a direction in which the reachable map evolves. The cone of variations is convex. This allows one to prove the following theorem: if there exist variationsv₁, ...,v_p such that int co{v₁, ...,v_p}, then the system is small-time locally controllable at the point of equilibrium. We provide a short proof of the main result of Sussmann (Ref. 12) and extend it to differential inclusions.This work was supported in part by FCAR of Canada.