Local controllability of control systems with feedback (Q1096566)

We prove sufficient conditions for the instantaneous local controllability of nonlinear (nonsmooth) control systems with feedback. We introduce for that purpose the high-order variation 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 variations \(v_ 1\),..., \(v_ p\) such that \(\theta\) \(\in int co\{v_ 1,...,v_ p\}\), then the system is small-time locally controllable at the point of equilibrium. We provide a short proof of the main result due to \textit{H. Sussmann} [see, e.g., SIAM J. Control Optimization 16, 790-802 (1978; Zbl 0391.93004)] and extend it to differential inclusions.