On the Small-time Local Controllability

The convexity of a set of "control variations" is one of the crucial properties needed to prove sufficient controllability conditions or necessary optimality conditions. Heuristically, if one can construct control variations in all possible directions, then the considered control system is small-time locally controllable. As it was shown by R.-M. Bianchini and M. Kawski ["Needle variations that cannot be summed", SIAM J. Control Optimization 42 (2003) 218--238] the cones generated by needle variations may fail to be convex. The purpose of the present paper is to define a convex set of high-order control variations and to prove a sufficient controllability condition. The proof is based on a general Lie series formalism. One illustrative example is also presented.