Sufficient Conditions for Local Controllability with Unbounded Controls

Local controllability of analytic affine control systems Σ with an arbitrary number of controls is studied, without any restriction on the dimension of the Lie algebra T' generated by the input vector fields. A sufficient condition for local controllability at a point is obtained, first for scalar input systems and then generalized in two different ways to the multi-input case. In all versions, this involves the computation of Lie brackets at the point under consideration and the verification of a criterion of the type considered by Sussmann (this Journal, 16 (1978), pp. 790–802). As an application, controllability of systems defined on Rⁿ and with constant input vector fields is studied.