Geometric Local Controllability: Second-Order Conditions

In a geometric point of view, a nonlinear control system, a#ne in the controls, is thought of as an a#ne subbundle of the tangent bundle of the state space. In deriving conditions for local controllability from this point of view, one should describe those properties of the a#ne subbundle that either ensure or prohibit local controllability. In this paper, second-order conditions of this nature are provided. The techniques involve a fusion of well-established analytical methods with di#erential geometric ideas. 1