Controllability Via Chronological Calculus

. We demonstrate how the chronological formalism, in particular exponential product expansions and combinatorial features of Viennot-Hall bases lead to much streamlined proofs of high order conditions for controllability and optimality which are essential at singular points where classical techniques necessarily fail. 1 Introduction Nonlinear control systems typically feature singular states where standard linearization techniques fail completely. Such states are distinguished as singular points of the distributions spanned by the system vector fields. The best known example may be the angular velocity equations of a rigid body whose uncontrolled drift vector field is purely quadratic, i.e. its linearization about the rest point is trivial. Nonetheless, higher order nonlinear conditions still provide for controllability with fewer controls than states. Controllability is just one, but a critical step towards final objectives like regulation, feedback stabilization, tracking, disturban..