Noncommutative Power Series and Formal Lie-algebraic Techniques in Nonlinear Control Theory

In nonlinear control, it is helpful to choose a formalism well suited to computations involving solutions of controlled differential equations, exponentials of vector fields, and Lie brackets. We show by means of an example -- the computation of control variations that give rise to the Legendre-Clebsch condition -- how a good choice of formalism, based on expanding diffeomorphisms as products of exponentials, can simplify the calculations. We then describe the algebraic structure underlying the formal part of these calculations, showing that it is based on the theory of formal power series, Lie series, the Chen series -- introduced in control theory by M. Fliess -- and the formula for the dual basis of a Poincaré-Birkhoff-Witt basis arising from a generalized Hall basis of a free Lie algebra