Mathematics

Local properties of a control system $\dot x=f_0(x)+\sum_{j=1}^d u_jf_j(x)$ are completely determined by the structure of the Lie Algebra $\Ggi$ generated by the vector fields $f_0,f_1,\ldots,f_d$. Of particular importance, from applications point of view, are nilpotent systems. This fact motivates the efforts of the research in approximating a general system with a nilpotent one ("nilpotent approximation"). Moreover it is fundamental for an approximating nilpotent system to have a "simple" and "compact" form, i.e. canonical form. In this work we consider the class of abstract nilpotent Lie Algebras freely generated by a set of $d$ elements. We determine those nilpotent Lie Algebras stable under transformation of the generators and build the relative multiplication table. Finally we go back to nilpotent Lie Algebras of vector fields and for them we provide a canonical coordinate representation.