On decompositions and integral representation of solutions for affine control systems (Q1319478)

The paper deals with the problem of representing the solutions of affine nonlinear control systems of the form \[ {\textstyle {{dx} \over {dt}}}= f(x)+ \sum_{i=1}^ n u_ i g^ i(x), \qquad x\in \mathbb{R}^ n, \quad u_ i \in \mathbb{R}^ m \] by solving a gradient system \[ {\textstyle {{\partial G} \over {\partial t_ j}}}= X^ j (p;G), \qquad j=1,\dots, M, \quad G(0,x)=x \] and considering \(p\) as the new control guided by a controllable system \[ {\textstyle {{dp} \over {dt}}}= \sum_{i=1}^ n u_ i q^ i(p), \qquad p(0)=0. \] The main result consists in a decomposition formula for the trajectories when the \(C^ \infty\) vector fields \(g^ 1,\dots, g^ m\) are not commuting and the Lie algebra \(L(g^ 1,\dots, g^ m)\) is finitely generated. The case in which the vector fields \(q^ 1,\dots, q^ m\) are analytic is also considered.