Solvable Approximations to Control Systems

This paper is concerned with extending certain results of Rothschild and Stein [Acts. Math., 137 (1976), pp. 247–320] and Goodman [Lecture Notes in Mathematics 562, Springer-Verlag, New York, 1976], in which a finite set of vector fields $g_1 , \cdots ,g_m $ are lifted to vector fields approximated by generators of a free nilpotent Lie algebra. We wish to add a vector field f, to the set $g_1 , \cdots ,g_m $ and lift these vector fields to ones on a finite dimensional vector space V, approximated by generators F, $G_1 , \cdots ,G_m $ of a solvable Lie algebra, in which $ad^j F(G_i )$ generate a nilpotent, but not free, ideal. This procedure is accomplished in the context of a nonlinear control system, with outputs, in which f vanishes at the initial state, and in such a way that the output functions lift to the state space V, to define a system whose input output map is the same as the original system. The approximating system is obtained from a suitable realization of a truncation of the Volterra series ...