On Local Controllability

Let X, Y be real analytic vector fields on a real analytic n-dimensional manifold M. Consider a control system on M with dynamics described by \[(1) {{dx} / {dt}} = X(x(t)) + u(t)Y(x(t)), x(0) = p,\]where an admissible control is Lebesgue measurable with values | u(t) | 1. The relationship between the map \[(2) (s_1 , ⋯ ,s_n )→ (exp s_1 Y) ∘ ⋯ ∘ (exp s_n ({ad}n - 1 X,Y)) ∘ (exp tX)(p)\] being one-one on a neighborhood of 0 ∈ Rⁿ for each 0 < t < ε and other known necessary conditions for local controllability are studied. In dimension $ n =2 $, many necessary conditions are equivalent, and also sufficient. For n 3, the map (2) being locally one-one implies many necessary conditions are satisfied, but these need not be sufficient. Examples which illustrate what occurs geometrically are given.