Large-Time Local Controllability via Homogeneous Approximations

If all points in a neighborhood of a rest solution of an $n$-dimensional, affine control system can be attained in some sufficiently large time $t_1 0$, we say that the system is large-time locally controllable at the rest solution. Sufficient conditions for large-time local controllability are given in terms of small-time local controllability of homogeneous approximating systems. The major result, Theorem 3, is a geometric test for large-time local controllability in terms of the (coordinate-free) structure of Lie products of the vector fields which define the system, evaluated at the rest solution. Large-time local controllability has implications for the problem of the existence of an asymptotically stabilizing feedback control.