If a smooth nonlinear affine control system has a controllable linear
approximation, a standard technique for constructing a smooth (linear)
asymptotically stabilizing feedbackcontrol is via the
LQR (linear, quadratic, regulator) method. The nonlinear system may
not have a controllable linear approximation, but instead may be shown
to be small (or large) time locally controllable via a high order,
homogeneous approximation. In this case one can attempt to construct
an asymptotically stabilizing feedback control as the optimal control,
relative to a cost functional with homogeneous integrand, for the
approximating system. Necessary, and some sufficient, conditions
for the existence of a smooth (real analytic), stabilizing feedback
control of this form are given. For some systems which satisfy
these necessary conditions, the specific form of a stabilizing
control is established.