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.