  This paper derives nonlinear feedback control synthesis for general control
affine systems using second-order actions---the second-order needle variations
of optimal control---as the basis for choosing each control response to the
current state. A second result of the paper is that the method provably
exploits the nonlinear controllability of a system by virtue of an explicit
dependence of the second-order needle variation on the Lie bracket between
vector fields. As a result, each control decision necessarily decreases the
objective when the system is nonlinearly controllable using first-order Lie
brackets. Simulation results using a differential drive cart, an underactuated
kinematic vehicle in three dimensions, and an underactuated dynamic model of an
underwater vehicle demonstrate that the method finds control solutions when the
first-order analysis is singular. Lastly, the underactuated dynamic underwater
vehicle model demonstrates convergence even in the presence of a velocity
field.
