  This paper derives nonlinear feedback control synthesis for general control
affine systems using second-order actions---the 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. Moreover, the simulated examples demonstrate superior
convergence when compared to synthesis based on first-order needle variations.
Lastly, the underactuated dynamic underwater vehicle model demonstrates the
convergence even in the presence of a velocity field.
