  We leverage second-order information for tuning of inverse optimal
controllers for a class of discrete-time nonlinear input-affine systems. For
this, we select the input penalty matrix, representing a tuning knob, to yield
the Hessian of the Lyapunov function of the closed-loop dynamics. This draws a
link between second-order methods known for their high speed of convergence and
the tuning of inverse optimal stabilizing controllers to achieve a fast decay
of the closed-loop trajectories towards a steady state. In particular, we
ensure quadratic convergence, a feat that is otherwise not achieved with a
constant input penalty matrix. To balance trade-offs, we suggest a practical
implementation of the Hessian and validate this numerically on a network of
phase-coupled oscillators that represent voltage source controlled power
inverters.
