  In this paper the infinite horizon optimal regulation problem is solved
online for a deterministic control-affine nonlinear dynamical system using the
state following (StaF) kernel method to approximate the value function. Unlike
traditional methods that aim to approximate a function over a large compact
set, the StaF kernel method aims to approximate a function in a small
neighborhood of a state that travels within a compact set. Simulation results
demonstrate that stability and approximate optimality of the control system can
be achieved with significantly fewer basis functions than may be required for
global approximation methods.
