  This manuscript presents an algorithm for obtaining an approximation of
nonlinear high order control affine dynamical systems, that leverages the
controlled trajectories as the central unit of information. As the fundamental
basis elements leveraged in approximation, higher order control occupation
kernels represent iterated integration after multiplication by a given
controller in a vector valued reproducing kernel Hilbert space. In a
regularized regression setting, the unique optimizer for a particular
optimization problem is expressed as a linear combination of these occupation
kernels, which converts an infinite dimensional optimization problem to a
finite dimensional optimization problem through the representer theorem.
Interestingly, the vector valued structure of the Hilbert space allows for
simultaneous approximation of the drift and control effectiveness components of
the control affine system. Several experiments are performed to demonstrate the
effectiveness of the approach.
