  This work considers the problem of approximating initial condition and
time-dependent optimal control and trajectory surfaces using multivariable
Fourier series. A modified Augmented Lagrangian algorithm for translating the
optimal control problem into an unconstrained optimization one is proposed and
two problems are solved: a quadratic control problem in the context of
Newtonian mechanics, and a control problem arising from an odd-circulant game
ruled by the replicator dynamics. Various computational results are presented.
Use of automatic differentiation is explored to circumvent the elaborated
gradient computation in the first-order optimization procedure. Furthermore,
mean square error bounds are derived for the case of one and two-dimensional
Fourier series approximations, inducing a general bound for problems of $n$
dimensions.
