  In this paper we study the problem of approximating the general solution to
an optimal control problem whose dynamics arise from a $2\times 2$
skew-symmetric evolutionary game with arbitrary initial condition. Our approach
uses a Fourier approximation method and generalizes prior work in the use of
orthogonal function approximation for optimal control. At the same time we cast
the fitting problem in the context of a non-standard feedforward neural network
and derive the back-propagation operator in this context. An example of the
efficacy of this approach is provided and generalizations are discussed.
