A Multidimensional Fourier Approximation of Optimal Control Surfaces

TL;DR

This paper introduces a multidimensional Fourier series approach for approximating optimal control surfaces, employing a modified Augmented Lagrangian algorithm and automatic differentiation, with applications demonstrated in mechanics and game theory.

Contribution

It presents a novel method for approximating control surfaces using multivariable Fourier series and a modified optimization algorithm, including error bounds and computational techniques.

Findings

Effective approximation of control surfaces demonstrated in mechanics and game theory.

Automatic differentiation reduces gradient computation complexity.

Derived mean square error bounds for Fourier series approximations.

Abstract

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.