Approximation of Optimal Control Surfaces for $2\times 2$ Skew-Symmetric Evolutionary Game Dynamics

TL;DR

This paper introduces a Fourier-based approximation method for solving optimal control problems in $2\times 2$ skew-symmetric evolutionary games, integrating neural network techniques for improved solution fitting.

Contribution

It generalizes orthogonal function approximation for optimal control and incorporates neural network back-propagation in this context.

Findings

Effective approximation demonstrated on example

Generalization of prior orthogonal function methods

Integration of neural network techniques enhances solution fitting

Abstract

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.