Fourier Approximations of Optimal Control Surfaces and Their Applications
Open Access
- Author:
- Reimao Maggi Nicolosi Rocha, Gabriel
- Graduate Program:
- Industrial Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 06, 2023
- Committee Members:
- Steven Landry, Program Head/Chair
Christopher Griffin, Co-Chair & Dissertation Advisor
Constantino Lagoa, Outside Unit, Field & Minor Member
Terry Friesz, Co-Chair & Dissertation Advisor
Brian Swenson, Special Member
Paul Griffin, Major Field Member - Keywords:
- Optimal Control
Numerical Optimal Control
Fourier Approximation
Evolutionary Games
Odd-Circulant Games
Learning
Direct Methods in Optimal Control
Fourier Neural Network
Augmented Lagrangian
Optimal Control Surfaces
Stochastic Bass Equation
Skew-Symmetric Evolutionary Games
Global Collocation - Abstract:
- We introduce a multidimensional Fourier series approximation of optimal control and trajectory surfaces. Inspired by pseudospectral methods in numerical optimal control and by the architecture of feedforward neural networks, we explore the use of a truncated Fourier decomposition to approximate initial condition and time-dependent optimal control laws resulting from the solution of optimal control problems (OCPs). Two first-order, backpropagation-inspired algorithms are introduced, wherein the infinite dimensional OCP is translated to a finite, unconstrained optimization problem. The use of automatic differentiation to perform gradient computation places our approach within a machine learning framework. Furthermore, our approximation scheme is shown to be robust under stochastic dynamics for the case of additive gaussian noise. Computational experiments for OCPs arising from evolutionary games, such as 2x2 skew-symmetric and odd-circulant games demonstrate the effectiveness of the proposed approach. Finally, mean square error bounds for the one and two-dimensional truncated Fourier series are derived.