Solving Infinite-Horizon Optimal Control Problems using the Extreme Theory of Functional Connections

TL;DR

This paper introduces a hybrid machine learning method combining the Theory of Functional Connections and Extreme Learning Machines to efficiently solve the Hamilton-Jacobi-Bellman PDE for infinite-horizon optimal control, with applications to spacecraft control.

Contribution

It develops a novel approach using X-TFC to solve HJB PDEs efficiently, ensuring boundary conditions are analytically satisfied and reducing training costs compared to PINNs.

Findings

Successfully applied to linear and nonlinear systems with known solutions.

Demonstrated effectiveness on spacecraft de-tumbling control tasks.

Reduced training time compared to traditional PINNs.

Abstract

This paper presents a physics-informed machine learning approach for synthesizing optimal feedback control policy for infinite-horizon optimal control problems by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation(PDE). The optimal control policy is derived analytically for affine dynamical systems with separable and strictly convex control costs, expressed as a function of the gradient of the value function. The resulting HJB-PDE is then solved by approximating the value function using the Extreme Theory of Functional Connections (X-TFC) - a hybrid approach that combines the Theory of Functional Connections (TFC) with the Extreme Learning Machine (ELM) algorithm. This approach ensures analytical satisfaction of boundary conditions and significantly reduces training cost compared to traditional Physics-Informed Neural Networks (PINNs). We benchmark the method on linear and non-linear systems with known analytical solutions as well as demonstrate its effectiveness on control tasks such as spacecraft optimal de-tumbling control.