Optimal Feedback Law Recovery by Gradient-Augmented Sparse Polynomial Regression

TL;DR

This paper introduces a sparse polynomial regression method leveraging gradient information to efficiently recover high-dimensional optimal feedback laws in nonlinear control, reducing sample complexity and model complexity.

Contribution

It presents a novel approach combining Hamilton-Jacobi-Bellman PDEs and Pontryagin's Maximum Principle with LASSO regression to improve feedback law computation.

Findings

Gradient information reduces training sample size.

Sparse polynomial models yield lower complexity feedback laws.

Method performs well in high-dimensional nonlinear control problems.

Abstract

A sparse regression approach for the computation of high-dimensional optimal feedback laws arising in deterministic nonlinear control is proposed. The approach exploits the control-theoretical link between Hamilton-Jacobi-Bellman PDEs characterizing the value function of the optimal control problems, and first-order optimality conditions via Pontryagin's Maximum Principle. The latter is used as a representation formula to recover the value function and its gradient at arbitrary points in the space-time domain through the solution of a two-point boundary value problem. After generating a dataset consisting of different state-value pairs, a hyperbolic cross polynomial model for the value function is fitted using a LASSO regression. An extended set of low and high-dimensional numerical tests in nonlinear optimal control reveal that enriching the dataset with gradient information reduces the number of training samples, and that the sparse polynomial regression consistently yields a feedback law of lower complexity.