Optimal feedback law recovery by gradient-augmented sparse polynomial regression

Azmi, Behzad; Kalise, Dante; Kunisch, Karl

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. © 2021 Behzad Azmi, Dante Kalise, and Karl Kunisch.

Citation

Azmi, B., Kalise, D., & Kunisch, K. (2021). Optimal feedback law recovery by gradient-augmented sparse polynomial regression. Journal of Machine Learning Research, 22, 1-32

• Journal Article Type: Article
• Acceptance Date: Dec 18, 2020
• Publication Date: Jan 1, 2021
• Deposit Date: Feb 26, 2021
• Publicly Available Date: Mar 12, 2021
• Journal: Journal of Machine Learning Research
• Print ISSN: 1532-4435
• Electronic ISSN: 1533-7928
• Publisher: Journal of Machine Learning Research
• Peer Reviewed: Peer Reviewed
• Volume: 22
• Pages: 1-32
• Public URL: https://nottingham-repository.worktribe.com/output/5352243
• Publisher URL: http://jmlr.org/papers/v22/20-755.html