# Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs

arXiv: https://arxiv.org/abs/1702.04400 · HTML: https://mlsanity.com/paper/1702.04400/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2017.

## Built on by (most cited first, 10 of 10)

- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation](https://mlsanity.com/paper/2112.14415/) (2021)
- [Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-dependent Riccati Equations](https://mlsanity.com/paper/2103.04091/) (2021)
- [Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2105.03336/) (2021)
- [Neural Network Optimal Feedback Control with Guaranteed Local Stability](https://mlsanity.com/paper/2205.00394/) (2022)
- [Actor-Critic Methods using Physics-Informed Neural Networks: Control of a 1D PDE Model for Fluid-Cooled Battery Packs](https://mlsanity.com/paper/2305.10952/) (2023)
- [HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models](https://mlsanity.com/paper/2409.09614/) (2024)
- [Neural network optimal feedback control with enhanced closed loop stability](https://mlsanity.com/paper/2109.07466/) (2021)
- [Bridging Autoencoders and Dynamic Mode Decomposition for Reduced-order Modeling and Control of PDEs](https://mlsanity.com/paper/2409.06101/) (2024)
- [Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs](https://mlsanity.com/paper/2606.23827/) (2026)

## Abstract (arXiv)

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the Successive Galerkin Approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear Generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.