# Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations

arXiv: https://arxiv.org/abs/1905.06276 · HTML: https://mlsanity.com/paper/1905.06276/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2019.

## Built on by (most cited first, 6 of 6)

- [Lessons from AlphaZero for Optimal, Model Predictive, and Adaptive Control](https://mlsanity.com/paper/2108.10315/) (2021)
- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [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)
- [Rates of Convergence in Certain Native Spaces of Approximations used in Reinforcement Learning](https://mlsanity.com/paper/2309.07383/) (2023)
- [Tailoring Reproducing Kernels for Optimal Control via Policy Iteration](https://mlsanity.com/paper/2510.22494/) (2025)

## Abstract (arXiv)

We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reduced-order model, we construct a robust feedback control based on the cH∞ control method, which requires the solution of an associated high-dimensional Hamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is tackled by means of a separable representation of the control system, and a polynomial approximation ansatz for the corresponding value function. Our method proves to be effective for the robust stabilization of nonlinear dynamics up to dimension d≈ 12. We assess the robustness and optimality features of our design over a class of nonlinear parabolic PDEs, including nonlinear advection and reaction terms. The proposed design yields a feedback controller achieving optimal stabilization and disturbance rejection properties, along with providing a modelling framework for the robust control of PDEs under parametric uncertainties.