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

## TL;DR

This paper introduces a numerical method for designing robust feedback controllers for high-dimensional nonlinear PDEs using Hamilton-Jacobi-Isaacs equations, enabling stabilization and disturbance rejection.

## Contribution

It develops a separable polynomial approximation approach to solve high-dimensional Hamilton-Jacobi-Isaacs PDEs for robust control of nonlinear PDEs.

## Key findings

- Effective stabilization of systems up to 12 dimensions
- Robust controllers achieve optimal stabilization and disturbance rejection
- Applicable to nonlinear parabolic PDEs with uncertainties

## Abstract

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 $\cH_{\infty}$ 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 $d\approx 12$. 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.