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

## TL;DR

This paper introduces a numerical method combining pseudospectral collocation and iterative Galerkin approximation to solve high-dimensional Hamilton-Jacobi-Bellman equations for feedback control of semilinear parabolic PDEs, achieving stabilization in systems up to 14 dimensions.

## Contribution

It presents a novel computational approach for high-dimensional HJB equations using polynomial approximation and iterative methods, enabling feedback control synthesis for complex PDE systems.

## Key findings

- Successfully approximated HJB equations in systems up to 14 dimensions.
- Developed a stable iterative scheme for nonlinear HJB equations.
- Demonstrated effectiveness in feedback control of semilinear parabolic PDEs.

## Abstract

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.