# A HJB-POD approach for the control of nonlinear PDEs on a tree structure

## TL;DR

This paper introduces a novel approach combining Hamilton-Jacobi-Bellman equations, model order reduction, and tree-structured algorithms to efficiently control nonlinear PDEs, with proven convergence and demonstrated numerical effectiveness.

## Contribution

It extends a tree-structured numerical method to nonlinear 2D PDEs, incorporating model order reduction and providing convergence guarantees.

## Key findings

- The method effectively reduces computational complexity.
- Numerical tests confirm the efficiency of the approach.
- Theoretical error estimates guarantee convergence.

## Abstract

The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation. Here, we aim to test the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.