# Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations

## TL;DR

This paper introduces a tensor decomposition method using tensor trains and Newton-like iterations to efficiently solve high-dimensional Hamilton-Jacobi-Bellman equations in optimal control, reducing computational complexity.

## Contribution

It presents a novel tensor train-based approach combined with iterative solving for high-dimensional HJB equations, addressing the curse of dimensionality.

## Key findings

- Effective in stabilizing Allen-Cahn and Fokker-Planck equations with over a hundred variables.
- Partial theoretical convergence analysis provided for linear-quadratic cases.
- Demonstrates polynomial scaling with respect to problem dimension.

## Abstract

A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton-like iterative method for the solution of the resulting nonlinear system. The tensor approximation leads to a polynomial scaling with respect to the dimension, partially circumventing the curse of dimensionality. A convergence analysis for the linear-quadratic case is presented. For nonlinear dynamics, the effectiveness of the high-dimensional control synthesis method is assessed in the optimal feedback stabilization of the Allen-Cahn and Fokker-Planck equations with a hundred of variables.