Dynamical low‐rank approximations of solutions to the Hamilton–Jacobi–Bellman equation

Abstract

We present a novel method to approximate optimal feedback laws for nonlinear optimal control based on low‐rank tensor train (TT) decompositions. The approach is based on the Dirac–Frenkel variational principle with the modification that the optimization uses an empirical risk. Compared to current state‐of‐the‐art TT methods, our approach exhibits a greatly reduced computational burden while achieving comparable results. A rigorous description of the numerical scheme and demonstrations of its performance are provided.

Editorial: Tensor numerical methods and their application in scientific computing and data science

“…A number of interesting numerical experiments are presented to demonstrate the benefit of over-relaxation in low-rank optimization. In particular the problems of matrix completion, low-rank solution of the Lyapunov matrix equation and solution of linear systems in the QTT tensor format have been considered.A novel method to approximate optimal feedback laws in the problem of nonlinear optimal control for dynamical systems by using the low rank TT tensor decomposition is proposed by Eigel et al 2 The feedback control is ubiquitous in real dynamical systems since the controlled system can in general not be expected to follow model predictions exactly. However, the computing an optimal feedback control law for nonlinear systems is inherently difficult since it requires solution of the so-called Hamilton-Jacobi-Bellmann equation, which is a high-dimensional nonlinear parabolic partial differential equation.…”

“…A novel method to approximate optimal feedback laws in the problem of nonlinear optimal control for dynamical systems by using the low rank TT tensor decomposition is proposed by Eigel et al 2 The feedback control is ubiquitous in real dynamical systems since the controlled system can in general not be expected to follow model predictions exactly. However, the computing an optimal feedback control law for nonlinear systems is inherently difficult since it requires solution of the so‐called Hamilton–Jacobi–Bellmann equation, which is a high‐dimensional nonlinear parabolic partial differential equation.…”