SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A1625-A1650, January 2021.
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.

中文翻译：

---

高维Hamilton-Jacobi-Bellman方程的张量分解方法

SIAM科学计算杂志，第43卷，第3期，第A1625-A1650页，2021年1月。
提出了一种张量分解方法，用于求解非线性动力学的最优反馈控制中产生的高维，完全非线性Hamilton-Jacobi-Bellman方程。该方法将针对值函数的张量列逼近与用于求解所得非线性系统的牛顿式迭代方法结合在一起。张量逼近导致相对于维度的多项式缩放，部分避开了维度的诅咒。给出了线性二次情形的收敛性分析。对于非线性动力学，在具有数百个变量的Allen-Cahn和Fokker-Planck方程的最佳反馈稳定性中评估了高维控制综合方法的有效性。