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.

翻译：在非线性动态的最佳反馈控制中产生的高维、完全非线性汉密尔顿-Jacobi-Bellman方程式的溶解分解方法。该方法将数值函数的温度列列近似近似值与解决由此产生的非线性系统的牛顿式迭接方法结合起来。高度近似值导致在维度上多米缩放,部分绕过了维度的诅咒。对线性赤道方程式的趋同分析。对于非线性动态,在Allen-Cahn和Fokker-Planck方程式与100个变量的最佳反馈稳定中评估了高度控制合成方法的有效性。