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.
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