A Tensor Decomposition Approach for High-Dimensional Hamilton-Jacobi-Bellman Equations
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A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control and estimation of nonlinear dynamics is presented. The proposed 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 effectiveness of tensor approximations circumvents the curse of dimensionality, solving Hamilton-Jacobi equations with more than 100 dimensions at modest cost. The linear scaling of the computational complexity with respect to the dimension allows to solve PDE-constrained optimal feedback control problems over high-dimensional state spaces. Numerical tests including the control of a 2D nonlinear reaction equation and the stabilization of a bilinear Fokker-Planck equation are presented.
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