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We introduce the tensor numerical method for solution of the $d$-dimensionaloptimal control problems with fractional Laplacian type operators inconstraints discretized on large $n^{\otimes d}$ tensor-product Cartesiangrids. The approach is based on the rank-structured approximation of the matrixvalued functions of the corresponding fractional finite difference Laplacian.We solve the equation for the control function, where the system matrixincludes the sum of the fractional $d$-dimensional Laplacian and its inverse.The matrix valued functions of discrete Laplace operator on a tensor grid arediagonalized by using the fast Fourier transform (FFT). Then the low rankapproximation of the $d$-dimensional tensors obtained by folding of thecorresponding large diagonal matrices of eigenvalues are computed, which allowsto solve the governing equation for the control function in a tensor-structuredformat. The existence of low rank canonical approximation to the class ofmatrix valued functions involved is justified by using the sinc quadratureapproximation method applied to the Laplace transform of the generatingfunction. The linear system of equations for the control function is solved bythe PCG iterative method with the rank truncation at each iteration step, wherethe low Kronecker rank preconditioner is precomputed. The right-hand side, thesolution vector, and the governing system matrix are maintained in therank-structured tensor format which beneficially reduces the numerical cost to$O(n\log n)$, outperforming the standard FFT based methods of complexity$O(n^3\log n)$ for 3D case. Numerical tests for the 2D and 3D control problemsconfirm the linear complexity scaling of the method in the univariate grid size$n$.

Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints