  We introduce the tensor numerical method for solution of the $d$-dimensional
optimal control problems with fractional Laplacian type operators in
constraints discretized on large $n^{\otimes d}$ tensor-product Cartesian
grids. The approach is based on the rank-structured approximation of the matrix
valued functions of the corresponding fractional finite difference Laplacian.
We solve the equation for the control function, where the system matrix
includes the sum of the fractional $d$-dimensional Laplacian and its inverse.
The matrix valued functions of discrete Laplace operator on a tensor grid are
diagonalized by using the fast Fourier transform (FFT). Then the low rank
approximation of the $d$-dimensional tensors obtained by folding of the
corresponding large diagonal matrices of eigenvalues are computed, which allows
to solve the governing equation for the control function in a tensor-structured
format. The existence of low rank canonical approximation to the class of
matrix valued functions involved is justified by using the sinc quadrature
approximation method applied to the Laplace transform of the generating
function. The linear system of equations for the control function is solved by
the PCG iterative method with the rank truncation at each iteration step, where
the low Kronecker rank preconditioner is precomputed. The right-hand side, the
solution vector, and the governing system matrix are maintained in the
rank-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 problems
confirm the linear complexity scaling of the method in the univariate grid size
$n$.
