ИСТИНА

In this report, we present the new statement of the optimal matrix Fourier filtering problem and give its mathematical basis: existence and Lipschitz continuous dependence on the filter for the energy class solution of the FDE, solvability of optimal filtering problem for various classes of matrix Fourier filters, and convergence of the gradient projection method for optimization of the target functional. We apply the Andronov-Hopf bifurcation approach to demonstrate wide range of possibilities for the use of matrix Fourier filtering as a novel tool for controlled pattern formation. We obtain general conditions on matrix filters giving rise to rotating or oscillating waves, present descriptive examples of matrix filters that provide the excitation of nontrivial bifurcation solutions, and demonstrate close correspondence between the results of direct numerical simulation and theoretically predicted shapes of the solutions.