Approximation of extremal solution of non-Fourier moment problem and optimal control for non-homogeneous vibrating systems

Trigonometric non-Fourier moment problems arise as a result of various control problem study. In current paper, the extremal solution, i.e. the one with the least L2-norm is searched for. Proposed is an algorithm that allows to change an infinite system of equations into the linear one with only a finite number of equations. The mentioned algorithm is based on the fact, that in the case of a Fourier moment problem, the extremal solution is periodic and easy to construct. The extremal solution of a non-Fourier moment problem close to a Fourier one is approximated by a sequence of solutions with periodicity disturbed in a finite number of equations. It is proved that this sequence of approximations converges to the desired extremal solution. The paper is concluded with the particular example whose consideration leads to a moment problem elaborated in the first part of the article.