Optimal control of non-homogeneous wave equation

摘要

While studying vibrations of non-homogeneous strings or chains a trigonometric non-Fourier moment problems arise. The existence of solutions of such problems is still researched by many authors. In current note, a particular solution, called optimal, 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 optimal solution is periodic and easy to construct. The optimal 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 solution sought for. The note is concluded with the application of proposed algorithm.