A new discretization method for solving an optimal vibrating controlled system on an arbitrary domain

Abstract

By setting a partition on the domain and approximating each part with a sector of a suitable circle, the system solution was identified as a trigonometric series with unknown coefficients. After bringing into account the harmony and smoothness of the solution, the problem was transferred into a new linear one in which it was involved with radon measures. Existence of the optimal solution for the new problem was proved automatically. Then, by some discretization schemes, it was also shown that how the optimal classical trajectory and control were identified simultaneously via the results of a finite linear programming. To see the advantages of this general, simple and linear new method, a numerical simulation was also presented.