A Solution Method for an Optimal Controlled Vibrating Circle Shell by Measure and Classical Trajectory

Abstract

The mean idea of this paper is to present a new combinatorial solution technique for the controlled vibrating circle shell systems. Based on the classical results of the wave equations on circle domains, the trajectory is considered as a finite trigonometric series with unknown coefficients in polar coordinates. Then, the problem is transferred to one in which its unknowns are a positive Radon measure and some positive coefficients. Extending the underlying space helps us to prove the existence of the solution. By using the density properties and some approximation schemes, the problem is deformed into a finite linear programming and the nearly optimal trajectory and control are identified simultaneously. A numerical example is also given.