Shooting and Discretization Method in Settling the Royalty Payment Problem

Abstract

Royalty payment has become one of the sources of income today. This research involved a non-classical Optimal Control problem (OCP), which is an economic application of the royalty problem. The first condition applied when the final state variable was unknown. The main goal was to maximize the functional performance index. However, the performance index was in terms of the unknown final state variable. Moreover, the unknown final state value produced a necessary boundary condition of the nonzero final shadow value. In this study, the three-stage royalty function was used and approximated into the continuous approximation of the hyperbolic tangent (tanh) procedure. This paper exhibits the output through shooting and discretization methods by manipulating the C++ and AMPL program language, respectively. The shooting method was constructed by combining the Newton's and Golden Section Search methods. At the end of the study, a validation process was conducted. This was done by comparing the shooting result with the discretization methods such as Euler, Runge–Kutta, Trapezoidal, and Hermite–Simpson methods. It is expected that the shooting method yields a more accurate optimal solution.