Nonstandard optimal control problem: case study in an economical application of royalty problem

This paper's focal point is on the nonstandard Optimal Control (OC) problem. In this matter, the value of the final state variable, y(T) is said to be unknown. Moreover, the Lagrangian integrand in the function is in the form of a piecewise constant integrand function of the unknown state value y(T). In addition, the Lagrangian integrand depends on the y(T) value. Thus, this case is considered as the nonstandard OC problem where the problem cannot be resolved by using Pontryagin’s Minimum Principle along with the normal boundary conditions at the final time in the classical setting. Furthermore, the free final state value, y(T) in the nonstandard OC problem yields a necessary boundary condition of final costate value, p(T) which is not equal to zero. Therefore, the new necessary condition of final state value, y(T) should be equal to a certain continuous integral function of y(T)=z since the integrand is a component of y(T). In this study, the 3-stage piecewise constant integrand system will be approximated by utilizing the continuous approximation of the hyperbolic tangent (tanh) procedure. This paper presents the solution by using the computer software of C++ programming and AMPL program language. The Two-Point Boundary Value Problem will be solved by applying the indirect method which will involve the shooting method where it is a combination of the Newton and the minimization algorithm (Golden Section Search and Brent methods). Finally, the results will be compared with the direct methods (Euler, Runge-Kutta, Trapezoidal and Hermite-Simpson approximations) as a validation process.