Navigating Non-Classical Optimal Control Problem: A Hybrid Shooting Approach with Discretization Validation

This paper presents a solution to the challenges encountered in the non-classical Optimal Control (OC) problems, where the final state variable is unknown, leading to a non-zero final shadow value. The main objective is to maximize the performance index, but the presence of a piecewise royalty function in the performance index makes it non-differentiable at certain time frames. Therefore, we adopted a continuous approach using the hyperbolic tangent (tanh) function to overcome this difficulty. To compute the unknown final state value, we employed a hybrid shooting method, which combines the Newton and Brent methods implemented in the C++ programming language. Since the final shadow value is non-zero, a new equation is mathematically required to continue the investigation. Thusly, we established a new natural boundary condition based on the fundamental theory proposed by previous researchers. At the same time, the validation process involved discretization methods such as Euler, Runge-Kutta, Trapezoidal and Hermite-Simpson approximations. The program was constructed in AMPL programming language with MINOS solver during the validation process. This study applied the proposed methods to fixed and three-stage piecewise royalty payments. The study expects the hybrid shooting method to produce a more accurate optimal result than the discretization method at the end of the investigation. This research highlights the significance of the fundamental theory in tackling real-world problems. In addition, the technique used here can serve as a stepping stone for future researchers exploring new mathematical approaches in real-world problem-solving. While at the same time, this ensures that the method remains up-to-date, making the academic field relevant for teaching and learning processes, especially in the domains of science and mathematics.