Modified Shooting with Discretization Validation for Non-Standard Optimal Control Problem: Case Study for Four-Stage Royalty Payment Problem

A modified shooting method augmented by discretization validation is presented in this paper to address the challenges inherent in non-standard optimal control problems. Specifically, a specific case study involving four-stage royalty payment functions is focused on, with the aim of effectively optimizing these complex, non-differentiable functions. The shooting method is adapted and enhanced to compute optimal solutions, and its accuracy is rigorously confirmed through discretization techniques. The primary objectives involve maximizing the performance index and determining optimal control strategies for scenarios where the final state variable remains unknown. In this study, the royalty payment is defined as a four-stage piecewise function. The incorporation of piecewise royalty functions introduces non-differentiability at specific time intervals, necessitating innovative approaches for finding optimal solutions. This, in turn, leads to the utilization of the continuous hyperbolic tangent (tanh) function to address the non-differentiability issue. A hybrid shooting method, combining the Newton and Golden Section Search methods, is employed in the C++ programming language to compute the unknown final state value. A new natural boundary condition, based on established theory, is introduced to further facilitate the investigation. Discretization methods such as Euler, Runge-Kutta, Trapezoidal, and Hermite-Simpson approximations are employed for validation. The validation process entails the use of the AMPL programming language with the MINOS solver. Comparative analyses reveal that the modified shooting method yields more accurate optimal results than discretization methods, thus demonstrating the method’s effectiveness in addressing non-standard optimal control problems. The significance of fundamental theories in addressing real-world challenges is underscored by this research, providing valuable insights for future researchers exploring mathematical approaches in similar contexts. The study contributes to the continued relevance of the academic field, particularly in science and mathematics education.