Solving a non‐standard Optimal Control royalty payment problem using a new modified shooting method

• Published in: Mathematical methods in the applied sciences 2025-01, Vol.48 (2), p.2665-2685
• Main Authors: Sufahani, Suliadi Firdaus, Ahmad, Wan Noor Afifah Wan, Jacob, Kavikumar, Shafie, Sharidan, Rahim, Ruzairi Abdul, Mohamad, Mahmod Abd Hakim, Rusiman, Mohd Saifullah, Roslan, Rozaini, Maarof, Mohd Zulariffin Md, Kamarudin, Muhamad Ali Imran
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Continuity (mathematics); direct method; economic application; Hyperbolic functions; indirect method; non‐standard optimal control; Optimal control; Performance indices; Pontryagin minimum principle; royalty problem; Runge-Kutta method
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.10457

This paper considers a non‐standard Optimal Control problem that has an application in economics. The primary focus of this research is to solve the royalty problem, which has been categorized as a non‐standard Optimal Control problem, where the final state value and its functional performance index value are unknown. A new continuous necessary condition is investigated for the final state value so that it will convert the final costate value into a non‐zero value. The research analyzes the seven‐stage royalty piecewise function, which is then approximated to continuous form with the help of the hyperbolic tangent function and solves the problem by using a new modified shooting method. This modified shooting method applies Sufahani–Ahmad–Newton–Brent–Royalty Algorithm and Sufahani‐Ahmad‐Powell‐Brent‐Royalty Algorithm. For a validation process, the results are compared with the existing methods such as Euler, Runge–Kutta, Trapezoidal, and Hermite–Simpson approximations, and the results show that the proposed method yields an accurate terminal state value.