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

Authors

  • Wan Noor Afifah Wan Ahmad Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia
  • Suliadi Firdaus Sufahani Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia
  • Mahmod Abd Hakim Mohamad Department of Mechanical Engineering, Center of Diploma, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia
  • Mohd Saifullah Rusiman Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia
  • Mohd Zulariffin Md Maarof Department of Sciences and Mathematics, Center of Diploma, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia
  • Muhamad Ali Imran Kamarudin School of Business Management, College of Business, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia
  • Naufal Ishartono Faculty of Education, Universitas Muhammadiyah Surakarta, Kabupaten Sukoharjo, Jawa Tengah 57169, Indonesia
  • Kah Howe Lee Ecube Global Pte Ltd, 1 Claymore Drive, 08-03, Orchard Tower Rear Block APT, 229594 Singapore
  • Asmala Ahmad Faculty of Information and Communication Technology, Universiti Teknikal Malaysia Melaka, 76100 Durian Tunggal, Melaka, Malaysia

DOI:

https://doi.org/10.37934/araset.58.1.3348

Keywords:

Discretization method, non-classical optimal control problem, piecewise royalty payment, shooting method

Abstract

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.

Downloads

Download data is not yet available.

Author Biographies

Wan Noor Afifah Wan Ahmad, Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia

hw180026@student.uthm.edu.my

Suliadi Firdaus Sufahani, Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia

suliadi@uthm.edu.my

Mahmod Abd Hakim Mohamad, Department of Mechanical Engineering, Center of Diploma, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia

hakim@uthm.edu.my

Mohd Saifullah Rusiman, Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia

saifulah@uthm.edu.my

Mohd Zulariffin Md Maarof, Department of Sciences and Mathematics, Center of Diploma, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Muar, Johor, Malaysia

zulariffin@uthm.edu.my

Muhamad Ali Imran Kamarudin, School of Business Management, College of Business, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia

aliimran@uum.edu.my

Naufal Ishartono, Faculty of Education, Universitas Muhammadiyah Surakarta, Kabupaten Sukoharjo, Jawa Tengah 57169, Indonesia

naufal.ishartono@ums.ac.id

Kah Howe Lee , Ecube Global Pte Ltd, 1 Claymore Drive, 08-03, Orchard Tower Rear Block APT, 229594 Singapore

kahhowe.lee@ecubed.me

Asmala Ahmad, Faculty of Information and Communication Technology, Universiti Teknikal Malaysia Melaka, 76100 Durian Tunggal, Melaka, Malaysia

asmala@utem.edu.my

Downloads

Published

2024-10-09

How to Cite

Wan Ahmad, W. N. A., Sufahani, S. F., Mohamad, M. A. H., Rusiman, M. S., Md Maarof, M. Z., Kamarudin, M. A. I., Ishartono, N., Lee , K. H., & Ahmad, A. (2024). Navigating Non-Classical Optimal Control Problem: A Hybrid Shooting Approach with Discretization Validation. Journal of Advanced Research in Applied Sciences and Engineering Technology, 33–48. https://doi.org/10.37934/araset.58.1.3348

Issue

Section

Articles

Most read articles by the same author(s)