NON-CLASSICAL OPTIMAL CONTROL PROBLEM: A CASE STUDY FOR CONTINUOUS APPROXIMATION OF FOUR-STEPWISE FUNCTION

Wan  Noor Afifah Wan Ahmad; Suliadi Firdaus Sufahani; Mahmod  Abd Hakim Mohamad; Mohd SaifullahRusiman; Mohd  Zulariffin Md Maarof; Muhamad  Ali lmran Kamarudin

Wan Noor Afifah Wan Ahmad Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Higher Educational Hub, 84600 Pagoh, Johor, Malaysia
Suliadi Firdaus Sufahani Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Higher Educational Hub, 84600 Pagoh, Johor, Malaysia
Mahmod Abd Hakim Mohamad Department of Mechanical Engineering, Center of Diploma, Universiti Tun Hussein Onn Malaysia, Pagoh Higher Educational Hub, 84600 Pagoh, Johor, Malaysia
Mohd SaifullahRusiman Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Higher Educational Hub, 84600 Pagoh, Johor, Malaysia
Mohd Zulariffin Md Maarof Department of Sciences and Mathematics, Center of Diploma, Universiti Tun Hussein Onn Malaysia, Pagoh Higher Educational Hub, 84600 Pagoh, Johor, Malaysia
Muhamad Ali lmran Kamarudin School of Business Management, College of Business, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia

Article ID: 2417

Keywords: discretization method; optimal control; shooting method

Abstract

The numerical properties of a contemporary optimal control problem (OCP) within the realm of financial aspects deviate from the conventional OCP framework. In our specific scenario, the final state condition is unknown, while the integrand exhibits a piecewise capacity that aligns with the unknown terminal state value. Since this is not a classical OCP, it cannot be solved using Pontryagin’s maximum approach under the expected end limit conditions. A free final state in the non-classical issue results in a critical limit condition of the final shadow value not being equal to zero. The new fundamental condition must be comparable to a particular necessary condition because the integrand is a part of the unidentified final state value. By employing the hyperbolic tangent (tanh) function, we showcase a continuous approximation of the piecewise constant integrand function. Furthermore, we tackle a specific scenario utilizing the shooting method in C++ programming language. This is by combining the Newton and Golden Section Search methods in the shooting technique to calculate the limiting free final state value in an external circle emphasis. Discretization methods such as Euler and Runge-Kutta approximations were used in the validation procedure. The program was constructed in AMPL programming language with MINOS solver.

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