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

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. Received: July 11, 2023Accepted: August 19, 2023