Direct and Indirect Methods for the Non-Standard Optimal Control Problem

Abstract

This paper comprehensively investigated direct and indirect methods to address the non-standard Optimal Control (OC) problem, specifically focusing on a case study involving piecewise function. Non-standard OC problems, deviating from standard formulations, present unique challenges that require specialized solution techniques. The primary aim of this research is to maximize the objective function. However, the presence of a piecewise function introduced non-differentiability at certain timeframes during the optimization process. Additionally, the unknown final state necessitates innovative approaches to find optimal solutions, as it leads to a non-zero costate value at the terminal time. The indirect approach was applied to handle these complexities; specifically, the shooting method, implemented in the C++ programming language, to compute the optimal solution. The case study centred around a non-standard OC problem concerning a two-stage piecewise function. To overcome the challenge of the non-differentiable piecewise function, a continuous approach was implemented by using hyperbolic tangent (tanh) modelling. We validated the performances with the direct methods, such as Euler and Runge-Kutta, through numerical experiments and comparisons—the validation of optimal results with direct methods involved utilizing the AMPL programming language with the MINOS solver. The direct method consists in parameterizing the control and state trajectories, transforming the OC problem into a nonlinear programming (NLP) problem. On the other hand, the indirect method utilizes Pontryagin’s Minimum Principle to derive the necessary conditions for optimality by solving the associated Two-Point Boundary Value Problem (TPBVP). The findings from the case study demonstrate that the indirect method yields a more accurate solution. The insights gained from this investigation contribute to a better understanding of OC techniques in non-standard scenarios, guiding researchers and practitioners in selecting appropriate methods for similar real-world problems. Moreover, this research enhances the comprehension of OC methods’ applicability to non-standard challenges, particularly in the domain of piecewise function problems. The outcomes of this study offer valuable insights for addressing complex optimization challenges in various disciplines and pave the way for further advancements in solving real-world non-standard OC problems.