Abstract
In relation to the non-standard Optimal Control problem, this paper focuses on maximizing the performance index. The model equation involved the royalty payment function that cannot be differentiated at a certain level. In addition, the state and costate at the final time are said to be unspecified and not equal to zero, respectively. This difficulty was avoided by using a continuous approach. The discrete royalty function is converted in this instance using the hyperbolic tangent function. After that, the issue was resolved indirectly by integrating the shooting method with the Newton and Golden Section Search methods. The program is constructed by using the C++ programming language. The shooting result was then compared with the indirect method, which involved the Runge–Kutta method. Both methods are expected to yield an accurate solution that is optimal (maximize) performance index. This brilliant idea can be implemented in real-world problems, especially in the academic field, so that the method used is up to date along with the current need.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ahmad A, Sakidin H, Dahlia A, Zetriuslita Z, Qudsi R (2020) Haze and its impact to paddy monitoring from remote sensing satellites. Int J Adv Trends Comput Sci Eng 9(4)
Betts JT (2010) Practical methods for optimal control using nonlinear programming. Advance in Design & Control, Society for Industrial & Applied Mathematics, Philadelphia, PA
Bryson AE (2018) Applied optimal control: optimization. Routledge, Estimation and Control
Cruz PAF, Torres DFM, Zinober ASI (2010) A non-classical class of variational problems. Int J Math Model Numer Optim 1(3): 227–236. Inderscience Publishers
Fourer R, Gay DM, Kernighan BW (1990) A modelling language for mathematical programming. Manag Sci 36(5):519–554
Malinowska AB, Torres DFM (2010) Natural boundary conditions in the calculus of variations. Math Methods Appl Sci 33(14):1712–1722. Wiley Online Library
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, Cambridge
Spence AM (1981) The learning curves and competition. Bell J Econ JSTOR 49–70
Zinober ASI (2010) Optimal control theory lecture notes (Unpublished). The University of Sheffield
Zinober ASI, Kaivanto K (2008) Optimal production subject to piecewise continuous royalty payment obligations. University of Sheffield
Benjamin, Passenberg Magnus, Kröninger Georg, Schnattinger Marion, Leibold Olaf, Stursberg Martin, Buss (2011) Initialization Concepts for Optimal Control of Hybrid Systems IFAC Proceedings Volumes 44(1) 10274-10280 10.3182/20110828-6-IT-1002.03012
Passenberg, B. (2012). Theory and Algorithms for Indirect Methods in Optimal Control of Hybrid Systems. Technische Universität München: Ph. D. Thesis.
Stryk OV & Bulirsch R (1992). Direct and Indirect Methods for Trajectory Optimization. Annals of Operational Research, 37(1). Springer, pp. 357-373.
Benson DA, Huntington GT, Thorvaldsen TP & Rao AV (2006). Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method. Journal of Guidance, Control & Dynamics, 29(6). pp. 1435-1440.
Acknowledgement
This research was supported by the Ministry of Higher Education (MOHE) through Fundamental Research Grant Scheme (FRGS/1/2021/STG06/UTHM/03/3). Thank you to Research Management Center (RMC), Universiti Tun Hussein Onn Malaysia (UTHM), for managing the research and publication process.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Ahmad, W.N.A.W. et al. (2023). Optimal Control Problem With Non-Standard Conditions: Direct and Indirect Approaches. In: Kaiser, M.S., Waheed, S., Bandyopadhyay, A., Mahmud, M., Ray, K. (eds) Proceedings of the Fourth International Conference on Trends in Computational and Cognitive Engineering. Lecture Notes in Networks and Systems, vol 618. Springer, Singapore. https://doi.org/10.1007/978-981-19-9483-8_10
Download citation
DOI: https://doi.org/10.1007/978-981-19-9483-8_10
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-19-9482-1
Online ISBN: 978-981-19-9483-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)