Nonstandard optimal control problem: case study in an economical application of royalty problem

(1) Wan Noor Afifah Wan Ahmad Mail (Universiti Tun Hussein Onn Malaysia, Malaysia)
(2) * Suliadi Firdaus Sufahani Mail (Universiti Tun Hussein Onn Malaysia, Malaysia)
(3) Alan Zinober Mail (University of Sheffield, United Kingdom)
(4) Azila M Sudin Mail (Universiti Tun Hussein Onn Malaysia, Malaysia)
(5) Muhaimin Ismoen Mail (Universiti Teknologi Brunei, Brunei Darussalam)
(6) Norafiz Maselan Mail (Universiti Teknologi Malaysia, Malaysia)
(7) Naufal Ishartono Mail (Universitas Muhammadiyah Surakarta, Indonesia)
*corresponding author


This paper's focal point is on the nonstandard Optimal Control (OC) problem. In this matter, the value of the final state variable, y(T) is said to be unknown. Moreover, the Lagrangian integrand in the function is in the form of a piecewise constant integrand function of the unknown state value y(T). In addition, the Lagrangian integrand depends on the y(T) value. Thus, this case is considered as the nonstandard OC problem where the problem cannot be resolved by using Pontryagin’s Minimum Principle along with the normal boundary conditions at the final time in the classical setting. Furthermore, the free final state value, y(T) in the nonstandard OC problem yields a necessary boundary condition of final costate value, p(T) which is not equal to zero. Therefore, the new necessary condition of final state value, y(T) should be equal to a certain continuous integral function of y(T)=z since the integrand is a component of y(T). In this study, the 3-stage piecewise constant integrand system will be approximated by utilizing the continuous approximation of the hyperbolic tangent (tanh) procedure. This paper presents the solution by using the computer software of C++ programming and AMPL program language. The Two-Point Boundary Value Problem will be solved by applying the indirect method which will involve the shooting method where it is a combination of the Newton and the minimization algorithm (Golden Section Search and Brent methods). Finally, the results will be compared with the direct methods (Euler, Runge-Kutta, Trapezoidal and Hermite-Simpson approximations) as a validation process.


Discretization method; Minimization technique; Nonstandard optimal control; Royalty problem; Shooting technique; Two-point boundary value problem



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[1] L. D. Berkovitz, Optimal Control Theory, 1974, vol. 12, doi: 10.1007/978-1-4757-6097-2.

[2] J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2010, doi: 10.1137/1.9780898718577.

[3] D. E. Kirk, Optimal control theory: an introduction. Courier Corporation, 2004, available at: Google Scholar.

[4] F. L. Lewis, D. Vrabie, and V. L. Syrmos, Optimal control. John Wiley & Sons, 2012, available at: Google Scholar.

[5] E. R. Pinch, Optimal control and the calculus of variations. Oxford University Press, 1995, available at: Google Scholar.

[6] W. N. A. W. Ahmad et al., “A comparative study for solving non-classical optimal control problem using euler, runge-kutta and shooting methods,” Far East J. Math. Sci., vol. 102, no. 10, pp. 2447–2458, Nov. 2017, doi: 10.17654/MS102102447.

[7] W. N. A. W. Ahmad, M. S. Rusiman, S. F. Sufahani, A. Zinober, M. Mohammad, and M. G. Kamardan, “A new combination of broyden-fletcher-goldfarb-shanno and brent techniques in shooting method for solving non-classical optimal control problem,” Far East J. Math. Sci., vol. 102, no. 11, pp. 2785–2796, Dec. 2017, doi: 10.17654/MS102112785.

[8] W. N. A. W. Ahmad, S. Sufahani, M. S. Rusiman, and M. Ali, “A Non-Classical Optimal Control Problem,” J. Sci. Technol., vol. 10, no. 1, 2018, available at : Google Scholar.

[9] W. N. A. W. Ahmad et al., “Continuous approximation of 3 stepwise functions in the non-standard optimal control problem,” Far East J. Math. Sci., vol. 102, no. 11, pp. 2797–2807, Dec. 2017, doi: 10.17654/MS102112797.

[10] W. N. A. W. Ahmad et al., “A non-standard optimal control problem using hyperbolic tangent,” Far East J. Math. Sci., vol. 102, no. 10, pp. 2435–2446, Nov. 2017, doi: 10.17654/MS102102435.

[11] U. Ledzewicz and H. Schättler, “Application of optimal control to a system describing tumor anti-angiogenesis,” in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Kyoto, Japan, 2006, pp. 478–484, available at : Google Scholar.

[12] R. L. OLLERTON, “Application of optimal control theory to diabetes mellitus,” Int. J. Control, vol. 50, no. 6, pp. 2503–2522, Dec. 1989, doi: 10.1080/00207178908953512.

[13] S. Sufahani and Z. Ismail, “The statistical analysis of the prevalence of pneumonia for children age 12 in west Malaysian hospital,” Appl. Math. Sci., vol. 8, pp. 5673–5680, 2014, doi: 10.12988/ams.2014.46407.

[14] W. N. A. W. Ahmad, S. F. Sufahani, and A. Zinober, “Solving Royalty Problem Through a New Modified Shooting Method,” Int. J. Recent Technol. Eng., vol. 8, no. 1, pp. 469–475, available at : Google Scholar.

[15] P. A. F. Cruz, D. F. M. Torres, and A. S. I. Zinober, “A non-classical class of variational problems,” Int. J. Math. Model. Numer. Optim., vol. 1, no. 3, p. 227, 2010, doi: 10.1504/IJMMNO.2010.031750.

[16] A. Zinober and S. Sufahani, “A non-standard optimal control problem arising in an economics application,” Pesqui. Operacional, vol. 33, no. 1, pp. 63–71, Apr. 2013, doi: 10.1590/S0101-74382013000100004.

[17] M. S. Rusiman, O. C. Hau, A. W. Abdullah, S. F. Sufahani, and N. A. Azmi, “An analysis of time series for the prediction of barramundi (ikan siakap) price in malaysia,” Far East J. Math. Sci., vol. 102, no. 9, pp. 2081–2093, Nov. 2017, doi: 10.17654/MS102092081.

[18] D. Jayeola, Z. Ismail, S. F. Sufahani, and D. P. Manliura, “Optimal method for investing on assets using black litterman model,” Far East J. Math. Sci., vol. 101, no. 5, pp. 1123–1131, Feb. 2017, doi: 10.17654/MS101051123.

[19] J. Z. Ben-Asher, Optimal Control Theory with Aerospace Applications, 2010, doi: 10.2514/4.867347, available at :

[20] E. Trélat, “Optimal Control and Applications to Aerospace: Some Results and Challenges,” J. Optim. Theory Appl., vol. 154, no. 3, pp. 713–758, Sep. 2012, doi: 10.1007/s10957-012-0050-5.

[21] D. Lastomo, H. Setiadi, and M. R. Djalal, “Optimization pitch angle controller of rocket system using improved differential evolution algorithm,” Int. J. Adv. Intell. Informatics, vol. 3, no. 1, pp. 27–34, Mar. 2017, doi: 10.26555/ijain.v3i1.83.

[22] R. Fourer, D. M. Gay, and B. W. Kernighan, “A Modeling Language for Mathematical Programming,” Manage. Sci., vol. 36, no. 5, pp. 519–554, May 1990, doi: 10.1287/mnsc.36.5.519.

[23] B. Passenberg, “Theory and algorithms for indirect methods in optimal control of hybrid systems,” Technische Universität München, 2012, available at: Google Scholar.

[24] M. Posa, C. Cantu, and R. Tedrake, “A direct method for trajectory optimization of rigid bodies through contact,” Int. J. Rob. Res., vol. 33, no. 1, pp. 69–81, Jan. 2014, doi: 10.1177/0278364913506757.

[25] K. Ratkovic, “Limitations in direct and indirect methods for solving optimal control problems in growth theory,” Industrija, vol. 44, no. 4, pp. 19–46, 2016, doi: 10.5937/industrija44-10874.

[26] O. von Stryk and R. Bulirsch, “Direct and indirect methods for trajectory optimization,” Ann. Oper. Res., vol. 37, no. 1, pp. 357–373, Dec. 1992, doi: 10.1007/BF02071065.

[27] B. Passenberg, M. Kröninger, G. Schnattinger, M. Leibold, O. Stursberg, and M. Buss, “Initialization Concepts for Optimal Control of Hybrid Systems,” IFAC Proc. Vol., vol. 44, no. 1, pp. 10274–10280, Jan. 2011, doi: 10.3182/20110828-6-IT-1002.03012.

[28] D. A. Benson, G. T. Huntington, T. P. Thorvaldsen, and A. V. Rao, “Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method,” J. Guid. Control. Dyn., vol. 29, no. 6, pp. 1435–1440, Nov. 2006, doi: 10.2514/1.20478.

[29] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed. New York, NY, USA: Cambridge University Press, 2007, available at:

[30] A. B. Malinowska and D. F. M. Torres, “Natural boundary conditions in the calculus of variations,” Math. Methods Appl. Sci., vol. 33, no. 14, pp. 1712–1722, Sep. 2010, doi: 10.1002/mma.1289.

[31] A. M. Spence, “The Learning Curve and Competition,” Bell J. Econ., vol. 12, no. 1, p. 49, 1981, doi: 10.2307/3003508.

[32] A. S. I. Zinober and K. Kaivanto, “Optimal production subject to piecewise continuous royalty payment obligations,” Univ. Sheff., 2008, available at : Google Scholar.

[33] N. Garofalo and F.-H. Lin, “Unique continuation for elliptic operators: A geometric-variational approach,” Commun. Pure Appl. Math., vol. 40, no. 3, pp. 347–366, May 1987, doi: 10.1002/cpa.3160400305.

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