In bang-bang optimal control problems, the control function is inherently piecewise constant. This feature creates substantial difficulties for the standard Legendre-Gauss-Radau pseudospectral method, which relies on polynomial approximation for the control function. This study introduces a simplified approach that seamlessly integrates sigmoid-based control parameterization with the traditional Legendre-Gauss-Radau pseudospectral method. This integration enables precise approximation of discontinuous control profiles while maintaining the polynomial approximation for state variables. The proposed method significantly minimizes the number of decision variables in the optimization problem while precisely determining both the number and locations of switching points. This leads to notable enhancements in computational efficiency and solution accuracy. Numerical experiments conducted on two benchmark problems, a bridge crane system and a robotic arm control problem, demonstrate the exceptional precision and efficiency of the proposed method. Despite its simplicity, the method delivers results that are on par with those produced by more advanced and intricate techniques.
[1] Y. M. Agamawi, W. W. Hager, and A. V. Rao, Mesh refinement method for solving bang-bang optimal control problems using direct collocation, AIAA Scitech 2020 Forum, 1(Part F) (2020), 1–25.
[2] V. Arakelian, Y. Lu, and J. Geng, Application of the “bang-bang” law in robot manipulators for the reduction of inertial forces and input torques, In the Mechanisms and Machine Science, Springer Nature, Switzerland, (2023), 149–158.
[3] F. Assassa and W. Marquardt, Dynamic optimization using adaptive direct multiple shooting, Computers and Chemical Engineering, 60 (2014), 242–259.
[4] M. Athans and P. L. Falb, Optimal Control: An Introduction to the Theory and Its Applications, Dover Publications, 2013.
[5] D. Baleanu, A. Jajarmi, S. S. Sajjadi, and D. Mozyrska, A new fractional model and optimal control of a tumorimmune surveillance with non-singular derivative operator, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(8) (2019), 083127.
[6] R. Baltensperger and M. R. Trummer, Spectral differencing with a twist, SIAM Journal on Scientific Computing, 24(5) (2003), 1465–1487.
[7] D. A. Benson, G. T. Huntington, T. P. Thorvaldsen, and A. V. Rao, Direct trajectory optimization and costate estimation via an orthogonal collocation method, Journal of Guidance, Control, and Dynamics, 29(6) (2006), 1435–1440.
[8] R. Bertrand and R. Epenoy, New smoothing techniques for solving bang-bang optimal control problems - numerical results and statistical interpretation, Optimal Control Applications and Methods, 23(4) (2002), 171–197.
[9] J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Society for Industrial and Applied Mathematics, second edition, 2010.
[10] H. G. Bock and K. J. Plitt, Multiple shooting algorithm for direct solution of optimal control problems, IFAC Proceedings Volumes, 17(2) (1985), 1603–1608.
[11] A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Taylor & Francis, 1975.
[12] W. Cai, L. Yang, and Y. Zhu, Bang-bang optimal control for differentially flat systems using mapped pseudospectral method and analytic homotopic approach, Optimal Control Applications and Methods, 37(6) (2016), 1217–1235.
[13] R. G. Deshmukh, D. A. Spencer, and S. Dutta, Investigation of direct force control for aerocapture at neptune, Acta Astronautica, 175 (2020), 375–386.
[14] J. D. Eide, W. W. Hager, and A. V. Rao, Modified legendre–gauss–radau collocation method for optimal control problems with nonsmooth solutions, Journal of Optimization Theory and Applications, 191(2-3) (2021), 600–633.
[15] G. Elnagar, M. A. Kazemi, and M. Razzaghi, The pseudospectral legendre method for discretizing optimal control problems, IEEE Transactions on Automatic Control, 40(10) (1995), 1793–1796.
[16] P. J. Enright and B. A. Conway, Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, Journal of Guidance, Control, and Dynamics, 15(4) (1992), 994–1002.
[17] Z. Fathi and B. Bidabad, On the geometry of zermelo’s optimal control trajectories, AUT Journal of Mathematics and Computing, 3(1) (2022), 1–10.
[18] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1996.
[19] Z. Foroozandeh, M. Shamsi, V. Azhmyakov, and M. Shafiee, A modified pseudospectral method for solving trajectory optimization problems with singular arc, Mathematical Methods in the Applied Sciences, 40(5) (2017), 1783–1793.
[20] Z. Foroozandeh, M. Shamsi, and M. D. R. De Pinho, A hybrid direct–indirect approach for solving the singular optimal control problems of finite and infinite order, Iranian Journal of Science and Technology, Transaction A: Science, 42(3) (2018), 1545–1554.
[21] Z. Foroozandeh, M. Shamsi, and M. D. R. De Pinho, A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems, International Journal of Control, 92(7) (2019), 1551–1566.
[22] Z. Gao and H. Baoyin, Using homotopy method to solve bang-bang optimal control problems, Springer Optimization and Its Applications, 76 (2013), 243–256.
[23] D. Garg, M. Patterson, W. W. Hager, A. V. Rao, D. A. Benson, and G. T. Huntington, A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46(11) (2010), 1843–1851.
[24] D. Garg, M. A. Patterson, C. Francolin, C. L. Darby, G. T. Huntington, W. W. Hager, and A. V. Rao, Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a radau pseudospectral method, Computational Optimization and Applications, 49(2) (2011), 335–358.
[25] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation, OUP Oxford, 2004.
[26] J. K. Geiser and D. A. Matz, Optimal angle of attack control for aerocapture maneuvers, AIAA Scitech 2022 Forum, (2022), 1–14.
[27] M. Gerdts, Direct shooting method for the numerical solution of higher-index dae optimal control problems, Journal of Optimization Theory and Applications, 117(2) (2003), 267–294.
[28] S. S. Haykin, Neural Networks and Learning Machines, Number v. 10 in Neural networks and learning machines, Prentice Hall, 2009.
[29] J. C. C. Henriques, J. M. Lemos, L. Eca, L. M. C. Gato, and A. F. O. Falcao, A high-order discontinuous galerkin method with mesh refinement for optimal control problems, Automatica, 85 (2017), 70–82.
[30] L. Hou-Yuan and Z. Chang-Yin, Optimization of low-thrust trajectories using an indirect shooting method without guesses of initial costates, Chinese Astronomy and Astrophysics, 36(4) (2012), 389–398.
[31] N. Hritonenko, N. Kato, and Y. Yatsenko, Optimal control of investments in old and new capital under improving technology, Journal of Optimization Theory and Applications, 172(1) (2017), 247–266.
[32] A. Jajarmi and M. Hajipour, An efficient finite difference method for the time-delay optimal control problems with time-varying delay, Asian Journal of Control, 19(2) (2017), 554–563.
[33] A. Jajarmi, N. Pariz, A. Vahidian Kamyad, and S. Effati, A novel modal series representation approach to solve a class of nonlinear optimal control problems, International Journal of Innovative Computing, Information and Control, 7(2) (2011), 501–510.
[34] R. Khanduzi, A. Ebrahimzadeh, and Z. Ebrahimzadeh, A combined bernoulli collocation method and imperialist competitive algorithm for optimal control of sediment in the dam reservoirs, AUT Journal of Mathematics and Computing, 5(1) (2024), 71–90.
[35] D. Kim, Pricing strategies of monopoly platform for technology transition in a two-sided market, 2012 Proceedings of PICMET ’12: Technology Management for Emerging Technologies, (2012), 165–172.
[36] D. E. Kirk, Optimal Control Theory: An Introduction, Dover Publications, 2004.
[37] Q. Lin, R. Loxton, and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10(1) (2014), 275–309.
[38] Z. Liu, S. Li, and K. Zhao, Extended multi-interval legendre-gauss-radau pseudospectral method for mixed-integer optimal control problem in engineering, International Journal of Systems Science, 52(5) (2021), 928–951.
[39] D. J. Lynch, K. M. Lynch, and P. B. Umbanhowar, The soft-landing problem: Minimizing energy loss by a legged robot impacting yielding terrain, IEEE Robotics and Automation Letters, 5(2) (2020), 3658–3665.
[40] M. Massaro, S. Lovato, M. Bottin, and G. Rosati, An optimal control approach to the minimum-time trajectory planning of robotic manipulators, Robotics, 12(3) (2023), 1–24.
[41] H. Maurer, C. Buskens, J. H. R. Kim, and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Applications and Methods, 26(3) (2021), 129–156.
[42] H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control, SIAM Journal on Control and Optimization, 42(6) (2004), 2239–2263.
[43] M. A. Mehrpouya, M. Shamsi, and M. Razzaghi, A combined adaptive control parametrization and homotopy continuation technique for the numerical solution of bang-bang optimal control problems, ANZIAM Journal, 56(1) (2014), 48–65.
[44] M. A. Mehrpouya, A modified pseudospectral method for indirect solving a class of switching optimal control problems, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 234(9) (2020), 1531–1542.
[45] M. A. Mehrpouya and S. Fallahi, A modified control parametrization method for the numerical solution of bangbang optimal control problems, JVC/Journal of Vibration and Control, 21(12) (2015), 2407–2415.
[46] M. A. Mehrpouya and M. Khaksar-E Oshagh, An efficient numerical solution for time switching optimal control problems, Computational Methods for Differential Equations, 9(1) (2021), 225–243.
[47] E. R. Pager and A. V. Rao, Method for solving bang-bang and singular optimal control problems using adaptive radau collocation, Computational Optimization and Applications, 81(3) (2022), 857–887.
[48] A. V. Rao, A survey of numerical methods for optimal control, Advances in the Astronautical Sciences, 135 (2009), 497–528.
[49] I. M. Ross, A Primer on Pontryagin’s Principle in Optimal Control, Collegiate Publishers, 2015.
[50] I. M. Ross and M. Karpenko, A review of pseudospectral optimal control: From theory to flight, Annual Reviews in Control, 36(2) (2012), 182–197.
[51] M. Shamsi, A modified pseudospectral scheme for accurate solution of bang-bang optimal control problems, Optimal Control Applications and Methods, 32(6) (2011), 668–680.
[52] C. Silva and E. Trelat, Smooth regularization of bang-bang optimal control problems, IEEE Transactions on Automatic Control, 55(11) (2010), 2488–2499.
[53] X. Tang and J. Chen, Direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using collocation at the flipped legendre-gauss-radau points, IEEE/CAA Journal of Automatica Sinica, 3(2) (2016), 174–183.
[54] K. Wang, Z. Chen, Z. Wei, F. Lu, and J. Li, A new smoothing technique for bang-bang optimal control problems, AIAA Scitech 2024 Forum, (2024), 1–14.
[55] J. A. C. Weideman and S. C. Reddy, A matlab differentiation matrix suite, ACM Transactions on Mathematical Software, 26(4) (2000), 465–519.
Samadi, F. (2025). Enhancing the Legendre-Gauss-Radau pseudospectral method with sigmoid-based control parameterization for solving bang-bang optimal control problems. Computational Methods for Differential Equations, 13(3), 802-814. doi: 10.22034/cmde.2025.65162.2982
MLA
Samadi, F. . "Enhancing the Legendre-Gauss-Radau pseudospectral method with sigmoid-based control parameterization for solving bang-bang optimal control problems", Computational Methods for Differential Equations, 13, 3, 2025, 802-814. doi: 10.22034/cmde.2025.65162.2982
HARVARD
Samadi, F. (2025). 'Enhancing the Legendre-Gauss-Radau pseudospectral method with sigmoid-based control parameterization for solving bang-bang optimal control problems', Computational Methods for Differential Equations, 13(3), pp. 802-814. doi: 10.22034/cmde.2025.65162.2982
CHICAGO
F. Samadi, "Enhancing the Legendre-Gauss-Radau pseudospectral method with sigmoid-based control parameterization for solving bang-bang optimal control problems," Computational Methods for Differential Equations, 13 3 (2025): 802-814, doi: 10.22034/cmde.2025.65162.2982
VANCOUVER
Samadi, F. Enhancing the Legendre-Gauss-Radau pseudospectral method with sigmoid-based control parameterization for solving bang-bang optimal control problems. Computational Methods for Differential Equations, 2025; 13(3): 802-814. doi: 10.22034/cmde.2025.65162.2982
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