American Institute of Mathematical Sciences

April  2016, 12(2): 781-810. doi: 10.3934/jimo.2016.12.781

VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems

 1 School of Automation Engineering, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu, Sichuan, 611731, China 2 Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, W.A. 6845 3 Department of Mathematics and Statistics, Curtin University, GPO Box U1987 Perth, Western Australia 6845 4 Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845 5 School of Business, Central South University, South Lushan Road, Changsha, Hunan, China 6 Department of Mathematics, University of Western Australia, Nedlands, Western Australia 6009, Australia

Received  November 2014 Revised  April 2015 Published  June 2015

The FORTRAN MISER software package has been used with great success over the past two decades to solve many practically important real world optimal control problems. However, MISER is written in FORTRAN and hence not user-friendly, requiring FORTRAN programming knowledge. To facilitate the practical application of powerful optimal control theory and techniques, this paper describes a Visual version of the MISER software, called Visual MISER. Visual MISER provides an easy-to-use interface, while retaining the computational efficiency of the original FORTRAN MISER software. The basic concepts underlying the MISER software, which include the control parameterization technique, a time scaling transform, a constraint transcription technique, and the co-state approach for gradient calculation, are described in this paper. The software structure is explained and instructions for its use are given. Finally, an example is solved using the new Visual MISER software to demonstrate its applicability.
Citation: Feng Yang, Kok Lay Teo, Ryan Loxton, Volker Rehbock, Bin Li, Changjun Yu, Leslie Jennings. VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems. Journal of Industrial & Management Optimization, 2016, 12 (2) : 781-810. doi: 10.3934/jimo.2016.12.781
References:
 [1] N. U. Ahmed, Dynamic Systems and Control with Applications,, World Scientific, (2006). doi: 10.1142/6262. Google Scholar [2] M. Athans and P. L. Falb, Optimal Control,, McGraw-Hill, (1966). Google Scholar [3] V. Azhmyakov, Optimal control of mechanical systems,, Differential Equations and Nonlinear Mechanics, (2007). Google Scholar [4] R. Bellman and R. E. Dreyfus, Dynamic Programming and Modern Control Theory,, Orlands, (1977). Google Scholar [5] A. E. Jr. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing, (1975). Google Scholar [6] C. Buskens, NUDOCCCS, FORTRAN-Subroutine NUDOCCCS (Numerical Discretisation method for Optimal Control problems with Constraints in Controls and States),, 2010. , (). Google Scholar [7] C. Buskens and H. Maurer, Nonlinear programming methods for real-time control of an industrial robot,, Journal of Optimization Theory and Applications, 107 (2000), 505. doi: 10.1023/A:1026491014283. Google Scholar [8] L. Cesari, Optimization: Theory and Applications,, Springer-Verlag, (1983). doi: 10.1007/978-1-4613-8165-5. Google Scholar [9] Q. Q. Chai, C. H. Yang, K. L. Teo and W. H. Gui, Optimal control of an industrial-scale evaporation process: Sodium aluminate solution,, Control Engineering Practice, 20 (2002), 618. doi: 10.1016/j.conengprac.2012.03.001. Google Scholar [10] B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance,, Springer, (2005). Google Scholar [11] M. Fikar, M. A. Latifi and Y. Creff, Optimal Changeover Profiles for an Industrial Depropanizer,, Chemical Engineering Science, 54 (1999), 2715. doi: 10.1016/S0009-2509(98)00375-3. Google Scholar [12] M. E. Fisher and L. S. Jennings, MATLAB MISER,, , (): 483. Google Scholar [13] P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, User's Guide for NPSOL 5.0: Fortran package for nonlinear programming,, 1986. , (). Google Scholar [14] W. E. Gruver and E. Sachs, Algorithmic Methods in Optimal Control,, Research Notes in Mathematics, (1981). Google Scholar [15] C. J. Goh and K. L. Teo, Control parameterization: a unified approach to optimal control problems with general constraints,, Automatica, 24 (1988), 3. doi: 10.1016/0005-1098(88)90003-9. Google Scholar [16] S. Gonzalez and A. Miele, Sequential gradient-restoration algorithm for optimal control problems with general boundary conditions,, Journal of Optimization Theory and Applications, 26 (1978), 395. doi: 10.1007/BF00933463. Google Scholar [17] G. R. Duan, D. K. Gu and B. Li, Optimal control for final approach of rendezvous with non-cooperative target,, Pacific Journal of Optimization, 6 (2010), 3157. Google Scholar [18] P. Howlett, The optimal control of a train,, Annals of Operations Research, 98 (2000), 65. doi: 10.1023/A:1019235819716. Google Scholar [19] H. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials,, Journal of the Franklin Institute, 339 (2002), 479. doi: 10.1016/S0016-0032(02)00028-5. Google Scholar [20] L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3.3 Optimal Control Software Version: Theory and User Manual,, the University of Western Australia, (2004). Google Scholar [21] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica, 26 (1990), 371. doi: 10.1016/0005-1098(90)90131-Z. Google Scholar [22] C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591. doi: 10.3934/jimo.2012.8.591. Google Scholar [23] C. Jiang, K. L. Teo and G. R. Duan, A suboptimal feedback control for nonlinear time-varying systems with continuous inequality constraints,, Automatica, 48 (2012), 660. doi: 10.1016/j.automatica.2012.01.019. Google Scholar [24] C. Jiang, Q. Lin, C. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints,, Journal of Optimization Theory and Applications, 154 (2012), 30. doi: 10.1007/s10957-012-0006-9. Google Scholar [25] C. Y. Kaya and J. M. Martnez, Euler discretization and inexact restoration for optimal control,, Journal of Optimization Theory and Applications, 134 (2007), 191. doi: 10.1007/s10957-007-9217-x. Google Scholar [26] C. Y. Kaya and J. L. Noakes, Leapfrog for Optimal Control,, SIAM Journal on Numerical Analysis, (2008). doi: 10.1137/060675034. Google Scholar [27] M. I. Kamien and N. L. Schwartz, Dynamic Optimization - The Calculus of Variations and Optimal Control in Economics and Management,, North Holland, (1991). Google Scholar [28] T. T. Lam and Y. Bayazitoglu, Application of the sequential gradient restoration algorithm to terminal convective instability problems,, Journal of Optimization Theory and Applications, 49 (1986), 47. doi: 10.1007/BF00939247. Google Scholar [29] B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles,, Applied Mathematics and Computation, 224 (2013), 866. doi: 10.1016/j.amc.2013.08.092. Google Scholar [30] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem,, Journal of Optimization Theory and Applications, 151 (2011), 260. doi: 10.1007/s10957-011-9904-5. Google Scholar [31] B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear constrained optimal control problems,, Discrete and Continuous Dynamical Systems Series B, 16 (2011), 1101. doi: 10.3934/dcdsb.2011.16.1101. Google Scholar [32] B. Li, K. L. Teo and G. R. Duan, Optimal control computation for discrete time time-delayed optimal control problem with all-time-step inequality constraints,, International Journal of Innovative Computing, 6 (2010), 521. Google Scholar [33] B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minimax optimal control problems with applications,, Australian and New Zealand Industrial and Applied Mathematics Journal, 51 (2009), 162. doi: 10.1017/S1446181110000040. Google Scholar [34] C. J. Li, K. L Teo, B. Li and G. F. Ma, A constrained optimal pid-like controller design for spacecraft attitude stabilization,, Acta Astronautica, 74 (2011), 131. doi: 10.1016/j.actaastro.2011.12.021. Google Scholar [35] C. C. Lim and K. L. Teo, Optimal insulin infusion control to a mathematical blood glucoregulatory model with fuzzy parameters,, Cybernetics and Systems, 22 (1991), 1. doi: 10.1080/01969729108902267. Google Scholar [36] Q. Lin, R. Loxton and K. L. Teo, The control parameterization for nonlinear optimal control: A survey,, Journal of Industrial and Management Optimization, 10 (2014), 275. doi: 10.3934/jimo.2014.10.275. Google Scholar [37] R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter,, Automatica, 45 (2009), 973. doi: 10.1016/j.automatica.2008.10.031. Google Scholar [38] R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250. doi: 10.1016/j.automatica.2009.05.029. Google Scholar [39] R. Loxton, K. L. Teo, and V. Rehbock, Computational method for a class of switched system optimal control problems,, IEEE Transactions on Automatic Control, 54 (2009), 2455. doi: 10.1109/TAC.2009.2029310. Google Scholar [40] R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results,, Numerical Algebra, 2 (2012), 571. doi: 10.3934/naco.2012.2.571. Google Scholar [41] R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control,, Automatica, 49 (2013), 2652. doi: 10.1016/j.automatica.2013.05.027. Google Scholar [42] R. Loxton, Q. Lin and K. L. Teo, Switching time optimization for nonlinear switched systems: Direct optimization and the time scaling transformation,, Pacific Journal of Optimization, 10 (2014), 537. Google Scholar [43] R. Luus, Iterative Dynamic Programming,, Chapman & Hall/CRC, (2000). doi: 10.1201/9781420036022. Google Scholar [44] R. Luus and O. N. Okongwu, Towards practical optimal contorl of batch reactors,, Chemical Engineering Journal, 75 (1999), 1. Google Scholar [45] R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy,, World Scientific, (1994). Google Scholar [46] MATLAB - The Language of Technical Computing, http://mathworks.com/products/matlab/,, 2008., (). Google Scholar [47] H. Maurer, C. Buskens and G. Feichtinger, Solution techniques for periodic control problems: a case study in production planning,, Optimal Control Applications and Methods, 19 (1998), 185. doi: 10.1002/(SICI)1099-1514(199805/06)19:3<185::AID-OCA627>3.0.CO;2-E. Google Scholar [48] H. H. Mehne and A. H. Borzabadi, A numerical method for solving optimal control problems using state parametrization,, Numerical Algorithms, 42 (2006), 165. doi: 10.1007/s11075-006-9035-5. Google Scholar [49] A. Miele and T. Wang, Primal-dual properties of sequential gradient-restoration algorithms for optimal control problems, Part 2, General problem,, Journal of Mathematical Analysis and Applications, 119 (1986), 21. doi: 10.1016/0022-247X(86)90142-3. Google Scholar [50] H. J. Oberle and B. Sothmann, Numerical computation of optimal feed rates for a fed-batch fermentation model,, Journal of Optimization Theory and Applications, 100 (1999), 1. Google Scholar [51] R. Petzold and A. C. Hindmarsh, LSODA, Ordinary Differential Equation Solver for Stiff or Non-Stiff System,, 2005., (). Google Scholar [52] L. S. Pontryagin, V. G. Boltayanskii, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes,, CRC Press, (1987). Google Scholar [53] V. Rehbock and I. Livk, Optimal control of a batch crystallization process,, Journal of Industrial and Management Optimization, 3 (2007), 331. doi: 10.3934/jimo.2007.3.585. Google Scholar [54] Y. Sakawa and Y. Shindo, Optimal control of container cranes,, Automatica, 18 (1982), 257. doi: 10.1016/0005-1098(82)90086-3. Google Scholar [55] K. Schittkowski, NLPQLP: A new fortran implementation of a sequential quadratic programming algorithm for parallel computing,, 2010., (). Google Scholar [56] A. L. Schwartz, RIOTS-A Matlab toolbox for solving general optimal control problems,, 2008. , (). Google Scholar [57] Y. Shindo and Y. Sakawa, Local convergence of an algorithm for solving optimal control problems,, Journal of Optimization Theory and Applications, 46 (1985), 265. doi: 10.1007/BF00939285. Google Scholar [58] W. Sun and Y. X. Yuan, Optimization Theory and Methods,, Springer, (2006). Google Scholar [59] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Longman Scientific and Technical, (1991). Google Scholar [60] K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems,, Journal of Australian Mathematical Society, 40 (1999), 314. doi: 10.1017/S0334270000010936. Google Scholar [61] K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems,, Automatica, 29 (1993), 789. doi: 10.1016/0005-1098(93)90076-6. Google Scholar [62] K. L. Teo, and K. H. Wong, Nonlinearly constrained optimal control problems,, Journal of Australian Mathematical Society, 33 (1992), 517. doi: 10.1017/S0334270000007207. Google Scholar [63] K. L. Teo, C. J. Goh and C. C. Lim, A computational method for a class of dynamical optimization problems in which the terminal time is conditionally free,, IMA - Journal of Mathematical Control and Information, 6 (1989), 81. doi: 10.1093/imamci/6.1.81. Google Scholar [64] K. L. Teo and C. C. Lim, Time optimal control computation with application to ship steering,, Journal of Optimization Theory and Applications, 56 (1988), 145. doi: 10.1007/BF00938530. Google Scholar [65] N. S. Trahair and J. R. Booker, Optimum elastic columns,, International Journal of Mechanical Sciences, 12 (1970), 973. doi: 10.1016/0020-7403(70)90037-8. Google Scholar [66] O. von Stryk, Optimization of dynamic systems in industrial applications,, in Proc. 2nd European Congress on Intelligent Techniques and Soft Computing (EUFIT) (H.J. Zimmermann ed.), (1994), 347. Google Scholar [67] C. Z. Wu and K. L. Teo, Global impulsive optimal control computation,, Journal of Industrial and Management Optimization, 2 (2006), 435. doi: 10.3934/jimo.2006.2.435. Google Scholar [68] J. L. Zhou and A. Tits, User's guide for FFSQP version 3.7: A Fortran code for solving optimization programs, possibly minimax,with general inequality constraints and linear equality constraints, generating feasible iterates, (1997),, Institute for Systems Research, (2074), 92. Google Scholar

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References:
 [1] N. U. Ahmed, Dynamic Systems and Control with Applications,, World Scientific, (2006). doi: 10.1142/6262. Google Scholar [2] M. Athans and P. L. Falb, Optimal Control,, McGraw-Hill, (1966). Google Scholar [3] V. Azhmyakov, Optimal control of mechanical systems,, Differential Equations and Nonlinear Mechanics, (2007). Google Scholar [4] R. Bellman and R. E. Dreyfus, Dynamic Programming and Modern Control Theory,, Orlands, (1977). Google Scholar [5] A. E. Jr. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing, (1975). Google Scholar [6] C. Buskens, NUDOCCCS, FORTRAN-Subroutine NUDOCCCS (Numerical Discretisation method for Optimal Control problems with Constraints in Controls and States),, 2010. , (). Google Scholar [7] C. Buskens and H. Maurer, Nonlinear programming methods for real-time control of an industrial robot,, Journal of Optimization Theory and Applications, 107 (2000), 505. doi: 10.1023/A:1026491014283. Google Scholar [8] L. Cesari, Optimization: Theory and Applications,, Springer-Verlag, (1983). doi: 10.1007/978-1-4613-8165-5. Google Scholar [9] Q. Q. Chai, C. H. Yang, K. L. Teo and W. H. Gui, Optimal control of an industrial-scale evaporation process: Sodium aluminate solution,, Control Engineering Practice, 20 (2002), 618. doi: 10.1016/j.conengprac.2012.03.001. Google Scholar [10] B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance,, Springer, (2005). Google Scholar [11] M. Fikar, M. A. Latifi and Y. Creff, Optimal Changeover Profiles for an Industrial Depropanizer,, Chemical Engineering Science, 54 (1999), 2715. doi: 10.1016/S0009-2509(98)00375-3. Google Scholar [12] M. E. Fisher and L. S. Jennings, MATLAB MISER,, , (): 483. Google Scholar [13] P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, User's Guide for NPSOL 5.0: Fortran package for nonlinear programming,, 1986. , (). Google Scholar [14] W. E. Gruver and E. Sachs, Algorithmic Methods in Optimal Control,, Research Notes in Mathematics, (1981). Google Scholar [15] C. J. Goh and K. L. Teo, Control parameterization: a unified approach to optimal control problems with general constraints,, Automatica, 24 (1988), 3. doi: 10.1016/0005-1098(88)90003-9. Google Scholar [16] S. Gonzalez and A. Miele, Sequential gradient-restoration algorithm for optimal control problems with general boundary conditions,, Journal of Optimization Theory and Applications, 26 (1978), 395. doi: 10.1007/BF00933463. Google Scholar [17] G. R. Duan, D. K. Gu and B. Li, Optimal control for final approach of rendezvous with non-cooperative target,, Pacific Journal of Optimization, 6 (2010), 3157. Google Scholar [18] P. Howlett, The optimal control of a train,, Annals of Operations Research, 98 (2000), 65. doi: 10.1023/A:1019235819716. Google Scholar [19] H. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials,, Journal of the Franklin Institute, 339 (2002), 479. doi: 10.1016/S0016-0032(02)00028-5. Google Scholar [20] L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3.3 Optimal Control Software Version: Theory and User Manual,, the University of Western Australia, (2004). Google Scholar [21] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica, 26 (1990), 371. doi: 10.1016/0005-1098(90)90131-Z. Google Scholar [22] C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem,, Journal of Industrial and Management Optimization, 8 (2012), 591. doi: 10.3934/jimo.2012.8.591. Google Scholar [23] C. Jiang, K. L. Teo and G. R. Duan, A suboptimal feedback control for nonlinear time-varying systems with continuous inequality constraints,, Automatica, 48 (2012), 660. doi: 10.1016/j.automatica.2012.01.019. Google Scholar [24] C. Jiang, Q. Lin, C. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints,, Journal of Optimization Theory and Applications, 154 (2012), 30. doi: 10.1007/s10957-012-0006-9. Google Scholar [25] C. Y. Kaya and J. M. Martnez, Euler discretization and inexact restoration for optimal control,, Journal of Optimization Theory and Applications, 134 (2007), 191. doi: 10.1007/s10957-007-9217-x. Google Scholar [26] C. Y. Kaya and J. L. Noakes, Leapfrog for Optimal Control,, SIAM Journal on Numerical Analysis, (2008). doi: 10.1137/060675034. Google Scholar [27] M. I. Kamien and N. L. Schwartz, Dynamic Optimization - The Calculus of Variations and Optimal Control in Economics and Management,, North Holland, (1991). Google Scholar [28] T. T. Lam and Y. Bayazitoglu, Application of the sequential gradient restoration algorithm to terminal convective instability problems,, Journal of Optimization Theory and Applications, 49 (1986), 47. doi: 10.1007/BF00939247. Google Scholar [29] B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles,, Applied Mathematics and Computation, 224 (2013), 866. doi: 10.1016/j.amc.2013.08.092. Google Scholar [30] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem,, Journal of Optimization Theory and Applications, 151 (2011), 260. doi: 10.1007/s10957-011-9904-5. Google Scholar [31] B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear constrained optimal control problems,, Discrete and Continuous Dynamical Systems Series B, 16 (2011), 1101. doi: 10.3934/dcdsb.2011.16.1101. Google Scholar [32] B. Li, K. L. Teo and G. R. Duan, Optimal control computation for discrete time time-delayed optimal control problem with all-time-step inequality constraints,, International Journal of Innovative Computing, 6 (2010), 521. Google Scholar [33] B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minimax optimal control problems with applications,, Australian and New Zealand Industrial and Applied Mathematics Journal, 51 (2009), 162. doi: 10.1017/S1446181110000040. Google Scholar [34] C. J. Li, K. L Teo, B. Li and G. F. Ma, A constrained optimal pid-like controller design for spacecraft attitude stabilization,, Acta Astronautica, 74 (2011), 131. doi: 10.1016/j.actaastro.2011.12.021. Google Scholar [35] C. C. Lim and K. L. Teo, Optimal insulin infusion control to a mathematical blood glucoregulatory model with fuzzy parameters,, Cybernetics and Systems, 22 (1991), 1. doi: 10.1080/01969729108902267. Google Scholar [36] Q. Lin, R. Loxton and K. L. Teo, The control parameterization for nonlinear optimal control: A survey,, Journal of Industrial and Management Optimization, 10 (2014), 275. doi: 10.3934/jimo.2014.10.275. Google Scholar [37] R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter,, Automatica, 45 (2009), 973. doi: 10.1016/j.automatica.2008.10.031. Google Scholar [38] R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250. doi: 10.1016/j.automatica.2009.05.029. Google Scholar [39] R. Loxton, K. L. Teo, and V. Rehbock, Computational method for a class of switched system optimal control problems,, IEEE Transactions on Automatic Control, 54 (2009), 2455. doi: 10.1109/TAC.2009.2029310. Google Scholar [40] R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results,, Numerical Algebra, 2 (2012), 571. doi: 10.3934/naco.2012.2.571. Google Scholar [41] R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control,, Automatica, 49 (2013), 2652. doi: 10.1016/j.automatica.2013.05.027. Google Scholar [42] R. Loxton, Q. Lin and K. L. Teo, Switching time optimization for nonlinear switched systems: Direct optimization and the time scaling transformation,, Pacific Journal of Optimization, 10 (2014), 537. Google Scholar [43] R. Luus, Iterative Dynamic Programming,, Chapman & Hall/CRC, (2000). doi: 10.1201/9781420036022. Google Scholar [44] R. Luus and O. N. Okongwu, Towards practical optimal contorl of batch reactors,, Chemical Engineering Journal, 75 (1999), 1. Google Scholar [45] R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy,, World Scientific, (1994). Google Scholar [46] MATLAB - The Language of Technical Computing, http://mathworks.com/products/matlab/,, 2008., (). Google Scholar [47] H. Maurer, C. Buskens and G. Feichtinger, Solution techniques for periodic control problems: a case study in production planning,, Optimal Control Applications and Methods, 19 (1998), 185. doi: 10.1002/(SICI)1099-1514(199805/06)19:3<185::AID-OCA627>3.0.CO;2-E. Google Scholar [48] H. H. Mehne and A. H. Borzabadi, A numerical method for solving optimal control problems using state parametrization,, Numerical Algorithms, 42 (2006), 165. doi: 10.1007/s11075-006-9035-5. Google Scholar [49] A. Miele and T. Wang, Primal-dual properties of sequential gradient-restoration algorithms for optimal control problems, Part 2, General problem,, Journal of Mathematical Analysis and Applications, 119 (1986), 21. doi: 10.1016/0022-247X(86)90142-3. Google Scholar [50] H. J. Oberle and B. Sothmann, Numerical computation of optimal feed rates for a fed-batch fermentation model,, Journal of Optimization Theory and Applications, 100 (1999), 1. Google Scholar [51] R. Petzold and A. C. Hindmarsh, LSODA, Ordinary Differential Equation Solver for Stiff or Non-Stiff System,, 2005., (). Google Scholar [52] L. S. Pontryagin, V. G. Boltayanskii, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes,, CRC Press, (1987). Google Scholar [53] V. Rehbock and I. Livk, Optimal control of a batch crystallization process,, Journal of Industrial and Management Optimization, 3 (2007), 331. doi: 10.3934/jimo.2007.3.585. Google Scholar [54] Y. Sakawa and Y. Shindo, Optimal control of container cranes,, Automatica, 18 (1982), 257. doi: 10.1016/0005-1098(82)90086-3. Google Scholar [55] K. Schittkowski, NLPQLP: A new fortran implementation of a sequential quadratic programming algorithm for parallel computing,, 2010., (). Google Scholar [56] A. L. Schwartz, RIOTS-A Matlab toolbox for solving general optimal control problems,, 2008. , (). Google Scholar [57] Y. Shindo and Y. Sakawa, Local convergence of an algorithm for solving optimal control problems,, Journal of Optimization Theory and Applications, 46 (1985), 265. doi: 10.1007/BF00939285. Google Scholar [58] W. Sun and Y. X. Yuan, Optimization Theory and Methods,, Springer, (2006). Google Scholar [59] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Longman Scientific and Technical, (1991). Google Scholar [60] K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems,, Journal of Australian Mathematical Society, 40 (1999), 314. doi: 10.1017/S0334270000010936. Google Scholar [61] K. L. Teo, V. Rehbock and L. S. 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