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Approximate algorithms for unrelated machine scheduling to minimize makespan
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 |
References:
[1] |
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Singapore, 2006.
doi: 10.1142/6262. |
[2] |
M. Athans and P. L. Falb, Optimal Control, McGraw-Hill, 1966. |
[3] |
V. Azhmyakov, Optimal control of mechanical systems, Differential Equations and Nonlinear Mechanics, Volume 2007. |
[4] |
R. Bellman and R. E. Dreyfus, Dynamic Programming and Modern Control Theory, Orlands, Florida, Academic Press, 1977. |
[5] |
A. E. Jr. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, DC, 1975. |
[6] |
C. Buskens, NUDOCCCS, FORTRAN-Subroutine NUDOCCCS (Numerical Discretisation method for Optimal Control problems with Constraints in Controls and States), 2010. http://www.swmath.org/software/8606 |
[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), 505-527.
doi: 10.1023/A:1026491014283. |
[8] |
L. Cesari, Optimization: Theory and Applications, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[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), 618-628.
doi: 10.1016/j.conengprac.2012.03.001. |
[10] |
B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance, Springer, The Netherlands, 2005. |
[11] |
M. Fikar, M. A. Latifi and Y. Creff, Optimal Changeover Profiles for an Industrial Depropanizer, Chemical Engineering Science, 54 (1999), 2715-2720.
doi: 10.1016/S0009-2509(98)00375-3. |
[12] |
M. E. Fisher and L. S. Jennings, MATLAB MISER, http://www.acad.polyu.edu.hk/ majlee/AMA483-523/OCTmanual.pdf |
[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. http://web.stanford.edu/group/SOL/npsol.htm |
[14] |
W. E. Gruver and E. Sachs, Algorithmic Methods in Optimal Control, Research Notes in Mathematics, Vol. 47, Pitman (Advance Publishing Program), London, 1981. |
[15] |
C. J. Goh and K. L. Teo, Control parameterization: a unified approach to optimal control problems with general constraints, Automatica, 24 (1988), 3-18.
doi: 10.1016/0005-1098(88)90003-9. |
[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), 395-425.
doi: 10.1007/BF00933463. |
[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), 3157-3175. |
[18] |
P. Howlett, The optimal control of a train, Annals of Operations Research, 98 (2000), 65-87.
doi: 10.1023/A:1019235819716. |
[19] |
H. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials, Journal of the Franklin Institute, 339 (2002), 479-498.
doi: 10.1016/S0016-0032(02)00028-5. |
[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. |
[21] |
L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica, 26 (1990), 371-375.
doi: 10.1016/0005-1098(90)90131-Z. |
[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), 591-609.
doi: 10.3934/jimo.2012.8.591. |
[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), 660-665.
doi: 10.1016/j.automatica.2012.01.019. |
[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), 30-53.
doi: 10.1007/s10957-012-0006-9. |
[25] |
C. Y. Kaya and J. M. Martnez, Euler discretization and inexact restoration for optimal control, Journal of Optimization Theory and Applications, 134 (2007), 191-206.
doi: 10.1007/s10957-007-9217-x. |
[26] |
C. Y. Kaya and J. L. Noakes, Leapfrog for Optimal Control, SIAM Journal on Numerical Analysis, in press, 2008.
doi: 10.1137/060675034. |
[27] |
M. I. Kamien and N. L. Schwartz, Dynamic Optimization - The Calculus of Variations and Optimal Control in Economics and Management, North Holland, 1991. |
[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), 47-63.
doi: 10.1007/BF00939247. |
[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), 866-875.
doi: 10.1016/j.amc.2013.08.092. |
[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), 260-291.
doi: 10.1007/s10957-011-9904-5. |
[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), 1101-1117.
doi: 10.3934/dcdsb.2011.16.1101. |
[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, Information and Control, 6 (2010), 521-532. |
[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), 162-177.
doi: 10.1017/S1446181110000040. |
[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), 131-140.
doi: 10.1016/j.actaastro.2011.12.021. |
[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), 1-16.
doi: 10.1080/01969729108902267. |
[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), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[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), 973-980.
doi: 10.1016/j.automatica.2008.10.031. |
[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), 2250-2257.
doi: 10.1016/j.automatica.2009.05.029. |
[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), 2455-2460.
doi: 10.1109/TAC.2009.2029310. |
[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, Control and Optimization, 2 (2012), 571-599.
doi: 10.3934/naco.2012.2.571. |
[41] |
R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.
doi: 10.1016/j.automatica.2013.05.027. |
[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), 537-560. |
[43] |
R. Luus, Iterative Dynamic Programming, Chapman & Hall/CRC, Boca Raton, 2000.
doi: 10.1201/9781420036022. |
[44] |
R. Luus and O. N. Okongwu, Towards practical optimal contorl of batch reactors, Chemical Engineering Journal, 75 (1999), 1-9. |
[45] |
R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, 1994, 185pp. |
[46] |
MATLAB - The Language of Technical Computing, http://mathworks.com/products/matlab/, 2008. |
[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), 185-203.
doi: 10.1002/(SICI)1099-1514(199805/06)19:3<185::AID-OCA627>3.0.CO;2-E. |
[48] |
H. H. Mehne and A. H. Borzabadi, A numerical method for solving optimal control problems using state parametrization, Numerical Algorithms, 42 (2006), 165-169.
doi: 10.1007/s11075-006-9035-5. |
[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), 21-54.
doi: 10.1016/0022-247X(86)90142-3. |
[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), 1-13. |
[51] |
R. Petzold and A. C. Hindmarsh, LSODA, Ordinary Differential Equation Solver for Stiff or Non-Stiff System, 2005. |
[52] |
L. S. Pontryagin, V. G. Boltayanskii, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, CRC Press, 1987. |
[53] |
V. Rehbock and I. Livk, Optimal control of a batch crystallization process, Journal of Industrial and Management Optimization, 3 (2007), 331-348.
doi: 10.3934/jimo.2007.3.585. |
[54] |
Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.
doi: 10.1016/0005-1098(82)90086-3. |
[55] |
K. Schittkowski, NLPQLP: A new fortran implementation of a sequential quadratic programming algorithm for parallel computing, 2010. |
[56] |
A. L. Schwartz, RIOTS-A Matlab toolbox for solving general optimal control problems, 2008. http://mechatronics.ucmerced.edu/RIOTS |
[57] |
Y. Shindo and Y. Sakawa, Local convergence of an algorithm for solving optimal control problems, Journal of Optimization Theory and Applications, 46 (1985), 265-293.
doi: 10.1007/BF00939285. |
[58] |
W. Sun and Y. X. Yuan, Optimization Theory and Methods, Springer, 2006. |
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K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, England, 1991. |
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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, Series B, 40 (1999), 314-335.
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K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems, Automatica, 29 (1993), 789-792.
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K. L. Teo, and K. H. Wong, Nonlinearly constrained optimal control problems, Journal of Australian Mathematical Society, Series B, 33 (1992), 517-530.
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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), 81-95.
doi: 10.1093/imamci/6.1.81. |
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K. L. Teo and C. C. Lim, Time optimal control computation with application to ship steering, Journal of Optimization Theory and Applications, 56 (1988), 145-156.
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N. S. Trahair and J. R. Booker, Optimum elastic columns, International Journal of Mechanical Sciences, 12 (1970), 973-983.
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C. Z. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450.
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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, University of Maryland, Technical Report SRC-TR-92-107r5, College Park, MD 20742. |
show all references
References:
[1] |
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Singapore, 2006.
doi: 10.1142/6262. |
[2] |
M. Athans and P. L. Falb, Optimal Control, McGraw-Hill, 1966. |
[3] |
V. Azhmyakov, Optimal control of mechanical systems, Differential Equations and Nonlinear Mechanics, Volume 2007. |
[4] |
R. Bellman and R. E. Dreyfus, Dynamic Programming and Modern Control Theory, Orlands, Florida, Academic Press, 1977. |
[5] |
A. E. Jr. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing, DC, 1975. |
[6] |
C. Buskens, NUDOCCCS, FORTRAN-Subroutine NUDOCCCS (Numerical Discretisation method for Optimal Control problems with Constraints in Controls and States), 2010. http://www.swmath.org/software/8606 |
[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), 505-527.
doi: 10.1023/A:1026491014283. |
[8] |
L. Cesari, Optimization: Theory and Applications, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[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), 618-628.
doi: 10.1016/j.conengprac.2012.03.001. |
[10] |
B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance, Springer, The Netherlands, 2005. |
[11] |
M. Fikar, M. A. Latifi and Y. Creff, Optimal Changeover Profiles for an Industrial Depropanizer, Chemical Engineering Science, 54 (1999), 2715-2720.
doi: 10.1016/S0009-2509(98)00375-3. |
[12] |
M. E. Fisher and L. S. Jennings, MATLAB MISER, http://www.acad.polyu.edu.hk/ majlee/AMA483-523/OCTmanual.pdf |
[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. http://web.stanford.edu/group/SOL/npsol.htm |
[14] |
W. E. Gruver and E. Sachs, Algorithmic Methods in Optimal Control, Research Notes in Mathematics, Vol. 47, Pitman (Advance Publishing Program), London, 1981. |
[15] |
C. J. Goh and K. L. Teo, Control parameterization: a unified approach to optimal control problems with general constraints, Automatica, 24 (1988), 3-18.
doi: 10.1016/0005-1098(88)90003-9. |
[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), 395-425.
doi: 10.1007/BF00933463. |
[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), 3157-3175. |
[18] |
P. Howlett, The optimal control of a train, Annals of Operations Research, 98 (2000), 65-87.
doi: 10.1023/A:1019235819716. |
[19] |
H. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials, Journal of the Franklin Institute, 339 (2002), 479-498.
doi: 10.1016/S0016-0032(02)00028-5. |
[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. |
[21] |
L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica, 26 (1990), 371-375.
doi: 10.1016/0005-1098(90)90131-Z. |
[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), 591-609.
doi: 10.3934/jimo.2012.8.591. |
[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), 660-665.
doi: 10.1016/j.automatica.2012.01.019. |
[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), 30-53.
doi: 10.1007/s10957-012-0006-9. |
[25] |
C. Y. Kaya and J. M. Martnez, Euler discretization and inexact restoration for optimal control, Journal of Optimization Theory and Applications, 134 (2007), 191-206.
doi: 10.1007/s10957-007-9217-x. |
[26] |
C. Y. Kaya and J. L. Noakes, Leapfrog for Optimal Control, SIAM Journal on Numerical Analysis, in press, 2008.
doi: 10.1137/060675034. |
[27] |
M. I. Kamien and N. L. Schwartz, Dynamic Optimization - The Calculus of Variations and Optimal Control in Economics and Management, North Holland, 1991. |
[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), 47-63.
doi: 10.1007/BF00939247. |
[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), 866-875.
doi: 10.1016/j.amc.2013.08.092. |
[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), 260-291.
doi: 10.1007/s10957-011-9904-5. |
[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), 1101-1117.
doi: 10.3934/dcdsb.2011.16.1101. |
[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, Information and Control, 6 (2010), 521-532. |
[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), 162-177.
doi: 10.1017/S1446181110000040. |
[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), 131-140.
doi: 10.1016/j.actaastro.2011.12.021. |
[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), 1-16.
doi: 10.1080/01969729108902267. |
[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), 275-309.
doi: 10.3934/jimo.2014.10.275. |
[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), 973-980.
doi: 10.1016/j.automatica.2008.10.031. |
[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), 2250-2257.
doi: 10.1016/j.automatica.2009.05.029. |
[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), 2455-2460.
doi: 10.1109/TAC.2009.2029310. |
[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, Control and Optimization, 2 (2012), 571-599.
doi: 10.3934/naco.2012.2.571. |
[41] |
R. Loxton, Q. Lin and K. L. Teo, Minimizing control variation in nonlinear optimal control, Automatica, 49 (2013), 2652-2664.
doi: 10.1016/j.automatica.2013.05.027. |
[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), 537-560. |
[43] |
R. Luus, Iterative Dynamic Programming, Chapman & Hall/CRC, Boca Raton, 2000.
doi: 10.1201/9781420036022. |
[44] |
R. Luus and O. N. Okongwu, Towards practical optimal contorl of batch reactors, Chemical Engineering Journal, 75 (1999), 1-9. |
[45] |
R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, 1994, 185pp. |
[46] |
MATLAB - The Language of Technical Computing, http://mathworks.com/products/matlab/, 2008. |
[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), 185-203.
doi: 10.1002/(SICI)1099-1514(199805/06)19:3<185::AID-OCA627>3.0.CO;2-E. |
[48] |
H. H. Mehne and A. H. Borzabadi, A numerical method for solving optimal control problems using state parametrization, Numerical Algorithms, 42 (2006), 165-169.
doi: 10.1007/s11075-006-9035-5. |
[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), 21-54.
doi: 10.1016/0022-247X(86)90142-3. |
[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), 1-13. |
[51] |
R. Petzold and A. C. Hindmarsh, LSODA, Ordinary Differential Equation Solver for Stiff or Non-Stiff System, 2005. |
[52] |
L. S. Pontryagin, V. G. Boltayanskii, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, CRC Press, 1987. |
[53] |
V. Rehbock and I. Livk, Optimal control of a batch crystallization process, Journal of Industrial and Management Optimization, 3 (2007), 331-348.
doi: 10.3934/jimo.2007.3.585. |
[54] |
Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.
doi: 10.1016/0005-1098(82)90086-3. |
[55] |
K. Schittkowski, NLPQLP: A new fortran implementation of a sequential quadratic programming algorithm for parallel computing, 2010. |
[56] |
A. L. Schwartz, RIOTS-A Matlab toolbox for solving general optimal control problems, 2008. http://mechatronics.ucmerced.edu/RIOTS |
[57] |
Y. Shindo and Y. Sakawa, Local convergence of an algorithm for solving optimal control problems, Journal of Optimization Theory and Applications, 46 (1985), 265-293.
doi: 10.1007/BF00939285. |
[58] |
W. Sun and Y. X. Yuan, Optimization Theory and Methods, Springer, 2006. |
[59] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, England, 1991. |
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