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A new computational strategy for optimal control problem with a cost on changing control
1. | Department of Mathematics and Statistics, Curtin University, Perth, Australia, Australia |
2. | Department of Mathematics , Shanghai University, Shanghai, China |
References:
[1] |
B. Açikmeşe and L. Blackmore, Lossless convexification of a class of nonconvex optimal control problems for linear systems,, In Proceedings of the 2010 American control conference, (2010). Google Scholar |
[2] |
B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with non-convex control constraints,, Automatica, 47 (2011), 341.
doi: 10.1016/j.automatica.2010.10.037. |
[3] |
N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory,, Essex: Longman Scientific and Technical, (1988).
|
[4] |
N. U. Ahmed, Dynamic Systems and Control with Applications,, Singapore: World Scientific, (2006).
doi: 10.1142/6262. |
[5] |
J. M. Blatt, Optimal control with a cost of switching control,, Journal of the Australian Mathematical Society-Series B: Applied Mathematics, 19 (1976), 316.
|
[6] |
N. Banihashemi and C. Y. Kaya, Inexact restoration and adaptive mesh refinement for optimal control,, Journal of Industrial and Management Optimization, 10 (2014), 521.
doi: 10.3934/jimo.2014.10.521. |
[7] |
C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis, and real-time control,, Journal of Computational and Applied Mathematics, 120 (2000), 85.
doi: 10.1016/S0377-0427(00)00305-8. |
[8] |
W. N. Chen, J. Zhang, H. S. H. Chung, W. L. Zhong, W. G. Wu and Y. H. Shi, A novel set-based particle swarm optimization method for discrete optimization problems,, IEEE Transactions on Evolutionary Computation, 14 (2010), 278. Google Scholar |
[9] |
M. Gerdts, Global convergence of a non-smooth Newton method for control-state constrained optimal control problems,, SIAM Journal on Optimization, 19 (2008), 326.
doi: 10.1137/060657546. |
[10] |
M. Gerdts and M. Kunkel, A non-smooth Newton's method for discretized optimal control problems with state and control constraints,, Journal of Industrial and Management Optimization, 4 (2008), 247.
doi: 10.3934/jimo.2008.4.247. |
[11] |
R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints,, SIAM Review, 37 (1995), 181.
doi: 10.1137/1037043. |
[12] |
L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3 Optimal Control Software: Theory and User Manual,, version 3. University of Western Australia, (2004). Google Scholar |
[13] |
L. S. Jennings and K. L. Teo, A numerical algorithm for constrained optimal control problems with applications to harvesting,, in, (1990), 218.
doi: 10.1007/978-1-4684-6784-0_12. |
[14] |
C. H. Jiang, Q. Lin, C. J. 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. |
[15] |
C. Y. Kaya and J. L. Noakes, Computational method for time-optimal switching control,, Journal of Optimization Theory and Applications, 117 (2003), 69.
doi: 10.1023/A:1023600422807. |
[16] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems,, Dynamic Systems and Applications, 6 (1997), 243.
|
[17] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems,, Pacific Journal of Optimization, 7 (2011), 63.
|
[18] |
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. |
[19] |
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 (2014), 275.
doi: 10.3934/jimo.2014.10.275. |
[20] |
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. |
[21] |
R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints,, Automatica, 44 (2008), 2923.
doi: 10.1016/j.automatica.2008.04.011. |
[22] |
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. |
[23] |
D. G. Luenberger and Y. Y. Ye, Linear and Nonlinear Programming,, (3rd ed.). New York: Springer, (2008).
|
[24] |
J. Matula, On an extremum problem,, Journal of the Australian Mathematical Society-Series B: Applied Mathematics, 28 (1987), 376.
doi: 10.1017/S0334270000005464. |
[25] |
J. Nocedal and S. J. Wright, Numerical Optimization,, (2nd ed.). New York: Springer, (2006).
|
[26] |
H. L. Royden and P. M. Fitzpatrick, Real analysis,, (4th ed.). Boston: Prentice Hall, (2010). Google Scholar |
[27] |
Y. Sakawa and Y. Shindo, Optimal control of container cranes,, Automatica, 18 (1982), 257. Google Scholar |
[28] |
K. Schittkowski, NLPQLP: a fortran implementation of a sequential quadratic programming algorithm with distributed and non-monotone line search,, version 2.24. University of Bayreuth, (2007). Google Scholar |
[29] |
D. E. Stewart, A numerical algorithm for optimal control problems with switching costs,, Journal of the Australian Mathematical Society-Series B: Applied Mathematics, 34 (1992), 212.
doi: 10.1017/S0334270000008730. |
[30] |
K. L. Teo and C. J. Goh, On constrained optimization problems with non-smooth cost functions,, Applied Mathematics and Optimization, 17 (1988), 181.
doi: 10.1007/BF01443621. |
[31] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Essex: Longman Scientific and Technical, (1991).
|
[32] |
K. L. Teo and L. S. Jennings, Optimal control with a cost on changing control,, Journal of Optimization Theory and Applications, 68 (1991), 335.
doi: 10.1007/BF00941572. |
[33] |
R. J. Vanderbei, Case studies in trajectory optimization: trains, planes, and other pastimes,, Optimization and Engineering, 2 (2001), 215.
doi: 10.1023/A:1013145328012. |
[34] |
T. L. Vincent and W. J. Grantham, Optimality in Parametric Systems,, New York: John Wiley, (1981).
|
[35] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.
doi: 10.3934/jimo.2009.5.705. |
[36] |
Z. Y. Wu, F. S. Bai, H. W. J. Lee and Y. J. Yang, A filled function method for constrained global optimization,, Journal of Global Optimization, 39 (2007), 495.
doi: 10.1007/s10898-007-9152-2. |
[37] |
X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants,, IEEE Transactions on Automatic Control, 49 (2004), 2.
doi: 10.1109/TAC.2003.821417. |
[38] |
C. J. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation,, Journal of Global Optimization, 56 (2013), 503.
doi: 10.1007/s10898-012-9858-7. |
[39] |
C. J. Yu, K. L. Teo , L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems,, Journal of Industrial and Management Optimization, 6 (2010), 895.
doi: 10.3934/jimo.2010.6.895. |
[40] |
C. J. Yu, K. L. Teo and T. T. Tiow, Optimal control with a cost of changing control,, Australian Control Conference (AUCC), (2013), 20. Google Scholar |
[41] |
C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem,, Journal of Industrial Management and Optimization, 8 (2012), 485.
doi: 10.3934/jimo.2012.8.485. |
[42] |
F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li , C. J. Yu and L. Jennings, Visual MISER: An efficient user-friendly visual program for solving optimal control problems,, Journal of Industrial and Management Optimization, (2016).
doi: 10.3934/jimo.2016.12.781. |
[43] |
Y. Zhao and M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints,, Industrial and Engineering Chemistry Research, 50 (2011), 12678. Google Scholar |
show all references
References:
[1] |
B. Açikmeşe and L. Blackmore, Lossless convexification of a class of nonconvex optimal control problems for linear systems,, In Proceedings of the 2010 American control conference, (2010). Google Scholar |
[2] |
B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with non-convex control constraints,, Automatica, 47 (2011), 341.
doi: 10.1016/j.automatica.2010.10.037. |
[3] |
N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory,, Essex: Longman Scientific and Technical, (1988).
|
[4] |
N. U. Ahmed, Dynamic Systems and Control with Applications,, Singapore: World Scientific, (2006).
doi: 10.1142/6262. |
[5] |
J. M. Blatt, Optimal control with a cost of switching control,, Journal of the Australian Mathematical Society-Series B: Applied Mathematics, 19 (1976), 316.
|
[6] |
N. Banihashemi and C. Y. Kaya, Inexact restoration and adaptive mesh refinement for optimal control,, Journal of Industrial and Management Optimization, 10 (2014), 521.
doi: 10.3934/jimo.2014.10.521. |
[7] |
C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis, and real-time control,, Journal of Computational and Applied Mathematics, 120 (2000), 85.
doi: 10.1016/S0377-0427(00)00305-8. |
[8] |
W. N. Chen, J. Zhang, H. S. H. Chung, W. L. Zhong, W. G. Wu and Y. H. Shi, A novel set-based particle swarm optimization method for discrete optimization problems,, IEEE Transactions on Evolutionary Computation, 14 (2010), 278. Google Scholar |
[9] |
M. Gerdts, Global convergence of a non-smooth Newton method for control-state constrained optimal control problems,, SIAM Journal on Optimization, 19 (2008), 326.
doi: 10.1137/060657546. |
[10] |
M. Gerdts and M. Kunkel, A non-smooth Newton's method for discretized optimal control problems with state and control constraints,, Journal of Industrial and Management Optimization, 4 (2008), 247.
doi: 10.3934/jimo.2008.4.247. |
[11] |
R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints,, SIAM Review, 37 (1995), 181.
doi: 10.1137/1037043. |
[12] |
L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3 Optimal Control Software: Theory and User Manual,, version 3. University of Western Australia, (2004). Google Scholar |
[13] |
L. S. Jennings and K. L. Teo, A numerical algorithm for constrained optimal control problems with applications to harvesting,, in, (1990), 218.
doi: 10.1007/978-1-4684-6784-0_12. |
[14] |
C. H. Jiang, Q. Lin, C. J. 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. |
[15] |
C. Y. Kaya and J. L. Noakes, Computational method for time-optimal switching control,, Journal of Optimization Theory and Applications, 117 (2003), 69.
doi: 10.1023/A:1023600422807. |
[16] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems,, Dynamic Systems and Applications, 6 (1997), 243.
|
[17] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems,, Pacific Journal of Optimization, 7 (2011), 63.
|
[18] |
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. |
[19] |
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 (2014), 275.
doi: 10.3934/jimo.2014.10.275. |
[20] |
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. |
[21] |
R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints,, Automatica, 44 (2008), 2923.
doi: 10.1016/j.automatica.2008.04.011. |
[22] |
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. |
[23] |
D. G. Luenberger and Y. Y. Ye, Linear and Nonlinear Programming,, (3rd ed.). New York: Springer, (2008).
|
[24] |
J. Matula, On an extremum problem,, Journal of the Australian Mathematical Society-Series B: Applied Mathematics, 28 (1987), 376.
doi: 10.1017/S0334270000005464. |
[25] |
J. Nocedal and S. J. Wright, Numerical Optimization,, (2nd ed.). New York: Springer, (2006).
|
[26] |
H. L. Royden and P. M. Fitzpatrick, Real analysis,, (4th ed.). Boston: Prentice Hall, (2010). Google Scholar |
[27] |
Y. Sakawa and Y. Shindo, Optimal control of container cranes,, Automatica, 18 (1982), 257. Google Scholar |
[28] |
K. Schittkowski, NLPQLP: a fortran implementation of a sequential quadratic programming algorithm with distributed and non-monotone line search,, version 2.24. University of Bayreuth, (2007). Google Scholar |
[29] |
D. E. Stewart, A numerical algorithm for optimal control problems with switching costs,, Journal of the Australian Mathematical Society-Series B: Applied Mathematics, 34 (1992), 212.
doi: 10.1017/S0334270000008730. |
[30] |
K. L. Teo and C. J. Goh, On constrained optimization problems with non-smooth cost functions,, Applied Mathematics and Optimization, 17 (1988), 181.
doi: 10.1007/BF01443621. |
[31] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Essex: Longman Scientific and Technical, (1991).
|
[32] |
K. L. Teo and L. S. Jennings, Optimal control with a cost on changing control,, Journal of Optimization Theory and Applications, 68 (1991), 335.
doi: 10.1007/BF00941572. |
[33] |
R. J. Vanderbei, Case studies in trajectory optimization: trains, planes, and other pastimes,, Optimization and Engineering, 2 (2001), 215.
doi: 10.1023/A:1013145328012. |
[34] |
T. L. Vincent and W. J. Grantham, Optimality in Parametric Systems,, New York: John Wiley, (1981).
|
[35] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.
doi: 10.3934/jimo.2009.5.705. |
[36] |
Z. Y. Wu, F. S. Bai, H. W. J. Lee and Y. J. Yang, A filled function method for constrained global optimization,, Journal of Global Optimization, 39 (2007), 495.
doi: 10.1007/s10898-007-9152-2. |
[37] |
X. Xu and P. J. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants,, IEEE Transactions on Automatic Control, 49 (2004), 2.
doi: 10.1109/TAC.2003.821417. |
[38] |
C. J. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation,, Journal of Global Optimization, 56 (2013), 503.
doi: 10.1007/s10898-012-9858-7. |
[39] |
C. J. Yu, K. L. Teo , L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems,, Journal of Industrial and Management Optimization, 6 (2010), 895.
doi: 10.3934/jimo.2010.6.895. |
[40] |
C. J. Yu, K. L. Teo and T. T. Tiow, Optimal control with a cost of changing control,, Australian Control Conference (AUCC), (2013), 20. Google Scholar |
[41] |
C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem,, Journal of Industrial Management and Optimization, 8 (2012), 485.
doi: 10.3934/jimo.2012.8.485. |
[42] |
F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li , C. J. Yu and L. Jennings, Visual MISER: An efficient user-friendly visual program for solving optimal control problems,, Journal of Industrial and Management Optimization, (2016).
doi: 10.3934/jimo.2016.12.781. |
[43] |
Y. Zhao and M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints,, Industrial and Engineering Chemistry Research, 50 (2011), 12678. Google Scholar |
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