A new computational strategy for optimal control problem with a cost on changing control

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2016, 6(3): 339-364. doi: 10.3934/naco.2016016

A new computational strategy for optimal control problem with a cost on changing control

Yujing Wang ^1, , Changjun Yu ^2, and Kok Lay Teo ^1,

1.	Department of Mathematics and Statistics, Curtin University, Perth, Australia, Australia
2.	Department of Mathematics , Shanghai University, Shanghai, China

Received April 2015 Revised September 2016 Published September 2016

Abstract
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In this paper, we consider a class of optimal control problems where the cost function is the sum of the terminal cost, the integral cost and the full variation of control. Here, the full variation of a control is defined as the sum of the total variations of its components. By using the control parameterization technique in conjunction with the time scaling transformation, we develop a new computational algorithm for solving this type of optimal control problem. Rigorous convergence analysis is provided for the new method. For illustration, we solve two numerical examples to show the effectiveness of the proposed method.

Keywords: control variation, total variation., time scaling transformation, Optimal control problem, control parametrization.

Mathematics Subject Classification: Primary: 49J15, 34H05; Secondary: 93C9.

Citation: Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016

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, Baltimore, USA, 2010.
[2]	B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with non-convex control constraints, Automatica, 47 (2011), 341-347. 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), 316-332.
[6]	N. Banihashemi and C. Y. Kaya, Inexact restoration and adaptive mesh refinement for optimal control, Journal of Industrial and Management Optimization, 10 (2014), 521-542. 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), 85-108. 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), 278-300.
[9]	M. Gerdts, Global convergence of a non-smooth Newton method for control-state constrained optimal control problems, SIAM Journal on Optimization, 19 (2008), 326-350. 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), 247-270. 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), 181-218. 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.
[13]	L. S. Jennings and K. L. Teo, A numerical algorithm for constrained optimal control problems with applications to harvesting, in "Dynamics of Complex Interconnected Biological Systems", Birkhauser Boston, Boston, (1990), 218-234. 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), 30-53. 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), 69-92. 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), 243-262.
[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), 63-81.
[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), 866-875. 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), 275-309. 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, Control and Optimization, 2 (2012), 571-599. 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), 2923-2929. 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), 2250-2257. 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), 376-392. 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.
[27]	Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.
[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.
[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), 212-228. 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), 181-190. 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), 335-357. doi: 10.1007/BF00941572.
[33]	R. J. Vanderbei, Case studies in trajectory optimization: trains, planes, and other pastimes, Optimization and Engineering, 2 (2001), 215-243. 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), 705-718. 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), 495-507. 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), 2-16. 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), 503-518. 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), 895-910, 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), 20-25.
[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), 485-491. 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, doi:10.3934/jimo.2016.12.781, 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), 12678-12693.

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, Baltimore, USA, 2010.
[2]	B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with non-convex control constraints, Automatica, 47 (2011), 341-347. 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), 316-332.
[6]	N. Banihashemi and C. Y. Kaya, Inexact restoration and adaptive mesh refinement for optimal control, Journal of Industrial and Management Optimization, 10 (2014), 521-542. 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), 85-108. 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), 278-300.
[9]	M. Gerdts, Global convergence of a non-smooth Newton method for control-state constrained optimal control problems, SIAM Journal on Optimization, 19 (2008), 326-350. 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), 247-270. 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), 181-218. 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.
[13]	L. S. Jennings and K. L. Teo, A numerical algorithm for constrained optimal control problems with applications to harvesting, in "Dynamics of Complex Interconnected Biological Systems", Birkhauser Boston, Boston, (1990), 218-234. 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), 30-53. 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), 69-92. 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), 243-262.
[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), 63-81.
[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), 866-875. 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), 275-309. 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, Control and Optimization, 2 (2012), 571-599. 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), 2923-2929. 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), 2250-2257. 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), 376-392. 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.
[27]	Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.
[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.
[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), 212-228. 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), 181-190. 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), 335-357. doi: 10.1007/BF00941572.
[33]	R. J. Vanderbei, Case studies in trajectory optimization: trains, planes, and other pastimes, Optimization and Engineering, 2 (2001), 215-243. 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), 705-718. 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), 495-507. 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), 2-16. 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), 503-518. 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), 895-910, 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), 20-25.
[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), 485-491. 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, doi:10.3934/jimo.2016.12.781, 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), 12678-12693.

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