March  2022, 18(2): 1305-1320. doi: 10.3934/jimo.2021021

On the optimal control problems with characteristic time control constraints

Shanghai University, Shanghai, 200444, China

* Corresponding author: Shuxuan Su

Received  June 2020 Revised  December 2020 Published  March 2022 Early access  January 2021

Fund Project: This work is supported by National Natural Science Foundation of China(NSFC), Grant No.11871039 and Science and Technology Commission of Shanghai Municipality(STCSM), Grant No. 20JC1413900

In this paper, we consider a class of optimal control problems with control constraints on a set of characteristic time instants. By applying the control parameterization technique, these constraints are imposed on the subdomains that contain the characteristic time points. The values of the control functions as well as the lengths for their corresponding subdomains become decision variables. Time-scaling transformation is an effective technique to optimize the length of each subdomain in a new time horizon. However, the characteristic time instants in the original time horizon become variable time instants in the new time horizon, and hence the control constraints imposed on these characteristic time points are difficult to be formulated in the new time horizon. We propose a surrogate condition and show that the characteristic time control constraints will be satisfied once the surrogate condition holds. Moreover, this surrogate condition is easy to formulate in the new time horizon. The resulting approximate problem can be readily solved by many existing computational methods for solving constrained optimal control problems. Finally, we conclude this paper by solving two examples.

Citation: Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1305-1320. doi: 10.3934/jimo.2021021
References:
[1]

N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Longman Scientific and Technical, 1988.

[2]

M. Athans, Advances in Control Systems: Theory and Applications, vol. 11, Elsevier, 1966. doi: 10.1109/TAC.1966.1098358.

[3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.
[4]

X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin and J. G. Muga, Fast optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity, Physical Review Letters, 104 (2010), 063002. doi: 10.1103/PhysRevLett.104.063002.

[5]

X. Chen, E. Torrontegui and J. G. Muga, Lewis-riesenfeld invariants and transitionless quantum driving, Physical Review A, 83 (2011), 062116. doi: 10.1103/PhysRevA.83.062116.

[6]

X. Chen, E. Torrontegui, D. Stefanatos, J.-S. Li and J. G. Muga, Optimal trajectories for efficient atomic transport without final excitation, Physical Review A, 84 (2011), 043415. doi: 10.1103/PhysRevA.84.043415.

[7]

Z. GongC. Liu and Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042.

[8]

T.-Y. Huang, B. A. Malomed and X. Chen, Shortcuts to adiabaticity for an interacting bose–einstein condensate via exact solutions of the generalized ermakov equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30 (2020), 053131, 10 pp. doi: 10.1063/5.0004309.

[9]

L. Jennings, K. L. Teo, M. Fisher, C. J. Goh, L. S. Jennings and M. E. Fisher, MISER3 version 2, Optimal Control Software: Theory and User Manual, vol. 1, The University of Western Australia, Australia, 1997.

[10]

C. JiangK. XieC. YuM. YuH. WangY. He and K. L. Teo, A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit runge-kutta integration, Applied Mathematical Modelling, 58 (2018), 313-330.  doi: 10.1016/j.apm.2017.05.015.

[11]

Y. Kagan, E. L. Surkov and G. V. Shlyapnikov, Evolution of a bose-condensed gas under variations of the confining potential, Physical Review A, 54 (1996), R1753–R1756. doi: 10.1103/PhysRevA.54.R1753.

[12]

D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004.

[13]

L. KongC. YuK. L. Teo and C. Yang, Robust real-time optimization for blending operation of alumina production, Journal of Industrial and Management Optimization, 13 (2017), 1149-1167.  doi: 10.3934/jimo.2016066.

[14]

H. LeeK. TeoV. Rehbock and L. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems & Applications, 6 (1997), 243-262. 

[15]

J. Li, K. Sun and X. Chen, Shortcut to adiabatic control of soliton matter waves by tunable interaction, Scientific Reports, 6 (2016), 38258. doi: 10.1038/srep38258.

[16]

L. LiC. YuN. ZhangY. Bai and Z. Gao, A time-scaling technique for time-delay switched systems, Discrete and Continuous Dynamical Systems-S, 13 (2020), 1825-1843.  doi: 10.3934/dcdss.2020108.

[17]

Q. LinR. 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.

[18]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems B, 18 (2011), 59-76. 

[19]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependent stopping criteria, Automatica, 48 (2012), 2116-2129.  doi: 10.1016/j.automatica.2012.06.055.

[20]

C. LiuZ. GongE. Feng and H. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Journal of Industrial and Management Optimization, 5 (2009), 835-850.  doi: 10.3934/jimo.2009.5.835.

[21]

R. C. LoxtonK. 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]

P. MuL. Wang and C. Liu, A control parameterization method to solve the fractional-order optimal control problem, Journal of Optimization Theory and Applications, 187 (2020), 234-247.  doi: 10.1007/s10957-017-1163-7.

[23]

J. G. Muga, X. Chen, A. Ruschhaupt and D. Guéryodelin, Frictionless dynamics of bose–einstein condensates under fast trap variations, Journal of Physics B: Atomic Molecular and Optical Physics, 42 (2009), 241001. doi: 10.1088/0953-4075/42/24/241001.

[24]

J. G. Muga, X. Chen, E. Torrontegui, S. Ibá nez, I. Lizuain and A. Ruschhaupt, Shortcuts to quantum adiabatic processes, Journal of Physics Conference, 306 (2011), 012022. doi: 10.1088/1742-6596/306/1/012022.

[25]

M. NakazawaK. KurokawaH. Kubota and E. Yamada, Observation of the trapping of an optical soliton by adiabatic gain narrowing and its escape, Physical Review Letters, 65 (1990), 1881-1884.  doi: 10.1103/PhysRevLett.65.1881.

[26]

J. Nocedal, S. Wright, T. Mikosch, S. Resnick and S. Robinson, Numerical Optimization, Springer series in operations research and financial engineering, 1999.

[27]

L. Salasnich, A. Parola and L. Reatto, Effective wave equations for the dynamics of cigar-shaped and disk-shaped bose condensates, Physical Review A, 65 (2002), 043614. doi: 10.1103/PhysRevA.65.043614.

[28]

D. Stefanatos, Optimal shortcuts to adiabaticity for a quantum piston, Automatica, 49 (2013), 3079-3083.  doi: 10.1016/j.automatica.2013.07.020.

[29]

A. Tobalina, M. Palmero, S. Mart'inez-Garaot and J. G. Muga, Fast atom transport and launching in a nonrigid trap, Scientific Reports, 7 (2017), 5753. doi: 10.1038/s41598-017-05823-x.

[30]

E. Torrontegui, S. Ibá nez, X. Chen, A. Ruschhaupt, D. Guéry-Odelin and J. G. Muga, Fast atomic transport without vibrational heating, Physical Review A, 83 (2011), 013415. doi: 10.1103/PhysRevA.83.013415.

[31]

E. TorronteguiS. Ibá nezS. Martínez-GaraotM. ModugnoA. del CampoD. Guéry-OdelinA. RuschhauptX. Chen and J. G. Muga, Shortcuts to adiabaticity, Advances in Atomic, Molecular, and Optical Physics, 62 (2013), 117-169.  doi: 10.1016/B978-0-12-408090-4.00002-5.

[32]

L. WangJ. YuanC. Wu and X. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z.

[33]

Y. WangC. Yu and K. Teo, A new computational strategy for optimal control problem with a cost on changing control, Numerical Algebra, Control and Optimization, 6 (2016), 339-364.  doi: 10.3934/naco.2016016.

[34]

D. WuY. Bai and C. Yu, A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.  doi: 10.1016/j.automatica.2018.12.036.

[35]

F. YangK. L. TeoR. LoxtonV. RehbockB. LiC. Yu and L. Jennings, Visual miser: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.

[36]

C. YuQ. LinR. LoxtonK. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901.  doi: 10.1007/s10957-015-0783-z.

[37]

C. YuY. Wang and L. Li, Smoothing spline via optimal control under uncertainty, Applied Mathematical Modelling, 58 (2018), 203-216.  doi: 10.1016/j.apm.2017.07.062.

[38]

Y. ZhangC. YuY. Xu and Y. Bai, Minimizing almost smooth control variation in nonlinear optimal control problems, Journal of Industrial and Management Optimization, 16 (2020), 1663-1683.  doi: 10.3934/jimo.2019023.

show all references

References:
[1]

N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Longman Scientific and Technical, 1988.

[2]

M. Athans, Advances in Control Systems: Theory and Applications, vol. 11, Elsevier, 1966. doi: 10.1109/TAC.1966.1098358.

[3] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.
[4]

X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin and J. G. Muga, Fast optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity, Physical Review Letters, 104 (2010), 063002. doi: 10.1103/PhysRevLett.104.063002.

[5]

X. Chen, E. Torrontegui and J. G. Muga, Lewis-riesenfeld invariants and transitionless quantum driving, Physical Review A, 83 (2011), 062116. doi: 10.1103/PhysRevA.83.062116.

[6]

X. Chen, E. Torrontegui, D. Stefanatos, J.-S. Li and J. G. Muga, Optimal trajectories for efficient atomic transport without final excitation, Physical Review A, 84 (2011), 043415. doi: 10.1103/PhysRevA.84.043415.

[7]

Z. GongC. Liu and Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042.

[8]

T.-Y. Huang, B. A. Malomed and X. Chen, Shortcuts to adiabaticity for an interacting bose–einstein condensate via exact solutions of the generalized ermakov equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30 (2020), 053131, 10 pp. doi: 10.1063/5.0004309.

[9]

L. Jennings, K. L. Teo, M. Fisher, C. J. Goh, L. S. Jennings and M. E. Fisher, MISER3 version 2, Optimal Control Software: Theory and User Manual, vol. 1, The University of Western Australia, Australia, 1997.

[10]

C. JiangK. XieC. YuM. YuH. WangY. He and K. L. Teo, A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit runge-kutta integration, Applied Mathematical Modelling, 58 (2018), 313-330.  doi: 10.1016/j.apm.2017.05.015.

[11]

Y. Kagan, E. L. Surkov and G. V. Shlyapnikov, Evolution of a bose-condensed gas under variations of the confining potential, Physical Review A, 54 (1996), R1753–R1756. doi: 10.1103/PhysRevA.54.R1753.

[12]

D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004.

[13]

L. KongC. YuK. L. Teo and C. Yang, Robust real-time optimization for blending operation of alumina production, Journal of Industrial and Management Optimization, 13 (2017), 1149-1167.  doi: 10.3934/jimo.2016066.

[14]

H. LeeK. TeoV. Rehbock and L. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems & Applications, 6 (1997), 243-262. 

[15]

J. Li, K. Sun and X. Chen, Shortcut to adiabatic control of soliton matter waves by tunable interaction, Scientific Reports, 6 (2016), 38258. doi: 10.1038/srep38258.

[16]

L. LiC. YuN. ZhangY. Bai and Z. Gao, A time-scaling technique for time-delay switched systems, Discrete and Continuous Dynamical Systems-S, 13 (2020), 1825-1843.  doi: 10.3934/dcdss.2020108.

[17]

Q. LinR. 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.

[18]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems B, 18 (2011), 59-76. 

[19]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependent stopping criteria, Automatica, 48 (2012), 2116-2129.  doi: 10.1016/j.automatica.2012.06.055.

[20]

C. LiuZ. GongE. Feng and H. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Journal of Industrial and Management Optimization, 5 (2009), 835-850.  doi: 10.3934/jimo.2009.5.835.

[21]

R. C. LoxtonK. 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]

P. MuL. Wang and C. Liu, A control parameterization method to solve the fractional-order optimal control problem, Journal of Optimization Theory and Applications, 187 (2020), 234-247.  doi: 10.1007/s10957-017-1163-7.

[23]

J. G. Muga, X. Chen, A. Ruschhaupt and D. Guéryodelin, Frictionless dynamics of bose–einstein condensates under fast trap variations, Journal of Physics B: Atomic Molecular and Optical Physics, 42 (2009), 241001. doi: 10.1088/0953-4075/42/24/241001.

[24]

J. G. Muga, X. Chen, E. Torrontegui, S. Ibá nez, I. Lizuain and A. Ruschhaupt, Shortcuts to quantum adiabatic processes, Journal of Physics Conference, 306 (2011), 012022. doi: 10.1088/1742-6596/306/1/012022.

[25]

M. NakazawaK. KurokawaH. Kubota and E. Yamada, Observation of the trapping of an optical soliton by adiabatic gain narrowing and its escape, Physical Review Letters, 65 (1990), 1881-1884.  doi: 10.1103/PhysRevLett.65.1881.

[26]

J. Nocedal, S. Wright, T. Mikosch, S. Resnick and S. Robinson, Numerical Optimization, Springer series in operations research and financial engineering, 1999.

[27]

L. Salasnich, A. Parola and L. Reatto, Effective wave equations for the dynamics of cigar-shaped and disk-shaped bose condensates, Physical Review A, 65 (2002), 043614. doi: 10.1103/PhysRevA.65.043614.

[28]

D. Stefanatos, Optimal shortcuts to adiabaticity for a quantum piston, Automatica, 49 (2013), 3079-3083.  doi: 10.1016/j.automatica.2013.07.020.

[29]

A. Tobalina, M. Palmero, S. Mart'inez-Garaot and J. G. Muga, Fast atom transport and launching in a nonrigid trap, Scientific Reports, 7 (2017), 5753. doi: 10.1038/s41598-017-05823-x.

[30]

E. Torrontegui, S. Ibá nez, X. Chen, A. Ruschhaupt, D. Guéry-Odelin and J. G. Muga, Fast atomic transport without vibrational heating, Physical Review A, 83 (2011), 013415. doi: 10.1103/PhysRevA.83.013415.

[31]

E. TorronteguiS. Ibá nezS. Martínez-GaraotM. ModugnoA. del CampoD. Guéry-OdelinA. RuschhauptX. Chen and J. G. Muga, Shortcuts to adiabaticity, Advances in Atomic, Molecular, and Optical Physics, 62 (2013), 117-169.  doi: 10.1016/B978-0-12-408090-4.00002-5.

[32]

L. WangJ. YuanC. Wu and X. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z.

[33]

Y. WangC. Yu and K. Teo, A new computational strategy for optimal control problem with a cost on changing control, Numerical Algebra, Control and Optimization, 6 (2016), 339-364.  doi: 10.3934/naco.2016016.

[34]

D. WuY. Bai and C. Yu, A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.  doi: 10.1016/j.automatica.2018.12.036.

[35]

F. YangK. L. TeoR. LoxtonV. RehbockB. LiC. Yu and L. Jennings, Visual miser: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.

[36]

C. YuQ. LinR. LoxtonK. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901.  doi: 10.1007/s10957-015-0783-z.

[37]

C. YuY. Wang and L. Li, Smoothing spline via optimal control under uncertainty, Applied Mathematical Modelling, 58 (2018), 203-216.  doi: 10.1016/j.apm.2017.07.062.

[38]

Y. ZhangC. YuY. Xu and Y. Bai, Minimizing almost smooth control variation in nonlinear optimal control problems, Journal of Industrial and Management Optimization, 16 (2020), 1663-1683.  doi: 10.3934/jimo.2019023.

Figure 1.  Time-scaling function for pre-fixed terminal time
Figure 2.  Optimal control for Problem 1 with $ p = 9 $.
Figure 3.  Optimal state trajectories for Problem 1 obtained using the two different methods with $ p = 9 $.
Figure 4.  Optimal controls for Problem 2 with $ p = 5 $
Figure 5.  Optimal controls for Problem 2 with $ p = 6 $
Figure 6.  Optimal controls for Problem 2 with p = 7
Figure 7.  Optimal state trajectories for Problem 2 with different partition numbers
Table 1.  Optimal costs for Problem 1 obtained using the two different methods
Knots $ G_0^* $
Traditional control parameterization New time-scaling transformation
$ p=5 $ $ 110.8 $ $ 1.9531 $
$ p=7 $ $ 42.34 $ $ 1.6671 $
$ p=9 $ $ 21.79 $ $ 1.6053 $
Knots $ G_0^* $
Traditional control parameterization New time-scaling transformation
$ p=5 $ $ 110.8 $ $ 1.9531 $
$ p=7 $ $ 42.34 $ $ 1.6671 $
$ p=9 $ $ 21.79 $ $ 1.6053 $
Table 2.  Optimal costs for Problem 2 using different lower bounds on each $ \theta_l $ with different partition $ p $
Lower bounds on $ \theta_l\ (l=1, \ldots, p) $ $ J_0^* $
$ p=5 $ $ p=6 $ $ p=7 $
$ \geq 0.05 $ $ 12.189 $ $ 8.113 $ $ 4.572 $
$ \geq 0.1 $ $ 12.347 $ $ 8.258 $ $ 4.673 $
$ \geq 0.5 $ $ 13.147 $ $ 9.442 $ $ 5.506 $
$ \geq 1 $ $ 14.453 $ $ 10.924 $ $ 7.000 $
Lower bounds on $ \theta_l\ (l=1, \ldots, p) $ $ J_0^* $
$ p=5 $ $ p=6 $ $ p=7 $
$ \geq 0.05 $ $ 12.189 $ $ 8.113 $ $ 4.572 $
$ \geq 0.1 $ $ 12.347 $ $ 8.258 $ $ 4.673 $
$ \geq 0.5 $ $ 13.147 $ $ 9.442 $ $ 5.506 $
$ \geq 1 $ $ 14.453 $ $ 10.924 $ $ 7.000 $
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