doi: 10.3934/jimo.2019023

Minimizing almost smooth control variation in nonlinear optimal control problems

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

2. 

Department of Mathematics, Shanghai University, Baoshan 200444, Shanghai, China

3. 

Xingzhi College, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

* Corresponding author: Changjun Yu

Received  January 2018 Revised  March 2018 Published  March 2019

Fund Project: This paper is supported by NSFC grant 11871039, 11771275, the Scientific Research Project of Zhejiang Provincial Department of science and technology in China (Grant No. LGN19C040001), and the Scientific Research Project of Zhejiang Provincial Department of Education in China (Grant No. Y201329106)

In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach.

Citation: Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019023
References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, Singapore: World Scientific, 2006. doi: 10.1142/6262.

[2]

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.

[3]

D. Chang and Z. Wu, Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance, Journal of Industrial and Management Optimization, 11 (2015), 27-40. doi: 10.3934/jimo.2015.11.27.

[4]

M. Gerdts and M. Kunkel, A nonsmooth 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.

[5]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413.

[6]

Y. Han and Y. Gao, Determining the viability for hybrid control systems on a region with piecewise smooth boundary, Numerical Algebra, Control and Optimization, 5 (2015), 1-9. doi: 10.3934/naco.2015.5.1.

[7]

L. S. Jennings, M. E. Fisher, K. L. Teo, et al., MISER 3 Optimal Control Software: Theory and User Manual, version 3, University of Western Australia, 2004.

[8]

L. JenningsC. YuB. LiV. RehbockR. Loxton and F. Yang, Visual miser: an efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2015), 781-810. doi: 10.3934/jimo.2016.12.781.

[9]

J. KaartinenJ. HätönenH. Hyötyniemi and J. Miettunen, Machine-vision-based control of zinc flotation: A case study, Control Engineering Practice, 14 (2006), 1455-1466. doi: 10.1016/j.conengprac.2005.12.004.

[10]

C. T. Lawrence and A. L. Tits, A computationally efficient feasible sequential quadratic programming algorithm, SIAM Journal on Optimization, 11 (2006), 1092-1118. doi: 10.1137/S1052623498344562.

[11]

H. W. J. LeeK. L. TeoV. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261.

[12]

B. LiC. XuK. 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.

[13]

Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A surney, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275.

[14]

R. LoxtonQ. LinV. 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.

[15]

K. S. Peterson and A. G. Stefanopoulou, Extremum seeking control for soft landing of an electromechanical valve actuator, Automatica, 40 (2004), 1063-1069. doi: 10.1016/j.automatica.2004.01.027.

[16]

V. Rehbock, Tracking Control and Optimal Control, PhD thesis, University of Western Australia, Perth, 1994.

[17]

Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.

[18]

K. Schittkowski, NLPQL: A fortran subroutine solving constrained nonlinear programming problems, Ann. Oper. Res., 5 (1986), 485-500. doi: 10.1007/BF02739235.

[19]

K. L. Teo and L. S. Jennings, Nonlinear optimal control problems with continuous state inequality constraints, Journal of Optimization Theory and Application, 63 (1989), 1-22. doi: 10.1007/BF00940727.

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long-man Scientific and Technical, Essex, 1991.

[21]

W. X. Wang, Y. L. Shang, L. S. Zhang, et. al., Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446. doi: 10.1007/s11590-011-0427-7.

[22]

W. XuZ. G. FengJ. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193. doi: 10.1016/j.automatica.2016.12.002.

[23]

K. F. C. YiuY. Liu and K. L. Teo, A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229-238. doi: 10.1023/B:JOGO.0000015313.93974.b0.

[24]

C. YuK. L. TeoL. 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.

[25]

C. YuK. L. TeoL. 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.

show all references

References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, Singapore: World Scientific, 2006. doi: 10.1142/6262.

[2]

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.

[3]

D. Chang and Z. Wu, Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance, Journal of Industrial and Management Optimization, 11 (2015), 27-40. doi: 10.3934/jimo.2015.11.27.

[4]

M. Gerdts and M. Kunkel, A nonsmooth 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.

[5]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413.

[6]

Y. Han and Y. Gao, Determining the viability for hybrid control systems on a region with piecewise smooth boundary, Numerical Algebra, Control and Optimization, 5 (2015), 1-9. doi: 10.3934/naco.2015.5.1.

[7]

L. S. Jennings, M. E. Fisher, K. L. Teo, et al., MISER 3 Optimal Control Software: Theory and User Manual, version 3, University of Western Australia, 2004.

[8]

L. JenningsC. YuB. LiV. RehbockR. Loxton and F. Yang, Visual miser: an efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2015), 781-810. doi: 10.3934/jimo.2016.12.781.

[9]

J. KaartinenJ. HätönenH. Hyötyniemi and J. Miettunen, Machine-vision-based control of zinc flotation: A case study, Control Engineering Practice, 14 (2006), 1455-1466. doi: 10.1016/j.conengprac.2005.12.004.

[10]

C. T. Lawrence and A. L. Tits, A computationally efficient feasible sequential quadratic programming algorithm, SIAM Journal on Optimization, 11 (2006), 1092-1118. doi: 10.1137/S1052623498344562.

[11]

H. W. J. LeeK. L. TeoV. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261.

[12]

B. LiC. XuK. 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.

[13]

Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A surney, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275.

[14]

R. LoxtonQ. LinV. 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.

[15]

K. S. Peterson and A. G. Stefanopoulou, Extremum seeking control for soft landing of an electromechanical valve actuator, Automatica, 40 (2004), 1063-1069. doi: 10.1016/j.automatica.2004.01.027.

[16]

V. Rehbock, Tracking Control and Optimal Control, PhD thesis, University of Western Australia, Perth, 1994.

[17]

Y. Sakawa and Y. Shindo, Optimal control of container cranes, Automatica, 18 (1982), 257-266.

[18]

K. Schittkowski, NLPQL: A fortran subroutine solving constrained nonlinear programming problems, Ann. Oper. Res., 5 (1986), 485-500. doi: 10.1007/BF02739235.

[19]

K. L. Teo and L. S. Jennings, Nonlinear optimal control problems with continuous state inequality constraints, Journal of Optimization Theory and Application, 63 (1989), 1-22. doi: 10.1007/BF00940727.

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long-man Scientific and Technical, Essex, 1991.

[21]

W. X. Wang, Y. L. Shang, L. S. Zhang, et. al., Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446. doi: 10.1007/s11590-011-0427-7.

[22]

W. XuZ. G. FengJ. W. Peng and K. F. C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193. doi: 10.1016/j.automatica.2016.12.002.

[23]

K. F. C. YiuY. Liu and K. L. Teo, A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229-238. doi: 10.1023/B:JOGO.0000015313.93974.b0.

[24]

C. YuK. L. TeoL. 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.

[25]

C. YuK. L. TeoL. 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.

Figure 1.  Optimal control $ u_{1}(t) $.
Figure 2.  Optimal control function $ u_{2}(t) $.
Figure 3.  Optimal state trajectory $ x_{1}(t) $.
Figure 4.  Optimal state trajectory $ x_{2}(t) $.
Figure 5.  Optimal control $ x_{3}(t) $.
Figure 6.  Optimal state trajectory $ x_{4}(t) $.
Figure 7.  Optimal state trajectory $ x_{5}(t) $.
Figure 8.  Optimal control $ x_{6}(t) $.
[1]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

[2]

Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial & Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311

[3]

Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control & Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289

[4]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011

[5]

Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159

[6]

Thalya Burden, Jon Ernstberger, K. Renee Fister. Optimal control applied to immunotherapy. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 135-146. doi: 10.3934/dcdsb.2004.4.135

[7]

Ellina Grigorieva, Evgenii Khailov. Optimal control of pollution stock. Conference Publications, 2011, 2011 (Special) : 578-588. doi: 10.3934/proc.2011.2011.578

[8]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[9]

Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275

[10]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021

[11]

Gilles Carbou, Stéphane Labbé, Emmanuel Trélat. Smooth control of nanowires by means of a magnetic field. Communications on Pure & Applied Analysis, 2009, 8 (3) : 871-879. doi: 10.3934/cpaa.2009.8.871

[12]

Jorge San Martín, Takéo Takahashi, Marius Tucsnak. An optimal control approach to ciliary locomotion. Mathematical Control & Related Fields, 2016, 6 (2) : 293-334. doi: 10.3934/mcrf.2016005

[13]

C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435

[14]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455

[15]

Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030

[16]

Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415

[17]

Bavo Langerock. Optimal control problems with variable endpoints. Conference Publications, 2003, 2003 (Special) : 507-516. doi: 10.3934/proc.2003.2003.507

[18]

Alberto Bressan, Yunho Hong. Optimal control problems on stratified domains. Networks & Heterogeneous Media, 2007, 2 (2) : 313-331. doi: 10.3934/nhm.2007.2.313

[19]

François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41

[20]

M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure & Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (31)
  • HTML views (165)
  • Cited by (0)

Other articles
by authors

[Back to Top]