Research on the Application of Numerical Method in Control Theory

Yuanyuan Zhang

doi:doi:10.11648/j.acm.20170604.14

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Research on the Application of Numerical Method in Control Theory

Yuanyuan Zhang

Published in Applied and Computational Mathematics (Volume 6, Issue 4)

Received: 19 July 2017 Published: 19 July 2017

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Abstract

Time optimal control problems of ordinary differential equations have been of great interest for decades due to their practical applications. There are mainly two ways to compute optimal times. The first one is the Switching Time Optimization method, where the switching time is taken as extra unknowns and the optimization problems is solved by nonlinear programming technique. The second one is based on the first order necessary condition for optimal control. In this paper, we extend the numerical method given in [1] for the computation of the optimal time for the time optimal control problems. In the end some examples are provided to show the efficiency of the numerical method.

Published in	Applied and Computational Mathematics (Volume 6, Issue 4)
DOI	10.11648/j.acm.20170604.14
Page(s)	185-188
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Numerical Method, Ordinary Differential Equation, Time Optimal Control Problems

References

[1]	X. Lu, L. Wang and Q. Yan, “Computation of Time Optimal Control Problems governed by Linear Ordinary Differential Equations”, Journal of Scientific Computing, 2017, pp. 1-25.
[2]	V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993, pp. 74-98.
[3]	L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture Notes, Department of Mathematics, University of California, Berkeley, 2008, pp. 92-101.
[4]	H. O. Fattorini, Infinite Dimensional Linear Control Systems, the Time Optimal and Norm Optimal Control Problems, North-Holland Mathematics Studies 201, Elsevier, 2005, pp.48-67.
[5]	K. Ito and K. Kunisch, “Semismooth newton methods for time-optimal control for a class of ODES”, SIAM J. Control Optim., vol. 48, 2010, pp. 3997-4013.
[6]	C. Y. Kaya and J. L. Noakes, “Computations and time-optimal controls”, Optimal Control Applications and Methods, vol.17, 1996, pp.171-185.
[7]	C. Y. Kaya and J. L. Noakes, “Computational methods for time-optimal switching controls”, J. Optim. Theory Appl., vol. 117, 2003, pp. 69-92.
[8]	J. P. Lasalle, The Time Optimal Control Problem, Contributions to the Theory of Nonlinear Oscillations, Princeton University Press, Princeton, 1960, 1-24.
[9]	X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995, pp. 127-135.
[10]	P. Lin and G. Wang, “Blowup time optimal control for ordinary differential equations”, SIAM J. Control Optim., vol. 49, 2011, pp. 73-105.
[11]	E. Meier and A. E. Bryson, Efficient algorithms for time-optimal rigid spacecraft reorientation problem, Journal of Guidance, Control, and Dynamics, vol. 13, 1990, pp. 859-866.
[12]	T. Li and B. Rao. “A note on the exact synchronization by groups for a coupled system of wave equations”, Mathematical Methods in the Applied Sciences, vol. 38, 2015, pp. 241–246.
[13]	L. Hu, F. Ji and K. Wang. “Exact boundary controllability and exact boundary observability for a coupled system of quasilinear wave equations”, Chinese Annals of Mathematics, vol.34 (4), 2013, pp. 479–490.
[14]	B. Z. Guo, D. H. Yang and L. Zhang. “On optimal location of diffusion and related optimal control for null controllable heat equation”, Journal of Mathematical Analysis and Applications, vol. 433 (2), 2016, pp. 1333-1349.
[15]	M. Chen. “Bang bang property for time optimal control of the korteweg de vries burgers equation”, appl. math. opt., vol. 2, 2016, pp. 1-16.
[16]	L. Wang and Q. Yan. “Time Optimal Controls of Semilinear Heat Equation with Switching Control”, Journal of Theory and Applications, vol. 165 (1), 2015: 263-278.

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Yuanyuan Zhang. (2017). Research on the Application of Numerical Method in Control Theory. Applied and Computational Mathematics, 6(4), 185-188. https://doi.org/10.11648/j.acm.20170604.14

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ACS Style

Yuanyuan Zhang. Research on the Application of Numerical Method in Control Theory. Appl. Comput. Math. 2017, 6(4), 185-188. doi: 10.11648/j.acm.20170604.14

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AMA Style

Yuanyuan Zhang. Research on the Application of Numerical Method in Control Theory. Appl Comput Math. 2017;6(4):185-188. doi: 10.11648/j.acm.20170604.14

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@article{10.11648/j.acm.20170604.14,
  author = {Yuanyuan Zhang},
  title = {Research on the Application of Numerical Method in Control Theory},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {4},
  pages = {185-188},
  doi = {10.11648/j.acm.20170604.14},
  url = {https://doi.org/10.11648/j.acm.20170604.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170604.14},
  abstract = {Time optimal control problems of ordinary differential equations have been of great interest for decades due to their practical applications. There are mainly two ways to compute optimal times. The first one is the Switching Time Optimization method, where the switching time is taken as extra unknowns and the optimization problems is solved by nonlinear programming technique. The second one is based on the first order necessary condition for optimal control. In this paper, we extend the numerical method given in [1] for the computation of the optimal time for the time optimal control problems. In the end some examples are provided to show the efficiency of the numerical method.},
 year = {2017}
}

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TY  - JOUR
T1  - Research on the Application of Numerical Method in Control Theory
AU  - Yuanyuan Zhang
Y1  - 2017/07/19
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170604.14
DO  - 10.11648/j.acm.20170604.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 185
EP  - 188
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170604.14
AB  - Time optimal control problems of ordinary differential equations have been of great interest for decades due to their practical applications. There are mainly two ways to compute optimal times. The first one is the Switching Time Optimization method, where the switching time is taken as extra unknowns and the optimization problems is solved by nonlinear programming technique. The second one is based on the first order necessary condition for optimal control. In this paper, we extend the numerical method given in [1] for the computation of the optimal time for the time optimal control problems. In the end some examples are provided to show the efficiency of the numerical method.
VL  - 6
IS  - 4
ER  -

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Author Information

Yuanyuan Zhang

College of Science, China Three Gorges University, Yichang, China

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Plain Text BibTeX RIS

APA Style

Yuanyuan Zhang. (2017). Research on the Application of Numerical Method in Control Theory. Applied and Computational Mathematics, 6(4), 185-188. https://doi.org/10.11648/j.acm.20170604.14

Copy | Download

ACS Style

Yuanyuan Zhang. Research on the Application of Numerical Method in Control Theory. Appl. Comput. Math. 2017, 6(4), 185-188. doi: 10.11648/j.acm.20170604.14

Copy | Download

AMA Style

Yuanyuan Zhang. Research on the Application of Numerical Method in Control Theory. Appl Comput Math. 2017;6(4):185-188. doi: 10.11648/j.acm.20170604.14

Copy | Download

@article{10.11648/j.acm.20170604.14,
  author = {Yuanyuan Zhang},
  title = {Research on the Application of Numerical Method in Control Theory},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {4},
  pages = {185-188},
  doi = {10.11648/j.acm.20170604.14},
  url = {https://doi.org/10.11648/j.acm.20170604.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170604.14},
  abstract = {Time optimal control problems of ordinary differential equations have been of great interest for decades due to their practical applications. There are mainly two ways to compute optimal times. The first one is the Switching Time Optimization method, where the switching time is taken as extra unknowns and the optimization problems is solved by nonlinear programming technique. The second one is based on the first order necessary condition for optimal control. In this paper, we extend the numerical method given in [1] for the computation of the optimal time for the time optimal control problems. In the end some examples are provided to show the efficiency of the numerical method.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Research on the Application of Numerical Method in Control Theory
AU  - Yuanyuan Zhang
Y1  - 2017/07/19
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170604.14
DO  - 10.11648/j.acm.20170604.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 185
EP  - 188
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170604.14
AB  - Time optimal control problems of ordinary differential equations have been of great interest for decades due to their practical applications. There are mainly two ways to compute optimal times. The first one is the Switching Time Optimization method, where the switching time is taken as extra unknowns and the optimization problems is solved by nonlinear programming technique. The second one is based on the first order necessary condition for optimal control. In this paper, we extend the numerical method given in [1] for the computation of the optimal time for the time optimal control problems. In the end some examples are provided to show the efficiency of the numerical method.
VL  - 6
IS  - 4
ER  -

Copy | Download