Engineering Mathematics

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Relative Controllability for a Class of Linear Impulsive Systems

Received: 3 November 2022    Accepted: 15 December 2022    Published: 27 February 2023
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Abstract

Hybrid systems are systems that involve continuous and discrete event dynamical behaviors. A impulsive system is a special hybrid system. The continuous dynamics of impulsive systems are usually described by ordinary differential equations and the discrete event dynamics with instantaneously rapid jumps are described by switching laws. Various complex dynamical phenomena that can be modeled by impulsive systems arise in many areas of modern science and technology such as economics, physics, chemistry, biology, information science, radiotherapy, acupuncture, robotics, neural networks, automatic control, artificial intelligence, space technology, and telecommunications, etc. In modern control theory, controllability is one of the most important dynamical properties of considered impulsive systems, therefore, the controllability problem is regarded as one of the fundamental issues of impulsive systems. The basic questions for controllability of impulsive systems as well as for the ordinary systems without impulses and with control function are to obtain useful criteria that allow us to identify whether given dynamic systems are controllable or not. Up to now there have been being many investigation results for controllability of different kinds of impulsive systems with respect to the terminal state constraints of a point type. The purpose of this paper is to study relative controllability with respect to the terminal state constraint of a general type for a class of linear time-varying impulsive systems. In this paper, several types of criteria for relative controllability of such systems are established by a algebraic method, that is, specially speaking, by the matrix rank method. Some corresponding necessary and sufficient conditions for controllability of linear time-invariant impulsive systems are also obtained more compactly. Meanwhile, for given impulsive systems some equivalent relationships between different kinds of controllability are investigated and our criteria for relative controllability are compared with the existing results. A simple example is given to illustrate the utility of our criteria.

DOI 10.11648/j.engmath.20230701.11
Published in Engineering Mathematics (Volume 7, Issue 1, June 2023)
Page(s) 1-18
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), 2024. Published by Science Publishing Group

Keywords

Impulsive Systems, Impulsive Control, Complete Controllability, Null Controllability, Relative Controllability, Relative Null Controllability

References
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[2] R. K. George, A. K. Nandakumaran and A. Arapostathis, A note on controllability of impulsive systems, Journal of Mathematical Analysis and Applications, 241 (2000), 276-283.
[3] Z. H. Guan, T. H. Qian and X. H. Yu, On controllability and observability for a class of impulsive systems, Systems & Control Letters, 47 (2002), 247-257.
[4] Z. H. Guan, T. H. Qian and X. H. Yu, Controllability and observability of linear time-varying impulsive systems, IEEE Transaction on Circuits and Systems, 49, 8 (2002), 1198-1208.
[5] S. C. Ji, G. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Applied Mathematics and Computation, 217 (2011), 6981-6989.
[6] E. Joelianto and D. Williamson, Controllability of linear impulsive systems, Information, Decision and Control: Data and Information Fusion Symposium, Signal Processing and Communications Symposium and Decision and Control Symposium, Adelaide, SA, Australia, IEEE, 2002, 205-210.
[7] S. Leela, Farzana A. McRae and S. Sivasundaram, Controllability of impulsive differential equations, Journal of Mathematical Analysis and Applications, 177 (1993), 24-30.
[8] X. Z. Liu, Impulsive control and optimization, Applied Mathematics and Computation, 73 (1995), 77-98.
[9] B. M. Miller and E. Ya. Rubinovich, Impulsive control in continuous and discrete-continuous systems, Norwell, MA: Kluwer, 2003.
[10] S. Sivasundaram and Josaphat Uvah, Controllability of impulsive hybrid integro-differential systems, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1003-1009.
[11] Vijayakumar S. Muni and Raju K. George, Controllability of semelinear impulsive control systems with multiple time delays in control, IMA Journal of Mathematical Control and Information, 00 (2018), 1-31.
[12] G. M. Xie and L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Transactions on Automatic Control, 49, 6 (2004), 960-966.
[13] G. M. Xie and L. Wang, Controllability and observability of a class of linear impulsive systems, Journal of Mathematical Analysis and Applications, 304 (2005), 336-355.
[14] D. G. Xu, Controllability and observability of a class of piecewise linear impulsive control systems, Advances in Computer, Communication, Control & Automation, LNEE 121 (2011), 321-328.
[15] T. Yang, Impulsive control theory, Springer, 2001.
[16] Z. L. You, M. Feˇckan, J. R. Wang and D. O’Regan, Relative controllability of impulsive multi-delay differential systems, Nonlinear Analysis: Modeling and Control, 27, 1 (2022), 70-90.
[17] S. W. Zhao and J. T. Sun, Controllability and obserbility for a class of time-varying impulsive systems, Nonlinear Analysis: Real World Applications, 10 (2009), 1370-1380.
[18] S. W. Zhao and J. T. Sun, Controllability and obserbility for impulsive systems in complex fields, Nonlinear Analysis: Real World Applications, 11 (2010), 1513-1521.
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[20] S. W. Zhao, Z. H. Zhang, T. B. Wang and W. Q. Yu, Controllability for a class of time-varying controlled switching impulsive systems with time delays, IEEE Transactions on Cercuits and Systems-I Fundamental Theory and Applications, 49, 8 (2002), 1198-1208.
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  • APA Style

    Gwang Jin Kim, Su Song Om, Tae Gun Oh, Nam Chol Yu, Jin Sim Kim. (2023). Relative Controllability for a Class of Linear Impulsive Systems. Engineering Mathematics, 7(1), 1-18. https://doi.org/10.11648/j.engmath.20230701.11

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    ACS Style

    Gwang Jin Kim; Su Song Om; Tae Gun Oh; Nam Chol Yu; Jin Sim Kim. Relative Controllability for a Class of Linear Impulsive Systems. Eng. Math. 2023, 7(1), 1-18. doi: 10.11648/j.engmath.20230701.11

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    AMA Style

    Gwang Jin Kim, Su Song Om, Tae Gun Oh, Nam Chol Yu, Jin Sim Kim. Relative Controllability for a Class of Linear Impulsive Systems. Eng Math. 2023;7(1):1-18. doi: 10.11648/j.engmath.20230701.11

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  • @article{10.11648/j.engmath.20230701.11,
      author = {Gwang Jin Kim and Su Song Om and Tae Gun Oh and Nam Chol Yu and Jin Sim Kim},
      title = {Relative Controllability for a Class of Linear Impulsive Systems},
      journal = {Engineering Mathematics},
      volume = {7},
      number = {1},
      pages = {1-18},
      doi = {10.11648/j.engmath.20230701.11},
      url = {https://doi.org/10.11648/j.engmath.20230701.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20230701.11},
      abstract = {Hybrid systems are systems that involve continuous and discrete event dynamical behaviors. A impulsive system is a special hybrid system. The continuous dynamics of impulsive systems are usually described by ordinary differential equations and the discrete event dynamics with instantaneously rapid jumps are described by switching laws. Various complex dynamical phenomena that can be modeled by impulsive systems arise in many areas of modern science and technology such as economics, physics, chemistry, biology, information science, radiotherapy, acupuncture, robotics, neural networks, automatic control, artificial intelligence, space technology, and telecommunications, etc. In modern control theory, controllability is one of the most important dynamical properties of considered impulsive systems, therefore, the controllability problem is regarded as one of the fundamental issues of impulsive systems. The basic questions for controllability of impulsive systems as well as for the ordinary systems without impulses and with control function are to obtain useful criteria that allow us to identify whether given dynamic systems are controllable or not. Up to now there have been being many investigation results for controllability of different kinds of impulsive systems with respect to the terminal state constraints of a point type. The purpose of this paper is to study relative controllability with respect to the terminal state constraint of a general type for a class of linear time-varying impulsive systems. In this paper, several types of criteria for relative controllability of such systems are established by a algebraic method, that is, specially speaking, by the matrix rank method. Some corresponding necessary and sufficient conditions for controllability of linear time-invariant impulsive systems are also obtained more compactly. Meanwhile, for given impulsive systems some equivalent relationships between different kinds of controllability are investigated and our criteria for relative controllability are compared with the existing results. A simple example is given to illustrate the utility of our criteria.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - Relative Controllability for a Class of Linear Impulsive Systems
    AU  - Gwang Jin Kim
    AU  - Su Song Om
    AU  - Tae Gun Oh
    AU  - Nam Chol Yu
    AU  - Jin Sim Kim
    Y1  - 2023/02/27
    PY  - 2023
    N1  - https://doi.org/10.11648/j.engmath.20230701.11
    DO  - 10.11648/j.engmath.20230701.11
    T2  - Engineering Mathematics
    JF  - Engineering Mathematics
    JO  - Engineering Mathematics
    SP  - 1
    EP  - 18
    PB  - Science Publishing Group
    SN  - 2640-088X
    UR  - https://doi.org/10.11648/j.engmath.20230701.11
    AB  - Hybrid systems are systems that involve continuous and discrete event dynamical behaviors. A impulsive system is a special hybrid system. The continuous dynamics of impulsive systems are usually described by ordinary differential equations and the discrete event dynamics with instantaneously rapid jumps are described by switching laws. Various complex dynamical phenomena that can be modeled by impulsive systems arise in many areas of modern science and technology such as economics, physics, chemistry, biology, information science, radiotherapy, acupuncture, robotics, neural networks, automatic control, artificial intelligence, space technology, and telecommunications, etc. In modern control theory, controllability is one of the most important dynamical properties of considered impulsive systems, therefore, the controllability problem is regarded as one of the fundamental issues of impulsive systems. The basic questions for controllability of impulsive systems as well as for the ordinary systems without impulses and with control function are to obtain useful criteria that allow us to identify whether given dynamic systems are controllable or not. Up to now there have been being many investigation results for controllability of different kinds of impulsive systems with respect to the terminal state constraints of a point type. The purpose of this paper is to study relative controllability with respect to the terminal state constraint of a general type for a class of linear time-varying impulsive systems. In this paper, several types of criteria for relative controllability of such systems are established by a algebraic method, that is, specially speaking, by the matrix rank method. Some corresponding necessary and sufficient conditions for controllability of linear time-invariant impulsive systems are also obtained more compactly. Meanwhile, for given impulsive systems some equivalent relationships between different kinds of controllability are investigated and our criteria for relative controllability are compared with the existing results. A simple example is given to illustrate the utility of our criteria.
    VL  - 7
    IS  - 1
    ER  - 

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Author Information
  • Faculty of Mathematics, Kim Hyong Jik University of Education, Pyongyang, Democratic People’s Republic of Korea

  • Faculty of Mathematics, Kim Hyong Jik University of Education, Pyongyang, Democratic People’s Republic of Korea

  • Faculty of Mathematics, Kim Hyong Jik University of Education, Pyongyang, Democratic People’s Republic of Korea

  • School of Science and Engineering, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

  • International Technology Cooperation Center, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

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