Relative Controllability for a Class of Linear Impulsive Systems

Gwang Jin Kim; Su Song Om; Tae Gun Oh; Nam Chol Yu; Jin Sim Kim

doi:doi:10.11648/j.engmath.20230701.11

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Relative Controllability for a Class of Linear Impulsive Systems

Gwang Jin Kim, Su Song Om, Tae Gun Oh, Nam Chol Yu, Jin Sim Kim

Published in Engineering Mathematics (Volume 7, Issue 1)

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.

Published in	Engineering Mathematics (Volume 7, Issue 1)
DOI	10.11648/j.engmath.20230701.11
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.
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[7]	S. Leela, Farzana A. McRae and S. Sivasundaram, Controllability of impulsive differential equations, Journal of Mathematical Analysis and Applications, 177 (1993), 24-30.
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[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.
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[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.
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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

Gwang Jin Kim

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

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

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

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

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

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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

Copy | Download

@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}
}

Copy | Download

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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