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On Con-Numbers and Con-Matrices with Applications to Control Systems

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Abstract

In this paper, the concepts of con-numbers and con-matrices are proposed by the introduction of conjugate operators into the field of complex numbers, and some properties of these two concepts are derived. In addition, two potential applications of these two concepts are addressed in details. Specifically, a class of quantum systems is expressed in terms of the proposed con-numbers, and the ℝ-linear mapping is represented by con-matrices. These two applications imply the importance of the con-numbers and con-matrices. In addition, some methods are discussed on the construction of the conjugate-operator-induced numbers. Also, some further research directions are provided on these extended numbers and matrices with their applications to systems control.

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Acknowledgment

The author is very grateful for Prof. Daizhan Cheng from Academy of Mathematics and Systems Science, Chinese Academy of Sciences to bring the concept of hyperbolic numbers. With such a concept, the paper has been thoroughly revised, and the quality of this paper has been significantly improved. The author also would like to thank Dr. Zhiyuan Dong for providing the example of quantum systems.

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Authors and Affiliations

  1. Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China

    Aiguo Wu

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  1. Aiguo Wu
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Correspondence to Aiguo Wu.

Additional information

This work was supported by the National Natural Science Foundation of China for Excellent Young Scholars under Grant No. 61822305, Major Program of National Natural Science Foundation of China under Grant Nos. 61690210 and 61690212, the Fundamental Research Funds for the Central Universities under Grant No. HIT.BRETIV.201907, Shenzhen Municipal Basic Research Project for Discipline Layout with Project No. JCYJ20180507183437860, and Guangdong Natural Science Foundation under Grant Nos. 2020A1515011091 and 2019A1515011576.

This paper was recommended for publication by Editor QI Hongsheng.

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Wu, A. On Con-Numbers and Con-Matrices with Applications to Control Systems. J Syst Sci Complex 35, 32–57 (2022). https://doi.org/10.1007/s11424-021-0081-9

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  • DOI: https://doi.org/10.1007/s11424-021-0081-9

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