Probability Distribution and Statistical Properties of Reachability Sets in Controlled Linear Systems with Random Parameters

Authors

  • Alaa Hussein Hammadi College of Computer Science and Information Technology, University of Al-Qadisiyah, Al-Diwaniyah, 58002, Iraq
  • Muhammad Sajjad National University of Technology (NUTECH), Islamabad, Pakistan https://orcid.org/0000-0003-0006-1156 (unauthenticated)
  • Mushtaq K. Abdalrahem
  • Qin Xin

DOI:

https://doi.org/10.29020/nybg.ejpam.v19i1.7206

Keywords:

linear control systems \sep dynamical systems \sep probability \sep statistically invariant sets

Abstract

This research has two parts, in the first part: we study the statistical characteristics of the reachability set  $R(t,\sigma,X) $ of the control system
$\dot{x}=f\left(h^t\sigma,x,u\right), u\in U\left(h^t\sigma,x\right) \left(t,\sigma,x\right)\in R\times\mathrm{\Sigma}\times R^n,$ 
which is parameterized using a metric dynamic system $(\mathrm{\Sigma},P,\mu, h^t)$. 

We examine such characteristics as the relative absorption frequency and the upper and lower relative frequencies of absorption of the reachability set. For the above system, we also provide sufficient conditions under which a given set M is statistically invariant with respect to the controlled system. In addition, we compare the distribution function defined on the control linear space with the corresponding relative frequency. In the second part, we investigate the role of the distribution function and its relationship to the stability of the dynamical system under different types of distributions.

Author Biographies

  • Alaa Hussein Hammadi, College of Computer Science and Information Technology, University of Al-Qadisiyah, Al-Diwaniyah, 58002, Iraq
    • Professor
    • Research Director at University of Al-Qadisiyah, Iraq
  • Muhammad Sajjad, National University of Technology (NUTECH), Islamabad, Pakistan

    Dr. Muhammad Sajjad is a dedicated mathematician from Pakistan, currently serving as an assistant professor and director of the collaborative learning community at the National University of Technology (NUTECH), Islamabad. He earned his PhD from Quaid-i-Azam University, Islamabad, Pakistan, in 2024 and is an esteemed member of the American Mathematical Society (AMS). Recognised for his outstanding contributions to cybersecurity and cryptography, Dr. Sajjad has received several prestigious accolades, including two international research awards in cybersecurity and cryptography, the Computer Scientist Award from China, and the National Cryptology Award from NCCS Pakistan. His research spans multiple disciplines, including mathematics, electrical engineering, and computer science, with a strong emphasis on vector algebras, non-commutative and non-associative algebras, number theory, coding theory, channel coding, cryptography, cryptology, post-quantum cryptography, graph theory, and elliptic curves. He particularly focusses his work on designing efficient error-correcting codes, advancing data security, and exploring complex algebraic structures. Dr. Sajjad is an active contributor to the global academic community, having delivered talks at prestigious international conferences. His research collaborations extend to leading universities in South America, Denmark, and Saudi Arabia, further strengthening his impact on the field. He has undergone professional training in various aspects of academic management and possesses strong programming skills in Python, MATLAB, C++, and more. Over his academic career, he has taught 21 bachelor and master's courses and published 20 research papers. Through his dedication to teaching, research, and innovation, Dr. Muhammad Sajjad continues to shape the future of mathematical sciences and information security.

References

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Published

2026-02-21

Issue

Vol. 19 No. 1: (January 2026)

Section

Mathematical Statistics

License

Copyright (c) 2026 Alaa Hussein Hammadi, Muhammad Sajjad, Mushtaq K. Abdalrahem, Qin Xin

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How to Cite

Probability Distribution and Statistical Properties of Reachability Sets in Controlled Linear Systems with Random Parameters. (2026). European Journal of Pure and Applied Mathematics, 19(1), 7206. https://doi.org/10.29020/nybg.ejpam.v19i1.7206