Relative controllability of linear state-delay fractional systems


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Mahmudov N. I.

FRACTIONAL CALCULUS AND APPLIED ANALYSIS, vol.27, no.3, pp.987-1016, 2024 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 27 Issue: 3
  • Publication Date: 2024
  • Doi Number: 10.1007/s13540-024-00270-8
  • Journal Name: FRACTIONAL CALCULUS AND APPLIED ANALYSIS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, zbMATH
  • Page Numbers: pp.987-1016
  • Azerbaijan State University of Economics (UNEC) Affiliated: No

Abstract

In this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new alpha \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples.