Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders
type: Item
instance of: scholarly article
title: Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders
zbMATH Open document ID: 1465.93003
author: Tadeusz Kaczorek
author: Łukasz Sajewski
published in: International Journal of Applied Mathematics and Computer Science
publication date: 2021-05-03T00:00:00Z
review text: The subject is continuous time linear systems for vector functions \(x(t)\) containing fractional derivatives \((d^\alpha /dt^\alpha) x(t)\). The derivative is the \textit{Caputo fractional derivative} \[ \frac{d^\alpha}{dt^\alpha} x(t) = \frac{1}{\Gamma(n - \alpha)} \int_0^t \frac{1}{(t - \tau)^{\alpha - n + 1}} \frac{d^n}{d\tau^n} x(\tau) d\tau \quad (n - 1 < \alpha < n) \, . \tag{1} \] The choice of this derivative over the Riemann-Liouville version \[ \frac{d^\alpha}{dt^\alpha} x(t) = \frac{1}{\Gamma(n - \alpha)} \frac{d^n}{dt^n}\int_0^t \frac{1}{(t - \tau)^{\alpha - n + 1}} x(\tau) d\tau \quad (n - 1 < \alpha < n) \tag{2} \] is motivated by the fact that the Laplace transform of (1) involves \(x(0), \dots, x^{(n-1)}(0)\) rather than fractional derivatives of \(x(\cdot)\) at \(t = 0\) as in (2), thus allowing pointwise initial conditions. The particular system examined is \[ E \left[ \begin{matrix} \frac{d^\alpha}{dt^\alpha} x_1(t) \\ \frac{d^\beta}{dt^\beta} x_2(t) \end{matrix} \right] = A \left[ \begin{matrix} x_1(t) \\ x_2(t) \end{matrix} \right] \tag{3} \] for \(x(t) = (x_1(t), x_2(t))\). The system (3) is \textit{pointwise complete} for \(t = t_f\) if for every \(x_f\) there exists initial conditions such that the solution \(x(\cdot)\) satisfies \(x(t_f) = x_f\). The results are on completeness and on its absence, degeneracy. Roughly, they are the counterpart for systems of those in [\textit{T. Kaczorek} and \textit{A. Ruszewski}, Int. J. Appl. Math. Comput. Sci. 30, No. 2, 219--223 (2020; Zbl 1465.93095)] for the one-derivative system \(E (d^\alpha / dt^\alpha )x(t) = Ax(t)\).
reviewed by: Hector O. Fattorini
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 93C23
zbMATH DE Number: 7342320
zbMATH Keywords: pointwise completeness
zbMATH Keywords: pointwise degeneracy
zbMATH Keywords: fractional systems
zbMATH Keywords: descriptor systems
zbMATH Keywords: Caputo derivatives
MaRDI profile type: Publication
OpenAlex ID: W3119773948
cites work: Frequency response based curve fitting approximation of fractional-order PID controllers
cites work: Applications of the Drazin Inverse to Linear Systems of Differential Equations with Singular Constant Coefficients
cites work: Singular control systems
cites work: Synchronization of fractional-order discrete-time chaotic systems by an exact delayed state reconstructor: application to secure communication
cites work: Matrix pencil of closed-loop descriptor systems: infinite-eigenvalue assignment
cites work: Analysis and design of descriptor linear systems
cites work: Fractional positive linear systems
cites work: Positive linear systems with different fractional orders
cites work: Selected problems of fractional systems theory.
cites work: Pointwise completeness and pointwise degeneracy of positive fractional descriptor continuous-time linear systems with regular pencils
cites work: Absolute stability of a class of fractional positive nonlinear systems
cites work: Fractional linear systems and electrical circuits
cites work: Fundamental theorem of state feedback for singular systems
cites work: On properties of pointwise degenerate linear autonomous control systems. I.
cites work: Q3141902
cites work: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
cites work: Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders
DOI: 10.34768/AMCS-2020-0047
description: scientific article; zbMATH DE number 7342320