Controllability criteria for time–delay fractional systems with a retarded state

Abstract

The paper is concerned with time-delay linear fractional systems with multiple delays in the state. A formula for the solution of the discussed systems is presented and derived using the Laplace transform. Definitions of relative controllability with and without constraints for linear fractional systems with delays in the state are formulated. Relative controllability, both with and without constraints imposed on control values, is discussed. Various types of necessary and sufficient conditions for relative controllability and relative U -controllability are established and proved. Numerical examples illustrate the obtained theoretical results.

Controllability Problem of Fractional Neutral Systems: A Survey

“…It should be noticed that there are many unsolved problems for controllability concepts for different types of dynamical systems. The methodology presented in this paper may well be used in a research on controllability of stochastic dynamical systems [69], in a search of optimal control [70,71], for systems with constraints on control signal [11], and for dynamical systems with delay in state and control [12,72].…”

“…A standard approach is to transform the controllability problem into a fixed point problem for an appropriate operator in a functional space. There are many papers devoted to the controllability problem, in which authors used the theory of fractional calculus [3][4][5][6][7][8][9][10][11][12][13] and a fixed point approach [14][15][16][17][18][19][20][21][22][23].…”

Minimal positive realizations of linear continuous-time fractional descriptor systems: Two cases of an input-output digraph structure

“…Mathematical fundamentals of the fractional calculus are given by Nishimoto (1984), Miller and Ross (1993), Podlubny (1999), Das (2011), Ortigueira (2011) as well as Kaczorek and Sajewski (2014). Some other applications of fractional-order systems can be found in the works of Ionescu et al (2010), Magin et al (2011), Kaczorek (2011), Petras et al (2012), Vandoorn et al (2013), Podlubny et al (2014), , , Muresan et al (2016a), Sikora (2016), or Muresan et al (2016b).…”

Controllability of Fractional-order Dynamical System with Time Varying-Delays

“…In recent years, the controllability of fractionalorder dynamical systems has attracted the attention of many researchers because of the critical role it plays in their analysis. [41] established different types of necessary and sufficient conditions for the relative controllability and relative constrained controllability for both with and without retarded state for linear fractional-order systems. In [6] sufficient condition for the controllability of nonlinear integrodifferential system with implicit fractional derivative is obtained using the notion of the measure of noncompactness of a set and Darbo's fixed point theorem, while [2] used Schauder's fixed point theorem to prove sufficient conditions for the controllability of linear and nonlinear fractional-order dynamical systems in finite dimensional spaces.…”