Asymptotic behavior of solutions of linear multi-order fractional differential systems

Abstract

In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems.

Stability Analysis of Two-Dimensional Incommensurate Systems of Fractional-Order Differential Equations

“…If q 1 = q 2 , solving this system for a 11 and a 22 shows that the characteristic equation (3) has a pair of pure imaginary roots if and only if (a 11 , a 22 ) belongs to the curve Γ(δ, q 1 , q 2 ) given by Lemma 3.1.…”

On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

“…In order to get a succinct representation of the solution based on (20) and (25), it will be convenient to write:…”

“…While mixed index problems can, in some cases, be written in the form of linear sequential problems, namely ∑ R i=1 D β i t y = f (y), we claim that it is inappropriate to do so in many cases. We note that Diethelm et al [20] have very recently considered the asymptotic behaviour of certain linear multi-order fractional differential equations from a theoretical viewpoint.…”

“…We note that scalar linear sequential fractional problems have been considered whose solution can be described by multi-indexed Mittag-Leffler functions [16], and there are a number of articles on the numerical solution of multi-term fractional differential equations [17][18][19][20][21][22]. While mixed index problems can, in some cases, be written in the form of linear sequential problems, namely ∑ R i=1 D β i t y = f (y), we claim that it is inappropriate to do so in many cases.…”

Exact bounds for robust stability of output feedback controlled fractional‐order systems with single parameter perturbations

“…13 A fractional-order PI controller was provided for the closed-loop energy generation control in the work of Alagoz and Kaygusuz. 14 In addition, for servo system control, a fractional-order PID controller was proposed by using the multivariable multiobjective genetic algorithm in the work of Ren et al 15 Robust stability is a crucial issue in fractional-order control systems, 4,[16][17][18][19] because practical control systems are frequently affected by various types of perturbations and uncertainties such as interval uncertainties, 4,[20][21][22][23][24] norm bounded uncertainties, [25][26][27][28] polytopic uncertainties, 29 and structured uncertainties. [30][31][32] Some valuable research results on the robust stability of FOSs with different kinds of uncertainties were put forward in time domain.…”