Stability Analysis of Distributed Order Fractional Differential Equations

Abstract

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

Stabilization and passification of distributed-order fractional linear systems using methods of preservation

“…We extend recent results on stability of linear fractional distributed order systems in [22]. These results allow us to extend some classical methods for stabilization and passivation of LTI systems, to a new family of linear fractional distributed order systems, which includes among other the distributed order linear time invariant systems (DOLTIS) in [21].…”

“…On the other hand, stabilization (and more so passification) has been neglected in linear systems of distributed order, even when these latter systems have lately gained importance, as attested by the recent publication of a monograph on the subject [21], and the appearance of papers that treats the distributed order systems as generalizations of the fractional order ones [13,22]. Moreover, recently in [15] the impulse response of the distributed order integrator/differentiator and its asymptotic property is derived by using the complex path integral.…”

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

“…A nice survey on the stability (both linear and nonlinear) of fractional differential equations is given in Li and Zhang [27], while Saberi Najafi et al [28] have extended some of these stability results to distributed order fractional differential equations with respect to an order density function. Zhang et al [29] consider the stability of nonlinear fractional differential equations.…”

A Note About Stability of Fractional Retarded Linear Systems With Distributed Delays

“…The case of distributed delays for retarded type fractional system of is studied in [11], [14] and for neutral systems in [13] and [12]. Some stability results for fractional system with distributed order derivatives and constant delays are given in [9] and the case of distributed delays is considered in [1]. The works [11] and [13] are our first motivation to extend and improve some of the obtained there explicit type sufficient conditions for asymptotical stability of the zero solution of this systems.…”