Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

Derakhshan, Mohammadhossein; Aminataei, A.

doi:10.1155/2020/1896563

Cited by 5 publications

(9 citation statements)

References 20 publications

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“…Moreover, at the same time, in the domain of applied mathematics, those distributed-order fractional operators have started to be used, in a satisfactory way, to describe some complex phenomena modeling real world problems-see, for instance, works in viscoelasticity [7,8] and in diffusion [9]. Today, the study of distributed-order systems with fractional derivatives is a hot subject-see, e.g., [10][11][12] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Distributed-Order Non-Local Optimal Control

Ndaïrou

Torres

2020

Axioms

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Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.

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Section: Introductionmentioning

confidence: 99%

Distributed-Order Non-Local Optimal Control

Ndaïrou

Torres

2020

Axioms

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“…Remark 1. e GCD generalizes the classical derivative (θ � 1) and the conformable derivative ψ a (t, θ) � (t − a) 1− θ (see [6,22] for a � 0). Remark 2.…”

Section: Preliminariesmentioning

confidence: 99%

“…e presentation of a nonlinear system with the noninteger derivative is named fractional-order system (FOS). ere are many applications of FOSs on real-world fields, whether in signal processing, chemistry, electricity, thermal, or control theory, for example, observer design [1][2][3], finite-time stability [4], fault estimation [5], and asymptotic stability [6][7][8]. In fact, with regard to observer design, authors in [1] presented a fractional-order observer design for fractionalorder nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

“…In the context of fault estimation, Ibrahima N'Doye and Taous-Meriem Laleg-Kirati in [5] detailed the fractional-order adaptive fault estimation for a class of nonlinear fractional-order systems. Finally, with regard to asymptotic stability, in [6], authors presented the asymptotic stability of distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

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Controllability of Differential Systems with the General Conformable Derivative

et al. 2021

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In this paper, the controllability of differential systems with the general conformable derivative is studied. By elaborating the rank criterion and the conformable Gram criterion, sufficient and necessary conditions to investigate that a linear general conformable system is null completely controllable are given. We obtain a full generalization to the general conformable fractional-order system case. In addition, Krasnoselskii’s fixed point theorem to obtain a complete controllability result for a semilinear general conformable system is applied.

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“…Moreover, at the same time, in the domain of applied mathematics, those distributed-order fractional operators have started to be used, in a satisfactory way, to describe some complex phenomena modeling real world problem: see, for instance, works in viscoelasticity [7,8] and in diffusion [9]. Today, the study of distributed-order systems with fractional derivatives is a hot subject: see, e.g., [10][11][12] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Distributed-Order Non-Local Optimal Control

Ndairou,

Torres

2020

Preprint

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Distributed-order fractional non-local operators have been introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical systems constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin's maximum principle and a sufficient optimality condition under appropriate convexity assumptions.

show abstract

Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

Cited by 5 publications

References 20 publications

Distributed-Order Non-Local Optimal Control

Distributed-Order Non-Local Optimal Control

Controllability of Differential Systems with the General Conformable Derivative

Distributed-Order Non-Local Optimal Control

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