Razumikhin Stability Theorem for Fractional Systems with Delay

Băleanu, Dumitru; Sadati, S. J.; Ghaderi, Reza; Ranjbar, A.A.; Abdeljawad, Thabet; Jarad, Fahd

doi:10.1155/2010/124812

Cited by 54 publications

(49 citation statements)

References 15 publications

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“…Motivated by the application of fractional calculus in nonlinear systems, the Lyapunov direct method has been extended to fractionalorder systems by Li et al [21,22]. Similarly, Baleanu et al [23] and Wu et al [24] extended the theorem to fractional-order functional systems and fractional-order discrete systems. The fractional Lyapunov method generalizes the idea that the stability condition is derived by constructing a suitable Lyapunov function and calculating its fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

“…Hu et al [29] considered an integerorder derivative instead of the fractional-order derivative of a Lyapunov function to prove the revised Lyapunov stability theorems. Nevertheless, the proposed Lyapunov functions [21][22][23][24][26][27][28][29] are valid only for some fractional-order systems with special characteristics. In classic Lyapunov theory, the quadratic form is one of the most commonly used Lyapunov functions for general integer-order nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

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Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions

Zhuang

et al. 2018

Adv Differ Equ

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In this paper, we aim to solve the stabilization problem for a large class of fractional-order nonautonomous systems via linear state feedback control and adaptive control. By constructing quadratic Lyapunov functions and utilizing a new property for Caputo fractional derivative we derive some sufficient conditions for the global asymptotical stabilization of a class of fractional-order nonautonomous systems. We give two illustrative examples to validate the effectiveness of the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions

Zhuang

et al. 2018

Adv Differ Equ

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“…Much attention has been paid to fractional‐order systems recently, because many real world physical systems are well characterized by fractional‐order state equations. Stability is also a fundamental concern of control systems, especially fractional‐order control systems.…”

Section: Introductionmentioning

confidence: 99%

An analysis and design method for fractional-order linear systems subject to actuator saturation and disturbance

Shahri

Balochian

2015

Optim. Control Appl. Meth.

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Summary In this paper, two methods are proposed for investigating stability of fractional‐order systems under saturated linear feedback. The first stability condition can be used for fractional‐order linear systems with nonlinear element that is Lipschitz in state. The second stability condition has been achieved by exploring the special property of saturation using an auxiliary feedback matrix. For comparison of the two analyses, domains of attraction are applied using the proposed optimizations whose conditions can be expressed as Linear Matrix Inequalities in terms of all the varying parameters hence being appropriate for controller synthesis. Furthermore, the stability condition is achieved with regard to persistence disturbance. It is employed for determining the invariant sets of the given system by the proposed optimization problems. The results of the proposed analysis and methods are illustrated by three examples. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Based on the same idea, Li Y. etc [15] propose a Lyapunov direct method for fractional nonlinear system without delay. Baleanu D. [20] extends this method to fractional systems with time delay and proposes the LyapunovKrasovskii theorem for fractional systems. The key issue of the two methods is to design a positive function V .…”

Section: Introductionmentioning

confidence: 99%