On the regional controllability of the sub-diffusion process with Caputo fractional derivative

AbstractThis paper is devoted to the investigation of regional controllability of the fractional order sub-diffusion process. We first derive the equivalent integral equations of the abstract sub-diffusion systems with Caputo and Riemann-Liouville fractional derivatives by utilizing the Laplace transform. The new definitions of regional controllability of the system studied are introduced by extending the existence contributions. Then we analyze the regional controllability of the fractional order sub-diffusion system with minimum energy control in two different cases:

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Enlarged Controllability of Riemann–Liouville Fractional Differential Equations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4038450 ◽

2018 ◽

Vol 13 (9) ◽

Cited By ~ 5

Author(s):

Touria Karite ◽

Ali Boutoulout ◽

Delfim F. M. Torres

Keyword(s):

Minimum Energy ◽

Fractional Diffusion ◽

Liouville Type ◽

Energy Control ◽

Penalization Method ◽

Diffusion Systems ◽

Minimum Energy Control ◽

Hilbert Uniqueness Method ◽

Uniqueness Method ◽

Fractional Differential

We investigate exact enlarged controllability (EEC) for time fractional diffusion systems of Riemann–Liouville type. The Hilbert uniqueness method (HUM) is used to prove EEC for both cases of zone and pointwise actuators. A penalization method is given and the minimum energy control is characterized.

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Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

Mathematics ◽

10.3390/math9172084 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2084

Author(s):

Oscar Martínez-Fuentes ◽

Fidel Meléndez-Vázquez ◽

Guillermo Fernández-Anaya ◽

José Francisco Gómez-Aguilar

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Lyapunov Functions ◽

Fractional Derivatives ◽

Direct Method ◽

The Laplace Transform ◽

Uniformly Convergent ◽

The Stability ◽

Stability Results ◽

Stability Concept

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

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Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition

Bulletin of the Polish Academy of Sciences Technical Sciences ◽

10.1515/bpasts-2017-0006 ◽

2017 ◽

Vol 65 (1) ◽

pp. 45-51 ◽

Cited By ~ 3

Author(s):

T. Kaczorek

Keyword(s):

Fractional Derivative ◽

Linear Systems ◽

Continuous Time ◽

Minimum Energy ◽

Energy Control ◽

Continuous Time Systems ◽

Minimum Energy Control ◽

Definition Of ◽

Time Systems ◽

Time Linear

Abstract The Caputo-Fabrizio definition of the fractional derivative is applied to minimum energy control of fractional positive continuous- time linear systems with bounded inputs. Conditions for the reachability of standard and positive fractional linear continuous-time systems are established. The minimum energy control problem for the fractional positive linear systems with bounded inputs is formulated and solved.

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Fractional-Order Colour Image Processing

Mathematics ◽

10.3390/math9050457 ◽

2021 ◽

Vol 9 (5) ◽

pp. 457

Author(s):

Manuel Henriques ◽

Duarte Valério ◽

Paulo Gordo ◽

Rui Melicio

Keyword(s):

Image Processing ◽

Edge Detection ◽

Fractional Derivative ◽

Fractional Order ◽

Fractional Derivatives ◽

Grey Scale ◽

Colour Image ◽

Colour Image Processing ◽

Image Processing Algorithms ◽

Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽

10.1515/phys-2019-0088 ◽

2019 ◽

Vol 17 (1) ◽

pp. 850-856 ◽

Cited By ~ 2

Author(s):

Jun-Sheng Duan ◽

Yun-Yun Xu

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Fractional Derivatives ◽

Harmonic Excitation ◽

Steady State Response ◽

Vibration System ◽

Derivative Operator ◽

Vibration Equation ◽

State Response ◽

Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

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