Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation

Бутковский, А. Г.; Postnov, S. S.; Постнова, Елена Александровна

doi:10.1134/s0005117913050019

Cited by 30 publications

(21 citation statements)

References 129 publications

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“…It seems that Bai and Fang [20] initially study nonlinear fractional differential systems. With the development of fractional calculus theory, there are more and more contributions on various fractional differential systems [21][22][23][24][25][26][27][28][29][30]. Generally speaking, nonlinear alternative results via fixed point method [20,[22][23][24][26][27][28][29], coincidence degree theory [25], and monotone iterative method [30] are used to deal with fractional initial value and boundary problems.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems

Zhang

Wang

2015

J. Appl. Math. Comput.

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In this paper, we discuss nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems. By using vectorial version fixed point theorems and splitting the Lipschitz or linear growth conditions on the nonlinear terms into two parts and applying the techniques that use convergent to zero matrix and vector-valued norm via boundedness and continuity of Mittag-Leffler functions, two couple existence results for the solutions are presented in a complete generalized metric space or a generalized Banach space.

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Section: Introductionmentioning

confidence: 99%

Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems

Zhang

Wang

2015

J. Appl. Math. Comput.

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“…Now it is not hard to find very interesting and novel applications of fractional differential equations in physics, chemistry, biology, engineering, finance and other areas of sciences that have been developed in the last few decades. Some of the applications include: diffusion processes [8,9], mechanics of materials [10,11], combinatorics [12,13], inequalities [14], analysis [15], calculus of variations [16][17][18][19][20][21], signal processing [22], image processing [23], advection and dispersion of solutes in porous or fractured media [24], modeling of viscoelastic materials under external forces [25], bioengineering [26], relaxation and reaction kinetics of polymers [27], random walks [28], mathematical finance [29], modeling of combustion [30], control theory [31], heat propagation [32], modeling of viscoelastic materials [33] and even in areas such as psychology Non-local fractional derivatives are defined via fractional integrals [38][39][40], while the local fractional derivatives are defined via a limit-based approach [41,42]. A new class of controlled-derivative approach appeared in [43].…”

Section: Introductionmentioning

confidence: 99%

Applications of fractional calculus in solving Abel-type integral equations: Surface–volume reaction problem

Evans

Katugampola

Edwards

2017

Computers & Mathematics with Applications

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In this paper we consider a class of partial integro-differential equations of fractional order, motivated by an equation which arises as a result of modeling surface-volume reactions in optical biosensors. We solve these equations by employing techniques from fractional calculus; several examples are discussed. Furthermore, for the first time, we encounter an order of the fractional derivative other than 1 2 in an applied problem. Hence, in this paper we explore the applicability of fractional calculus in real-world applications, further strengthening the true nature of fractional calculus. 1 arXiv:1510.00408v2 [math.CA] 12 Nov 2016 R. M. Evans et al. / Computers & Mathematics with Applications 00 (2016) 1-19 2 [34,35]. The list is by no means complete. It is easy to find hundreds, if not thousands, of new applications in which the fractional calculus approach is more than welcome. This is the first in a series of papers that seeks to find further potential applications of fractional calculus in solving real-world problems, a journey that can benefit both the understanding of profound complexities in the application, and the field of fractional calculus itself. As an application of the theory developed in this paper, we consider the surface-volume reaction problem. The governing equations of the mathematical formulation of such models naturally give rise to a nonlinear equation that contains a fractional integral embedded in it, and which has no solutions to date. Thus, in this paper we both extend the theory of fractional calculus methods by considering equations motivated by modeling the surface-volume reactions, and explore another interpretation of the fractional integral.The remainder of this paper is organized as follows. The definitions and basic results are given in Section 2. In Section 3, we give the main results, which are generalizations of Abel's integral approach to the tautochrone problem. In Section 4, we give illustrative examples to motivate our approaches. One of the main examples is the surface-volume reaction problem that has several very interesting applications in mathematical biology and engineering [36]. Basic Definitions and Preliminary ResultsWe adopt definitions given in [37] or in the encyclopaedic book by Samko et al. [38] here. We begin by introducing the concept of a Riemann-Liouville fractional integral:Definition 2.1 ([37]). Let α > 0 with n − 1 < α ≤ n, n ∈ N, and a < x < b. The left-and right-Riemann-Liouville fractional integrals of order α of a function f are given byThe fractional integrals and derivatives also satisfy the following important properties: fractional operators are linear, that is, if L is a fractional integral or derivative, thenfor any functions f, g ∈ C n [a, b] or f, g ∈ L p (a, b) (as the case may be) and k ∈ R. For any α, β > 0, they also satisfy the following semigroup properties:Equation (5) will be the key property used to prove the main results of this paper. It also says that the Caputo derivative is the left-inverse of the Riemann-Liouville fract...

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“…In the recent two-three decades, systems described by noninteger-order equations have been attracting much attention of researchers, which is clear from the books and papers on the subject [1][2][3][4][5]. Control problems for such systems have been intensively studied over the last ten years, also with accumulation of many publications [4][5][6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…Control problems for such systems have been intensively studied over the last ten years, also with accumulation of many publications [4][5][6][7][8][9][10][11][12]. There exist examples of efficient practical realization of fractional-order control systems, where plants are described by fractional-order and integer-order equations [4][5][6][7][8][9]. Among specific applications, we mention fractional controllers used in power-supply systems, indoor climate systems, precision positioning systems, isotope separation systems and others [4][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Analysis of two optimal control problems for a fractional-order pendulum by the method of moments

Кубышкин

Postnov

2015

Autom Remote Control

Self Cite

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This paper studies two optimal control problems for a fractional-order pendulum in the case when admissible control actions belong to the class of square integrable functions on a segment. The first problem is to find control actions transferring a system to a given state with the minimum control norm under a fixed control time. The second problem is to find control actions transferring the system to a given state within the minimum time under a given constraint on the control norm. The authors demonstrate that the problem can be reduced to the problem of moments, as well as derive the feasible statement and solvability conditions for the latter. Solution of the problem is obtained analytically in the form of quadratures. A series of computing experiments are conducted and the qualitative features of system dynamics are discussed.

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Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation

Cited by 30 publications

References 129 publications

Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems

Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems

Applications of fractional calculus in solving Abel-type integral equations: Surface–volume reaction problem

Analysis of two optimal control problems for a fractional-order pendulum by the method of moments

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