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Journal ArticleDOI

Formulation of Euler–Lagrange equations for fractional variational problems

01 Aug 2002-Journal of Mathematical Analysis and Applications (Academic Press)-Vol. 272, Iss: 1, pp 368-379
TL;DR: In this article, the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum were developed for systems containing fractional derivatives, where the fractional derivative is described in the Riemann-Liouville sense.
About: This article is published in Journal of Mathematical Analysis and Applications.The article was published on 2002-08-01 and is currently open access. It has received 866 citations till now. The article focuses on the topics: Fractional calculus & Euler–Lagrange equation.
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Journal ArticleDOI
TL;DR: In this article, a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems are presented, where the performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs.
Abstract: Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.

661 citations


Cites background from "Formulation of Euler–Lagrange equat..."

  • ...Agrawal [31] extends variational calculus to fractional variational problems....

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01 Jan 2012

433 citations


Additional excerpts

  • ...[238] O....

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Journal ArticleDOI
TL;DR: In this article, the transversality conditions for fractional variational problems (FVPs) defined in terms of Riesz fractional derivatives (RFDs) are considered.
Abstract: This paper presents extensions of traditional calculus of variations for systems containing Riesz fractional derivatives (RFDs). Specifically, we present generalized Euler–Lagrange equations and the transversality conditions for fractional variational problems (FVPs) defined in terms of RFDs. We consider two problems, a simple FVP and an FVP of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives, functions and parameters, and to unspecified boundary conditions. For the second problem, we present Lagrange-type multiplier rules. For both problems, we develop the Euler–Lagrange-type necessary conditions which must be satisfied for the given functional to be extremum. Problems are considered to demonstrate applications of the formulations. Explicitly, we introduce fractional momenta, fractional Hamiltonian, fractional Hamilton equations of motion, fractional field theory and fractional optimal control. The formulations presented and the resulting equations are similar to the formulations for FVPs given in Agrawal (2002 J. Math. Anal. Appl. 272 368, 2006 J. Phys. A: Math. Gen. 39 10375) and to those that appear in the field of classical calculus of variations. These formulations are simple and can be extended to other problems in the field of fractional calculus of variations.

330 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed some basics of discrete fractional calculus such as Leibniz rule and summation by parts formula and derived Euler-Lagrange equation.

287 citations

Journal ArticleDOI
TL;DR: In this paper, the Riemann-Liouville Fractional Derivatives (RLFDs) were used to solve fractional optimal control problems (FOCPs).
Abstract: This paper deals with a direct numerical technique for Fractional Optimal Control Problems (FOCPs). In this paper, we formulate the FOCPs in terms of Riemann—Liouville Fractional Derivatives (RLFDs). It is demonstrated that right RLFDs automatically arise in the formulation even when the dynamics of the system is described using left RLFDs only. For numerical computation, the FDs are approximated using the Grunwald—Letnikov definition. This leads to a set of algebraic equations that can be solved using numerical techniques. Two examples, one time-invariant and the other time-variant, are considered to demonstrate the effectiveness of the formulation. Results show that as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system. The approach requires dividing of the entire time domain into several sub-domains. Further, as the sizes of the sub-domains are reduced, the solutions converge to unique solutions. However, the convergence is slow. A sch...

275 citations


Cites background or methods from "Formulation of Euler–Lagrange equat..."

  • ...In Agrawal (2002), Agrawal presented fractional Euler–Lagrange equations for Fractional Variational Problems (FVPs)....

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  • ...For simplicity in the discussion to follow, we briefly review the Fractional Euler–Lagrange Equation (FELE) presented in Agrawal (2002)....

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  • ...It is demonstrated in Agrawal (2002) that the solution of the above problem must satisfy F y t D b F a D t y 0 (4) where t D b y is the Right Riemann–Liouville Fractional Derivative (RRLFD) of order defined as t D b y t 1 n d dt n b t t n 1 y d (5) Equation (5) will play an important role in the…...

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References
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Book
01 Jan 1999

15,898 citations

Book
19 May 1993
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.

7,643 citations

Book
08 Dec 1993
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.

7,096 citations

Book
02 Mar 2000
TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.
Abstract: An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.

5,201 citations

BookDOI
01 Jan 1997
TL;DR: Panagiotopoulos, O.K.Carpinteri, B. Chiaia, R. Gorenflo, F. Mainardi, and R. Lenormand as mentioned in this paper.
Abstract: A. Carpinteri: Self-Similarity and Fractality in Microcrack Coalescence and Solid Rupture.- B. Chiaia: Experimental Determination of the Fractal Dimension of Microcrack Patterns and Fracture Surfaces.- P.D. Panagiotopoulos, O.K. Panagouli: Fractal Geometry in Contact Mechanics and Numerical Applications.- R. Lenormand: Fractals and Porous Media: from Pore to Geological Scales.- R. Gorenflo, F. Mainardi: Fractional Calculus: Integral and Differential Equations of Fractional Order.- R. Gorenflo: Fractional Calculus: some Numerical Methods.- F. Mainardi: Fractional Calculus: some Basic Problems in Continuum and Statistical Mechanics.

1,389 citations