Fractional embedding of differential operators and Lagrangian systems

By Jacky Cresson. 2006

215 citations recorded by the source corpus.

This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the stochastic embedding theory developed with Darses [C. R. Acad. Sci. Ser. I: Math 342, 333 (2006); (preprint IHES 06/27, p. 87, 2006)], we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provides a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e., the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski...

Subfields: Finance, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7f0ea395b78c0aa27b4442695f22e9e0a4d62b1f

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Citations in the corpus

Fractional Variational Calculus with Classical and Combined Caputo Derivatives (2011, methodology): “, non-integer) derivative operators were introduced: the Grunwald– Letnikow, Riemann–Liouville, Caputo [15, 17, 26], and the more recent notions of Cresson [10], Jumarie [16], or Klimek [18].”
Fractional calculus of variations for a combined Caputo derivative (2011, background): “, [25, 35, 40]), and the more recent notions of [26], [15], and [23, 24].”
A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials (2016, methodology): “In this paper we have illustrated the direct approach of a Rayleigh–Ritz method for FVPs.”
Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints (2020, unknown): “We refer for example to [2, 6, 7, 9, 12, 21, 41, 43, 57] and references therein.”
Fractional actionlike variational problems (2008, methodology): “Since various applications of FC are based on replacing the time derivative in dynamical equations with a fractional derivative, in §3.1 we introduce the double-weighted FVP with the recent fractional derivatives of J. Cresson [6].”
Necessary optimality conditions for fractional action‐like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β) (2007, unknown): “Darses [5] (see also [4]): Definition 2.”
Fractional Optimal Control Problems with Several State and Control Variables (2010, methodology): “The technique presented here can be applied to optimal control problems defined using other types of fractional derivatives, e.g., Caputo, Riesz, Cresson (see, e.g., Cresson (2007)) and to fractional action-like variational formulations (see El-Nabulsi and Torres (2008)).”
Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem (2007, methodology): “In a more general way, formulations of Noether’s theorem were proved for fractional problems with left and right Riemann-Liouville fractional derivatives [9, 13, 14, 15].”
A fractional calculus of variations for multiple integrals with application to vibrating string (2010, background): “Since then, many different forms of fractional operators were introduced: the Grunwald-Letnikov, Riemann-Liouville, Riesz, and Caputo fractional derivatives [31, 35, 39], and the more recent notions of Klimek [32], Cresson [14], and Jumarie [24, 25, 27, 28].”