A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations

Abstract

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Accepted for publication in the Journal of Mathematical Analysis and Applications

References in corpus (2)

• Quasi-Invariant Optimal Control Problems
• Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control

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• Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
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• Noether's Theorem on Time Scales
• The generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative
• A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String
• Variational Calculus with Conformable Fractional Derivatives
• Fractional calculus of variations for a combined Caputo derivative
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• Leitmann's direct method for fractional optimization problems
• Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem
• Noether's Symmetry Theorem for Variational and Optimal Control Problems with Time Delay
• The isoperimetric problem for Holderian curves
• Multiobjective fractional variational calculus in terms of a combined Caputo derivative
• Fractional Euler-Lagrange differential equations via Caputo derivatives
• Noether's Theorem for Nonsmooth Extremals of Variational Problems with Time Delay
• Noether's theorem for fractional variational problems of variable order
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• Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
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• Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense
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• Time-Fractional Optimal Control of Initial Value Problems on Time Scales
• The Legendre Condition of the Fractional Calculus of Variations
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• A Generalized Fractional Calculus of Variations
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• Variational integrator for fractional Euler-Lagrange equations
• Noether-type theorem for fractional variational problems depending on fractional derivatives of functions
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• Hight-order Noether's Theorem for Nonsmooth Extremals of Isoperimetric Variational Problems with Time Delay
• Conservation laws for time-fractional subdiffusion and diffusion-wave equations