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 (1)

• Quasi-Invariant Optimal Control Problems

Cited by in corpus (32)

• Discrete-Time Fractional Variational Problems
• Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives
• Fractional conservation laws in optimal control theory
• Calculus of variations with fractional derivatives and fractional integrals
• Fractional Action-Like Variational Problems
• Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem
• Fractional Noether's theorem in the Riesz-Caputo sense
• Necessary Optimality Conditions for Fractional Action-Like Integrals of Variational Calculus with Riemann-Liouville Derivatives of Order $(\alpha ,\beta )$
• Noether's Theorem on Time Scales
• The generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative
• Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory
• A formulation of the fractional Noether-type theorem for multidimensional Lagrangians
• Duality for the left and right fractional derivatives
• Leitmann's direct method for fractional optimization problems
• Noether's symmetry theorem for nabla problems of the calculus of variations
• Noether's Symmetry Theorem for Variational and Optimal Control Problems with Time Delay
• Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem
• The isoperimetric problem for Holderian curves
• Composition Functionals in Fractional Calculus of Variations
• Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
• Fractional Calculus on Time Scales
• Variable Order Fractional Variational Calculus for Double Integrals
• Fractional Noether's Theorem with Classical and Riemann-Liouville Derivatives
• A class of fractional optimal control problems and fractional Pontryagin's systems. Existence of a fractional Noether's theorem
• A Time scales Noether's theorem
• Stochastic Fractional HP Equations
• Variational problems with fractional derivatives: Invariance conditions and Nöther's theorem
• A variational approach to the analysis of non-conservative mechatronic systems
• Generalized fractional hybrid Hamilton Pontryagin equations
• The second Noether theorem on time scale
• Conservation laws for time-fractional subdiffusion and diffusion-wave equations
• Stochastic generalized fractional HP equations and applications