Discrete-Time Fractional Variational Problems

Abstract

We introduce a discrete-time fractional calculus of variations on the time scale $h\mathbb{Z}$, $h>0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when $h$ tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler-Lagrange fractional equation.

Submitted 24/Nov/2009; Revised 16/Mar/2010; Accepted 3/May/2010; for publication in Signal Processing.

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• Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations
• Fractional variational calculus for nondifferentiable functions
• Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform
• Combined Delta-Nabla Sum Operator in Discrete Fractional Calculus
• Fractional Calculus of Variations for Double Integrals