Discrete-Time Fractional Variational Problems

TL;DR

This paper develops a discrete-time fractional calculus of variations on the time scale $h ext{Z}$, establishing optimality conditions, and demonstrating convergence to classical and continuous fractional solutions.

Contribution

It introduces a new discrete-time fractional calculus of variations framework with necessary optimality conditions and convergence properties.

Findings

Solutions reduce to classical discrete-time solutions for integer fractional orders.

Solutions converge to continuous-time fractional solutions as $h$ approaches zero.

Legendre type condition helps eliminate false candidates in 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.