Necessary optimality conditions for fractional difference problems of the calculus of variations

Abstract

We introduce a discrete-time fractional calculus of variations. 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 the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.Comment: Submitted 09-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 27-March-2010; accepted 04-July-201