Necessaryoptimality conditions for fractional difference problems of thecalculus of variations

Nuno R. O. Bastos; Rui A. C. Ferreira; Delfim F. M. Torres

doi:10.3934/dcds.2011.29.417

Article Contents

2011, Volume 29, Issue 2: 417-437. Doi: 10.3934/dcds.2011.29.417

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

1.
Department of Mathematics, ESTGV, Polytechnic Institute of Viseu, 3504-510 Viseu
2.
Faculty of Engineering and Natural Sciences, Lusophone University of Humanities and Technologies, 1749-024 Lisbon
3.
Department of Mathematics, University of Aveiro, 3810-193 Aveiro

Received: April 30, 2009

Revised: February 28, 2010

Published: May 2011

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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.

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Mathematics Subject Classification: 49K05, 39A12, 26A33.

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