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      Discrete-Time Fractional Variational Problems

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          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.

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          Initial value problems in discrete fractional calculus

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            Fractional conservation laws in optimal control theory

            Using the recent formulation of Noether's theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler-Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum, and the fractional derivative of the state variable.
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              A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations

              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.
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                Author and article information

                Journal
                03 May 2010
                Article
                10.1016/j.sigpro.2010.05.001
                1005.0252
                9b9d43f5-a492-4e94-90b0-f5616537b56f

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                26A33, 39A12, 49K05
                Signal Process. 91 (2011), no. 3, 513--524
                Submitted 24/Nov/2009; Revised 16/Mar/2010; Accepted 3/May/2010; for publication in Signal Processing.
                math.OC

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