Abstract
This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.
Highlights
Fractional calculus refers to the integration and differentiation of a non-integer order and is as old as the classical calculus [1]
We are ready to present the main contributions of this work, by proving some generalizations of the fractional variational problem studied in [33]
One of the advantages of fractional derivatives is that, in many real problems, they better describe the dynamics of the problems compared to the classical derivative
Summary
For a detailed study on this subject, see [1,17]. In this present work, we consider fractional operators with respect to an arbitrary kernel (see [17] for the Riemann–Liouville sense and [18] for the Caputo sense). Fractional calculus of variations is a recent field that consists of minimizing or maximizing functionals that depend on fractional operators. Many papers were published on different topics of the fractional calculus of variations for different types of fractional operators
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