Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative

Abstract

a b s t r a c tThis paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrangian depending on the free end-points. The fractional derivatives are defined in the sense of Caputo.

Discrete-Time Fractional Optimal Control

“…Analysis of the Caputo derivative with the help of the infinite state approach has been presented in [41]. Several works have been published on the calculus of the variations where the fractional derivatives are taken in the sense of Caputo [42,43], Riesz-Caputo [44], and combined Caputo [45]. Malinowska et al [46] discussed advanced methods in the fractional calculus of variations.…”

Fractional Variational Problems Depending on Indefinite Integrals and with Delay

“…[10,11,16,17,20,27,30,33]). For some literature on what this paper concerns, we suggest the reader to [2,4,6,8,9,12,18,19,23,31] for fractional variational problems dealing with Caputo derivative, in [3] for Lagrangians depending on fractional integrals, and in [13,21] when presence of indefinite integrals. For a standard variational approach to systems in presence of time delay or more general topics, we suggest the interested reader to the papers [1,24,25,29], and for the fractional approach to [7,14].…”

Isoperimetric problems of the calculus of variations with fractional derivatives

“…This is possible through a powerful tool known as the Euler-Lagrange equation [33]. Recently the theory of the calculus of variations has been considered in the fractional context [7,8,10,11,12,13,14,23,25,27,32]. The fractional calculus allows to generalize the ordinary differentiation and integration to an arbitrary (non-integer) order, and provides a powerful tool for modeling and solving various problems in science and engineering [28,29,31].…”