Generalized fractional calculus with applications to the calculus of variations

We study operators that are generalizations of the classical Riemann–Liouville fractional integral, and of the Riemann–Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved, as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual. Such results are explored in the context of dynamic optimization, by considering problems of the calculus of variations with general fractional operators. Necessary optimality conditions of Euler–Lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated. Interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of Riemann–Liouville or Caputo. As an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem. Illustrative optimization problems are considered.