Generalized Euler—Lagrange Equations and Transversality Conditions for FVPs in terms of the Caputo Derivative

Abstract

This paper presents extensions to the traditional calculus of variations for systems containing Fractional Derivatives (FDs) defined in the Caputo sense. Specifically, two problems are considered, the simplest Fractional Variational Problem (FVP) and the FVP of Lagrange. Both specified and unspecified end conditions and end points are considered. Results of the first problem are extended to problems containing multiple FDs. For the second problem, we present a Lagrange type multiplier rule. For both problems, we develop generalized Euler—Lagrange type necessary conditions and the transversality conditions, which must be satisfied for the given functional to be extremum. The formulations show that one may require fractional boundary conditions even when the FVP has been formulated in terms of Caputo Fractional Derivatives (CFDs) only, and that the Riemann—Liouville Fractional Derivatives (RLFDs) automatically appear in the resulting equations. The relationships between the RLFDs and the CFDs are used to write the resulting differential equations purely in terms of the CFDs. Several examples are considered to demonstrate the application of the formulations. The formulations presented and the resulting equations are very similar to the formulations given for FVPs defined in terms of the RLFDs and to those that appear in the field of classical calculus of variations. The formulations presented are simple and can be extended to other problems in the field of fractional calculus of variations.