Abstract
Sufficient conditions for extremum of fractional variational problems are formulated with the help of Caputo fractional derivatives. The Euler–Lagrange equation is defined in the Caputo sense and Jacobi conditions are derived using this. Again, Wierstrass integral for the considered functional is obtained from the Jacobi conditions and the transversality conditions. Further, using the Taylor’s series expansion with Caputo fractional derivatives in the Wierstrass integral, the Legendre’s sufficient condition for extremum of the fractional variational problem is established. Finally, a suitable counterexample is presented to justify the efficacy of the fresh findings.
Highlights
Introduction and preliminariesThe fractional calculus deals with the problem of extremizing functionals, which are of non integer order and are differentiable
Variational fuctionals with a Lagrangian are considered by Odzijewicz et al [31] using classical as well as Caputo fractional derivatives
Almeida [9] considered several problems of calculus of variations depending upon Lagrange function on a Caputo type fractional derivative
Summary
The fractional calculus deals with the problem of extremizing functionals, which are of non integer order and are differentiable. It’s origin is more than three centuries old, and dates back to L’Hopital’s query to Leibniz about the significance of the fractional order derivative of a function. In his reply to the above question, Leibniz indicated it to be a paradox which can lead to useful consequences in the near future [36]. Several fractional derivatives in the sense of Grunwald Letnikov, Riemann–Liouville and Caputo et al are discussed in [32]. Legendre’s second order necessary optimality conditions for weak extremizers of the variational problems was subsequently established by Lazo and Torres [29], where the involved functionals are fractional differentiable in the sense of Riemann–Liouville.
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