Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives

Abstract

We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler-Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo derivatives. Normal and abnormal extremals are considered.

Submitted 6/March/2010 to Communications in Nonlinear Science and Numerical Simulation; revised 12/July/2010; accepted for publication 16/July/2010

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• Calculus of variations with fractional derivatives and fractional integrals
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Cited by in corpus (7)

• Calculus of variations with fractional derivatives and fractional integrals
• Fractional Noether's theorem in the Riesz-Caputo sense
• Fractional variational calculus for nondifferentiable functions
• Towards a combined fractional mechanics and quantization
• Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
• The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo
• Fractional Calculus of Variations for Double Integrals