The Legendre condition of the fractional calculus of variations

• Published in: Optimization Vol. 63; no. 8; pp. 1157 - 1165
• Main Authors: Lazo, Matheus J., Torres, Delfim F.M.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.08.2014; Taylor & Francis LLC
• Subjects: Calculus; Calculus of variations; calculus of variations and optimal control; Derivation; Derivatives; Eulers equations; Fractal analysis; fractional calculus; Functionals; Integer programming; Integrals; Lagrange multiplier; Legendre condition; Mathematical analysis; Mathematical models; Mathematical problems; Optimization; Riemann-Liouville and Caputo derivatives; second-order optimality condition; Studies
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2013.877908

Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler-Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives.