Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition - Application to fractional variational problems

Loïc Bourdin (1) , Tatiana Odzijewicz (2) , Delfim Torres (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 Warsaw School of Economics
3 Universidade de Aveiro

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator K and defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler–Lagrange type. Main results are illustrated with special cases, when K is a general kernel operator and, in particular, with K the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in doi:10.1155/2012/871912.

Mots clés: Fractional calculus; Necessary optimality condition; Euler–Lagrange equation; Calculus of variations; Existence of minimizers

Domaines: Optimisation et contrôle [math.OC]; Systèmes dynamiques [math.DS]

Loïc Bourdin, Tatiana Odzijewicz, Delfim Torres. Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition - Application to fractional variational problems. Differential and integral equations, 2014, 27 (7-8), pp.743-766. ⟨10.1155/2012/871912]⟩. ⟨hal-01253284v2⟩