Legendre’s Necessary Condition for Fractional Bolza Functionals with Mixed Initial/Final Constraints

Abstract

The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore, we were eager to elaborate a correct proof and it turns out that this goal is highly nontrivial, especially when considering final constraints. This paper is the result of our reflections on this subject. Precisely, we consider here a constrained minimization problem of a general Bolza functional that depends on a Caputo fractional derivative of order $$0 < \alpha \le 1$$ and on a Riemann–Liouville fractional integral of order $$\beta > 0$$ , the constraint set describing general mixed initial/final constraints. The main contribution of our work is to derive corresponding first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the transversality conditions and, of course, the Legendre condition. A detailed discussion is provided on the obstructions encountered with the classical strategy, while the new proof that we propose here is based on the Ekeland variational principle. Furthermore, we underline that some subsidiary contributions are provided all along the paper. In particular, we prove an independent and intrinsic result of fractional calculus stating that it does not exist a nontrivial function which is, together with its Caputo fractional derivative of order $$0< \alpha <1$$ , compactly supported. Moreover, we also discuss some evidences claiming that Riemann–Liouville fractional integrals should be considered in the formulation of fractional calculus of variations problems in order to preserve the existence of solutions.