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.

目前的工作主要受我们在文献中发现的一些缺陷的启发，这些发现在证明二阶勒让德必要优化条件的分数阶变分演算问题中存在一些缺陷。因此，我们渴望详细说明一个正确的证明，结果证明这个目标非常重要，尤其是在考虑最终约束时。这篇论文是我们对这个主题的反思的结果。准确地说，我们在这里考虑一般 Bolza 泛函的约束最小化问题，该问题取决于阶数\(0 < \alpha \le 1\)的 Caputo 分数阶导数和阶数\(\beta > 0 )的黎曼-刘维尔分数积分\)，描述一般混合初始/最终约束的约束集。我们工作的主要贡献是推导出相应的一阶和二阶必要最优性条件，即欧拉-拉格朗日方程、横向条件，当然还有勒让德条件。详细讨论了经典策略遇到的障碍，而我们在这里提出的新证明是基于 Ekeland 变分原理。此外，我们强调整篇论文都提供了一些辅助贡献。特别地，我们证明了分数阶微积分的一个独立的和内在的结果，表明它不存在一个非平凡的函数，即连同其阶\(0< \alpha <1\) 的Caputo 分数阶导数 , 紧凑支持。此外，我们还讨论了一些证据，声称在制定变分问题的分数阶微积分时应考虑 Riemann-Liouville 分数阶积分，以保持解的存在性。