论文标题
Legendre的分数Bolza功能具有混合初始/最终约束的必要条件
Legendre's necessary condition for fractional Bolza functionals with mixed initial/final constraints
论文作者
论文摘要
目前的工作主要是由我们在文献中发现的一些缺陷的发现，这是在二阶Legendre的证据证明变异问题的分数计算所必需的最佳条件。因此，我们渴望详细说明正确的证据，事实证明，这个目标是高度不平凡的，尤其是在考虑最终约束时。本文是我们对这个主题的思考的结果。确切地说，我们在这里考虑了一般的Bolza功能的一个约束最小化问题，该问题取决于订单0 <$α$ $ \ le $ 1的caputo分数衍生物，以及riemann-liouville的分数$β$> 0的分数集成，该约束套件描述了一般混合初始/最终约束。我们工作的主要贡献是得出相应的一阶和二阶必要的最佳条件，即Euler-Lagrange方程，横向条件，当然还有Legendre条件。就经典策略遇到的障碍提供了详细的讨论，而我们在这里提出的新证明是基于Ekeland的各种原则。此外，我们强调了所有沿本文提供了一些辅助捐款。特别是，我们证明了分数演算的独立和内在结果，表明它不存在非平凡函数，该函数及其Caputo分数衍生物的订单0 <$α$ <1，紧凑地支持。此外，我们还讨论了一些证据，声称在制定变化问题的分数计算中应考虑Riemann-Liouville分数积分，以保留解决方案的存在。
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 < $α$ $\le$ 1 and on a Riemann-Liouville fractional integral of order $β$ > 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 < $α$ < 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.
