This paper expands the existing sufficiency results for the strong minimality of an extremal of the Bolza problem. We cover the case where the strict Legendre Clebsh conditions are not strictly verified. An efficient, easy to use, algorithm to prove minimality is provided. It can be used on solutions with bang-bang control and does not require any local controllability property. The interval where sufficiency is provided is maximized over a class of discontinuous Verification Functions determined solving a sequence of Riccati problems. This class of Verification Functions can be adapted to all kind of boundary conditions. The algorithms developed in this paper are applied to space trajectory extremals that exhibit bang-bang control.

中文翻译：

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Bolza 问题中强极小性的扩展充分条件：空间轨迹优化的应用

本文扩展了 Bolza 问题极值的强极小性的现有充分性结果。我们涵盖了严格的 Legendre Clebsh 条件没有得到严格验证的情况。提供了一种高效、易于使用的算法来证明最小性。它可用于具有 bang-bang 控制的解决方案，并且不需要任何本地可控性属性。提供充分性的区间在确定解决一系列 Riccati 问题的一类不连续验证函数上最大化。此类验证函数可以适应所有类型的边界条件。本文开发的算法适用于表现出 bang-bang 控制的空间轨迹极值。