Abstract In this paper, we are concerned with optimal control problems evolved on Riemannian manifolds, where the initial and final states satisfy some inequality and equality type constraints, and the control set is a separable metric space. We obtain the second order necessary conditions of integral and quasi-pointwise forms, both of which work for Pontryagin type critical controls and involve the curvature tensor. For this purpose, we need to develop the needle variation technique (up to the second order) in the framework of Riemannian manifolds, and show that along a critical variational direction, a suitably chosen set related to the second order variation is convex.

中文翻译：

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黎曼流形上端点约束最优控制问题的二阶必要条件

摘要 在本文中，我们关注在黎曼流形上演化的最优控制问题，其中初始状态和最终状态满足一些不等式和等式类型的约束，并且控制集是一个可分离的度量空间。我们获得了积分形式和准点形式的二阶必要条件，这两种形式都适用于庞特里亚金型临界控制并涉及曲率张量。For this purpose, we need to develop the needle variation technique (up to the second order) in the framework of Riemannian manifolds, and show that along a critical variational direction, a suitably chosen set related to the second order variation is convex.