机械系统最优控制中的高阶变分积分器
High order variational integrators in the optimal control of mechanical systems
AI总结：
本文提出两种高阶变分积分器，用于求解有限维机械系统最优控制问题，证明了状态和控制变量的收敛性，并证明了对偶化与离散化可交换。
摘要：
近年来，设计用于机械系统仿真和优化的数值方法越来越多地关注结构保持方案。变分积分器是一种保持动量和辛结构的方案。本文开发了两种高阶变分积分器，区别在于底层近似空间的维度，并探讨了其在有限维最优控制问题中的应用。证明了近似问题的状态和控制变量的收敛性。通过分析最优控制问题的伴随系统及其离散化版本，证明了对于这些特定积分器，对偶化与离散化可交换。
英文摘要：
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.
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