Mathematical Sciences: Lipschitz Stability and Its Application to Numerical Analysis in Optimal Control

9404431 Hager/Dontchev 本研究将追求最近的应用程序的隐函数型结果集值映射在无限维设置的数值算法的最优控制问题和Lipschitz性质的解决方案。 这项工作认为，控制问题的状态和控制约束，有许多开放的问题。将研究控制约束的对数障碍方法，并推导出离散化的误差估计。 本研究将发展并分析数值演算法，以解决具有控制与状态限制的最佳控制问题。特别是，可能的Lipschitz性能的最优解将进行调查。知识的Lipschitz连续性的解决方案是重要的，例如，在研究收敛行为的各种数值逼近的解决方案。该研究的预期应用包括计算机辅助几何设计中的问题，以及存在障碍物时柔性结构（如机器人或起重机）的最优运动规划。

9404431 Hager/Dontchev This research will pursue the applications of recent implicit function-type results for set-valued maps in an infinite dimensional setting to numerical algorithms for optimal control problems and the Lipschitz properties of their solutions. This work considers control problems with state and control constraints for which there are many open questions. Log-barrier approaches to control constraints will be investigated, and error estimates for discretizations will be derived. This research will develop and analyze numerical algorithms for solving optimal control problems with control and state constraints. In particular, the possible Lipschitz properties of optimal solutions will be investigated. Knowledge of the Lipschitz continuity of the solution is important, for instance, in the study of the convergence behavior of various numerical approximations for the solution. Anticipated applications of the research include problems in computer aided geometric design, and optimal motion planning of a flexible structure (such as a robot or a crane) in the presence of obstacles.