Differential-Geometric and Nonsmooth Methods in Deterministic Finite-Dimensional Control

该项目涉及属于或与确定性有限维非线性控制理论相关的数学问题的研究，继续首席研究员之前在该领域的广泛理论和应用问题方面的工作，目的是解决最优控制中的基本开放问题（特别是高阶非光滑最优性必要条件）。以及具有状态空间约束的最优控制)，这构成了重大的数学挑战，以及开发具有广泛潜在应用于新问题的工具。所使用的方法将是微分几何控制理论，特别是几何内禀版本的庞特里亚金极大原理、高阶最优性条件理论和非光滑分析。基本的研究策略将是使用多值微分、流和广义抽象变分，以及针变分理论来研究几乎下半连续集值映射。可能，实际分析地图的理论及其相关的分层也将发挥作用。在这项工作中要研究的问题包括在工程和机械应用中出现的大类优化问题，例如：(a)控制机械手和其他机器人系统，这在数学上相当于“非完整系统”的寻路（也就是说，人们只能直接控制少数方向的系统，但通过结合基本运动，人们也可以有效地实现许多其他方向的运动，就像汽车一样，它不能侧向移动，但可以停车-好像它可以侧向移动-通过结合向前和向后的运动和转弯）；(b)对化工厂的管制，这导致涉及所谓“单一管制”的问题；(c)对水下航行器的管制（目前正在进行的工作，已经发表了一份出版物）。

The project deals with research on mathematical problems belonging or related to deterministic, finite-dimensional nonlinear control theory, continuing the principal investigator's previous work in this area on a broad class of theoretical and applied questions, with the objective of solving fundamental open problems in optimal control (in particular on high-order nonsmooth necessary conditions for optimality, and optimal control with state space constraints) that pose significant mathematical challenges, and developing tools with a wide range of potential applications to new problems. The methods used will be those of differential-geometric control theory, especially geometrically intrinsic versions of the Pontryagin Maximum Principle, theories of high-order optimality conditions, and nonsmooth analysis. The basic research strategy will be to use multivalued differentials, flows, and generalized abstract variations, as well as the theory of needle variations for almost lower semicontinuous set-valued maps. Possibly, the theory of real-analytic maps and their associated stratifications will play a role as well.The questions to be studied in this work include large classes of optimization problems that occur in engineering and mechanical applications, such as: (a) control of robotic hands and other robotic systems, which mathematically amounts to path-finding for "non-holonomic systems" (that is, systems where one can directly control only a small number of directions, but one can effectively achieve motion in many other directions as well by combining the basic motions, as in the case of a car, which cannot move sideways but can be parked---as if it could move sideways---by combining forward and backward motions and turns), (b) control of chemical plants, which leads to problems involving so-called "singular controls", (c) control of underwater vehicles (in work now in progress, that has already lead to one publication).