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

该项目致力于研究属于或与确定性、有限维非线性控制理论有关的数学问题，继续主要研究者在这一领域广泛的理论和应用问题上的工作，目的是解决最优控制中的基本公开问题(特别是关于高阶非光滑的最优性必要条件，以及具有状态空间约束的最优控制)，并开发具有广泛潜在应用的新问题的工具。所使用的方法将是微分几何控制理论，特别是庞特里亚金最大值原理的几何内在版本、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，以及几乎下半连续集值映射的针变分理论。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)机械手和其他机器人系统的控制，这在数学上相当于“非完整系统”(即，其中一个人只能直接控制少量方向，但一个人也可以通过组合基本运动来有效地实现许多其他方向的运动，例如汽车，它不能横向移动，但可以停车-好像它可以横向移动-通过结合向前和向后运动和转弯)，(B)对化工厂的控制，这导致了涉及所谓的“单一控制”的问题，(C)对水下航行器的控制(正在进行中，这已经导致了一份出版物)。

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).