Nonsmooth and Geometric Methods in Nonlinear Control

本研究的主要目的是利用PI近年来发展起来的方法，在系统地使用广义微分代替经典微分，用流代替向量场，用抽象变分代替标准针变分的基础上，推导出最优控制理论中最小值的新必要条件。这将实现有限维庞特里亚金极大原理的各种现有版本的统一，将它们全部合并到一个结果中，此外，它将在范围上更普遍，也将适用于混合问题。其他几个相关的研究方向将继续进行：轨迹族最优的充分条件，使用V. Boltyainskii的“规则综合”思想的扩展（由于PI与B. Piccoli合作），斜积流的遍历性质（与M. Nerurkar合作），与退化拉格朗日问题对应的一阶Bellman方程的粘度解，以及值函数的亚分析性。这项工作的动机是需要强大的新的和可用的工具来研究曲线优化问题。这类问题出现在许多科学领域，从工程中自然产生的更传统的自动控制问题（例如电厂、飞机或各种机械装置的控制）到最近在生物和医学问题中的应用（例如寻找几种药物组合的最佳方法）。所有这些问题的共同之处是：(a)它们涉及寻找影响（即“控制”）系统行为的策略，以便使其以最佳方式实现预期目标（“最优控制”），或者至少尽可能接近该目标，(b)它们具有明确的动态结构(例如，人们不仅需要知道每种药物的剂量，但也包括具体的时间顺序，这是要做的)。寻找最优控制问题的解决方案已经变得更加困难，因为现有的数学方法涉及不同技术的集合，不能组合成一个单一的理论。这意味着大多数现实生活中的问题都不适合现有技术的框架，特别是当这些问题是“混合”的时候，即它们将连续的方面与离散的方面结合在一起。PI提出的研究的最终目标是开发出能够通过单一的、系统的、用户友好的方法来解决大量此类问题的工具。

The main purpose of the research is to derive new necessary conditions for a minimum in optimal control theory, using the approach developed by the PI in recent years, based on the systematic use of generalized differentials instead of classical differentials, of flows instead of vector fields, and of abstract variations instead of the standard needle variations. This will achieve a unification of the various existing versions of the finite-dimensional Pontryagin Maximum Principle, incorporating them all into a single result, which, in addition, will be more general in scope and will also apply to hybrid problems. Several other related lines of research will be pursued: sufficient conditions for an optimum for families of trajectories, using extensions ---due to the PI in collaboration with B. Piccoli---of V. Boltyainskii's idea of a "regular synthesis," ergodic properties of skew-product flows (in collaboration with M. Nerurkar), viscosity solutions of first-order Bellman equations corresponding to problems with degenerate Lagrangians, and subanalyticity of value functions.This work is motivated by the need for powerful new and usable tools for studying curve optimization problems. Such problems occur in many areas of science, ranging from the more traditional automatic control questions that arise naturally in engineering (e.g. control of power plants, aircraft, or various mechanical devices) to the more recent applications in biological and medical problems (e.g. the search for optimal ways to administer combinations of several medications). The common aspects of all these problems are (a) that they involve the search for strategies for influencing (i.e., "controlling") the behavior of a system so as to get it to achieve a desired goal in the best possible way ("optimal control") or, at least, to come as close as possible to that goal, and (b) that they have a definite dynamical structure (for example, one needs to know not only how much of each medication to administer, but also the specific time sequence in which this is to be done). The search for solutions to optimal control problems has been made more difficult by the fact that the existing mathematical methods involve a collection of different techniques that cannot be combined into a single theory. This means that most real-life problems do not fit within the framework of existing techniques, especially when these problems are "hybrid," in the sense that they combine continuous aspects with discrete ones. The ultimate goal of the PI's proposed research is to produce tools that will make it possible to attack large classes of such problems by means of a single, systematic, user-friendly approach.