Nonsmooth and Geometric Methods in Nonlinear Control

研究的主要目的是导出最优控制理论中最小值的新的必要条件，使用PI近年来开发的方法，基于系统地使用广义微分而不是经典微分，流而不是向量场，抽象变化而不是标准针变化。 这将实现有限维庞特里亚金最大值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，这将在范围上更一般，并且也将适用于混合问题。 其他几个相关的研究线将继续：充分条件的最佳家庭的轨迹，使用扩展-由于PI与B合作。V. Boltyainskii的“规则综合”思想的Piccoli，斜积流的遍历性质（与M。Nerurkar），退化Lagrange函数问题对应的一阶Bellman方程的粘性解，以及值函数的次解析性。这项工作的动机是研究曲线优化问题需要强大的新的可用工具。 这些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（例如发电厂，飞机或各种机械设备的控制）到生物和医学问题中的更近期应用（例如寻找最佳方法来管理几种药物的组合）。 所有这些问题的共同点是：（a）它们涉及寻求影响的战略（即，“控制”）系统的行为，以便使其以最佳可能的方式（“最佳控制”）实现期望的目标，或者至少尽可能接近该目标，以及（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给药量，而且需要知道这要在其中进行的特定时间顺序）。 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，这些技术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不适合现有技术的框架，特别是当这些问题是“混合”的时候，在这个意义上，它们将联合收割机连续的方面与离散的方面结合起来。 PI提出的研究的最终目标是生产工具，使人们有可能通过一个单一的，系统的，用户友好的方法来攻击大类的问题。

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.