Subdifferentials and Their Applications to Control Theory

本项目的主要目标是研究数学控制理论的几个重要领域，在这些领域中，非光滑性是问题固有的，即使原始问题是由连续甚至光滑数据定义的，通常的Lipschitz连续性或连续性假设也必须用下半连续性假设代替。作为这类研究的重要工具，我们还建议进一步发展次微分的基本理论和微积分。这些方法拟研究的控制问题包括：(1)无界控制和微分包含系统的可控性；(2)双层优化问题；(3)具有初值扰动和自然系统扰动的无限视界问题；(4)以拟线性椭圆型偏微分方程为起点的分布控制问题的最优性必要条件；(5)延迟控制问题的值函数和灵敏度分析；(6)离散随机变量控制问题的动态规划方法。上述提出的控制问题自然会在资源管理、柔性制造系统调度、运输调度等行业和管理问题中产生。这些控制问题已经用简化模型进行了研究，其中假定系统性能没有突变。然而，在现实生活中，这种突然的变化不仅是不可避免的，而且往往是问题的最重要特征之一。这些问题的不连续方面将用最近的次微分理论方法来研究。还将根据需要开发新的方法。这个项目的研究结果可能会为涉及类似不连续行为的其他应用提供启示。

9704203 Zhu The main goals of this project are to investigate several important areas of mathematical control theory in which nonsmoothness is intrinsic to the problem and the usual assumptions of Lipschitz continuity or continuity must be replaced by assumptions of lower semicontinuity even if the original problem is defined by continuous or even smooth data. As an important tool for such investigations, it is also proposed to further develop a basic theory and a calculus of subdifferentials. The control problems that are intended to be investigated by these methods include: (1) Controllability properties of unbounded control and differential inclusion systems; (2) Bilevel optimization problems; (3) Infinite horizon problems with both initial-value and natural system perturbations; (4) Necessary optimality conditions for distributed control problems starting with those defined by quasilinear elliptic partial differential equations; (5) Value function and sensitivity analysis for delayed control problems; and (6) Dynamical programming approach to control problems with discrete stochastic variables. The control problems proposed above naturally arise in resource management, flexible manufacturing system scheduling, transportation scheduling and other industry and management problems. These control problems have been studied with simplified models in which the system performance is assumed to be free of abrupt changes. However, in real life problems such abrupt changes are not only unavoidable but rather often among the most important features of the problem. The discontinuous aspects of these problems will be studied with recent methods of the subdifferential theory. New methods will also be developed as needed. Research results in this project are likely to shed light on other applications that involve similar discontinuous behaviors.