Fully Convex and Nonlinear Control Theory

控制理论中的线性-二次型模型是工程设计中许多强大方法的鼻祖，它仍然被广泛应用于许多其他学科。这主要是因为可以以一种相对容易的方式以非常理想的形式产生问题的解决方案。两个主要缺点是：(1)并非所有系统都能被线性-二次模型充分逼近；(2)为了有效地实现结果，状态空间必须相对较小。这一提议中的完全凸模型极大地扩大了线性-二次范式的适用范围，同时仍然保持了它的许多吸引人的特征。特别是，完全凸问题允许对控制变量进行硬约束，这一情况经常出现在应用程序中，但在传统框架中没有考虑到，因此将部分缓解缺点(1)。该提议寻求将哈密顿-雅可比理论的现代趋势与完全凸控制模型在理论和计算上统一起来。为了缓解(2)，该提议寻求开发基于丰富的凸对偶结构的高效算法。第二个目标包括详细分析非线性系统的渐近稳定性，特别是李雅普诺夫函数的构造和使用。最优控制理论是一门数学学科，在科学、工程、经济和工业中有着广泛的应用。现有的理论不断地受到这些应用的挑战，这随后为开发更复杂的数学工具提供了动力。这一建议的目的是在理论和计算方面取得进展，建立一个在应用中广泛使用但尚未充分发挥其潜力的数学模型。拟议目标的适用性将是提高一个重要的标准建模工具的效率和扩大其范围。

9972241WolenskiThe linear-quadratic model in control theory is the originator of many of the powerful methods in engineering design, and it is still widely used in many other disciplines as well. This is mainly because a solution to the problem can be produced in a relatively easy way in a highly desirable form. Two major drawbacks are that (1) not all systems can be adequately approximated by linear-quadratic models and (2) the state space must be relatively small for efficient implementation of the results. The fully convex model featured in this proposal significantly broadens the applicable scope of the linear-quadratic paradigm while still maintaining many of its attractive features. In particular, the fully convex problem permits hard constraints on the control variables, a situation often present in applications but not accounted for in the traditional framework, and thus will partly appease the drawback (1). The proposal seeks to unite modern trends in Hamilton-Jacobi theory with the fully convex control model both theoretically and computationally. To alleviate (2), the proposal seeks to develop efficient algorithms based on the rich convex duality structure. A secondary objective involves a detailed analysis on aspects of asymptotic stability of nonlinear systems, in particular the construction and use of Lyapunov functions.Optimal control theory is a mathematical discipline with far reaching applications throughout science, engineering, economics, and industry. The existing theory is constantly being challenged by these applications, which subsequently provides motivation for the development of more sophisticated mathematical tools. This proposal aims at making theoretical and computational progress in a mathematical model that is widely used in applications but which has by no means reached its full potential. The applicability of the proposed goals would be an increase in the efficiency and a broadening of the scope of an important standard modeling tool.