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有力方法的鼻祖，它仍然广泛地应用于许多其他学科。 这主要是因为可以以相对容易的方式以高度期望的形式产生问题的解决方案。 两个主要的缺点是：（1）不是所有的系统都可以用线性二次模型来近似;（2）为了有效地实现结果，状态空间必须相对较小。 该建议中的全凸模型显著拓宽了线性二次范式的适用范围，同时仍然保持了其许多有吸引力的特征。 特别是，全凸问题允许对控制变量进行硬约束，这种情况经常出现在应用中，但在传统框架中没有考虑到，因此将部分缓解缺点（1）。 该提案旨在将Hamilton-Jacobi理论的现代趋势与完全凸控制模型在理论和计算上结合起来。 为了缓解（2），该提案寻求开发基于丰富的凸对偶结构的有效算法。 第二个目标是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在科学、工程、经济和工业中有着广泛应用的数学学科。 现有的理论不断受到这些应用的挑战，随后为开发更复杂的数学工具提供了动力。 这一提议的目的是在一个数学模型中取得理论和计算上的进展，该模型在应用中得到广泛使用，但绝没有充分发挥其潜力。 拟议目标的适用性将是提高效率和扩大一个重要的标准建模工具的范围。

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.