Nonsmooth methods in optimal control theory

最优控制理论中的非光滑方法

• 批准号: 0405132
• 负责人: Peter Wolenski
• 金额: $ 18.4万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405132&HistoricalAwards=false
• 关键词: Nonsmooth; methods; optimal; control; theory

The main thrust of this research is to study problem formulations in optimal control theory for which the extant mathematical tools are not adequately developed. The approach relies on methods of nonsmooth analysis, which is an extension of classical calculus that systematically handles derivative-like properties of functions that may not be differentiable in the usual sense. Its development was largely motivated by problems in optimization, where inequality constraints and min/max operations are ubiquitous, but which do not preserve classical differentiability. This project investigates four related problem areas. (1) Fully convex control, which is on the one hand quite special since it requires convexity assumptions in the state and velocity. But on the other hand, it is a natural generalization of the linear-quadratic regulator, which is the workhorse of optimal control in applications. Problems with state constraints, impulse trajectories, and infinite horizon will be studied from this viewpoint. (2) One-sided Lipschitz dynamics, where the data may have non-Lipschitz behavior but only in a dissipative manner. Such structure in the dynamics is present in the modeling of dry friction. (3) Time-delay problems. (4) Impulsive systems, in which the states evolve according to two time scales and can jump over short time intervals.To reflect the desirability of certain behaviors of engineered systems in preference to others, aspects of optimization are often included in mathematical models, and dynamic optimization problems arise. Optimal control theory offers deep insight into the nature of these problems. Because the world appears to have many more nonsmooth characteristics than previously imagined, new theoretical challenges in optimal control have emerged. The development of nonsmooth analysis was largely motivated by these considerations, and now consists of a substantial body of results that is being increasingly utilized by engineers. This research project will broaden the range of application of nonsmooth analytic tools, and the results will allow control engineers to employ mathematical models that are more accurate and realistic than those in current use.

本研究的主旨是研究最优控制理论中现存的数学工具还没有得到充分发展的问题。 该方法依赖于非光滑分析的方法，这是经典微积分的扩展，系统地处理在通常意义上可能不可微的函数的类似导数的性质。 它的发展很大程度上是由优化问题推动的，其中不等式约束和最小/最大运算无处不在，但不保持经典的可微性。 该项目调查了四个相关的问题领域。 (1)完全凸控制，这是一方面非常特殊，因为它需要凸性假设的状态和速度。 但另一方面，它是线性二次型调节器的自然推广，线性二次型调节器是应用中最优控制的主力。 从这个观点出发，我们将研究具有状态约束、脉冲轨迹和无限视界的问题。 (2)单侧Lipschitz动力学，其中数据可能具有非Lipschitz行为，但仅以耗散方式。 这种结构的动力学是目前在建模的干摩擦。 (3)时间延迟问题。 (4)脉冲系统，其状态按照两个时间尺度演化，并且可以在很短的时间间隔内跳跃。为了反映工程系统的某些行为优先于其他行为的可取性，数学模型中经常包含优化的方面，从而产生动态优化问题。 最优控制理论为这些问题的本质提供了深刻的见解。 由于世界似乎比以前想象的具有更多的非光滑特性，因此出现了最优控制的新理论挑战。 非光滑分析的发展在很大程度上是出于这些考虑，现在包括大量的结果，越来越多地被工程师使用。 该研究项目将拓宽非光滑分析工具的应用范围，其结果将允许控制工程师采用比目前使用的数学模型更准确和更现实的数学模型。