Nonsmooth methods in optimal control theory

本研究的主要目的是研究最优控制理论中现有数学工具尚未充分开发的问题公式。该方法依赖于非光滑分析方法，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导数性质。它的发展在很大程度上是由优化问题所驱动的，其中不等式约束和最小/最大操作是普遍存在的，但它们不能保持经典的可微性。这个项目调查了四个相关的问题领域。(1)全凸控制，这一方面是非常特殊的，因为它需要在状态和速度上的凸性假设。但另一方面，它是线性二次型调节器的自然推广，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。这种动力学结构存在于干摩擦模型中。(3)时滞问题。(4)脉冲系统，其中状态根据两个时间尺度演化，并且可以在短时间间隔内跳跃。为了反映工程系统的某些行为优于其他行为的可取性，优化的各个方面通常包含在数学模型中，并且出现了动态优化问题。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征，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由这些考虑所推动的，现在包含了大量的结果，这些结果越来越多地被工程师所利用。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使用比当前使用的数学模型更准确和更现实的数学模型。

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.