Nonsmooth methods in optimal control theory

这项研究的主旨是研究最优控制理论中现有数学工具不充分开发的问题公式。这种方法依赖于非光滑分析方法，非光滑分析方法是经典微积分的扩展，它系统地处理可能在通常意义上不可微的函数的导数性质。它的发展在很大程度上是由最优化中的问题推动的，在最优化中，不等式约束和最小/最大运算无处不在，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)完全凸控制，一方面，这是相当特殊的，因为它需要状态和速度的凸性假设。但另一方面，它是线性二次型调节器的自然推广，而线性二次型调节器是应用中最优控制的主力。我们将从这个角度来研究状态约束、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，但仅以耗散的方式存在。动力学中的这种结构存在于干摩擦的建模中。(3)时滞问题。(4)脉冲系统，其中状态按两个时间尺度演化，并且可以跳过短时间间隔。为了反映工程系统某些行为优先于其他行为的可取性，数学模型中往往包含优化方面，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。由于世界上的非平稳特征似乎比之前想象的要多得多，最优控制领域出现了新的理论挑战。非平稳分析的发展在很大程度上是由这些考虑推动的，现在由大量的结果组成，越来越多的工程师使用这些结果。这一研究项目将拓宽非光滑分析工具的应用范围，其结果将使控制工程师能够采用比目前使用的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.