Analysis and Computation for Optimal Control Problems with Pointwise State and Control Constraints

9803755 Dunn物理系统的状态响应于可变的外部施加的控制力、电压、温度、利率政策、资源分配策略等而演变，通常由微分方程或近似差分方程来建模。当允许在时间或空间上应用控制输入的方式中进行某些选择时，相关的优化问题自然会出现。这些最优控制问题在变分法中有经典的先驱，也可以被看作是函数空间或有限维序列空间中的特殊结构的非线性规划。最困难的最优控制问题对控制变量和状态变量施加严格的逐点界或其他不等式约束。这一类的普通问题可以很容易地涉及到数百或数千个具有相当多约束条件的变量，并且当以当前公认的数学格式进行转换时，通常会严重缩放，以至于最简单和最容易实现的迭代优化方法实际上是无能为力的。更复杂的牛顿和拟牛顿算法确实为几乎奇异局部极小点附近的不良标度提供了计算对策;但是，在此情况下，牛顿尺度方法是昂贵的，不能保证良好的非局部收敛性，而迭代仍然远离局部极小值。建议的研究将解决这些问题的理论和计算评估的标准优化计划实施的新的非标准数学框架的最优控制问题逐点状态和控制约束。替代框架取代局部微分或差分形式的状态演化方程的非局部集成的形式，并取代原始变量集的状态变量的新的人工变量，符合状态时，所有的约束条件。最近的实验与混合增广拉格朗日投影方法概述了这一建议表明真实的和重要的分析和计算的优势，在新的配方。拟议研究的目标是获得更多在此环境中实现算法的经验，并开发更清晰的最优性条件和其他所需的数学工具，以更深入地了解观察到的行为并改进计算方法。在这方面的改进必须有一个直接和重大的实际影响，因为大规模的计算挑战性的最优控制问题出现在许多物理设置。

9803755DunnPhysical systems whose states evolve in response to variable externallyapplied controlling forces, voltages, temperatures, interest rate policies,resource allocation strategies, and the like, are commonly modeled bydifferential equations or approximating difference equations. Relatedoptimization problems arise naturally when some choice is permitted inthe way the control inputs are applied in time or space. These optimal control problems have classic precursors in the Calculus of Variations, and may also be viewed as specially structured nonlinear programs in function spaces orfinite-dimensional sequence spaces. The most difficult optimal controlproblems enforce strict pointwise bounds or other inequality constraintson the control and state variables. Run-of-the-mill problems in thiscategory can easily entail hundreds or thousands of variables with comparablymany constraints, and when cast in the currently accepted mathematical format,are often so badly scaled that the simplest and most readily implementediterative optimization methods are effectively incapacitated. The moresophisticated Newtonian and quasi-Newtonian algorithms do providecomputational countermeasures for bad scaling near almost-singular local minimizers; however, Newtonian scaling procedures are costly and do notguarantee good nonlocal convergence properties while the iterates are stillfar from a local minimizer.The proposed investigation would address these issues with theoretical andcomputational evaluations of standard optimization schemes implemented ina new nonstandard mathematical framework for optimal control problemswith pointwise state and control constraints. The alternative framework replaces local differential or difference forms of the state evolution equations by nonlocal integrated forms, and replaces state variables in the primal variable set by new artificial variables that coincide with the state when all constraints are met. Recent experiments with hybrid augmented Lagrangian projection methods outlined in this proposal indicate real and important analytical and computational advantages in the new formulation. The goal of the proposed study is to gain more experience with algorithm implementations in this setting, and to develop sharpened optimality conditions and other mathematical tools needed to achieve a deeper understanding of the observed behavior and improve the computational methods. Improvements in this area must have an immediate and significant practical impact, since large-scale computationally challenging optimal control problems arise in many physical settings.