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

具有逐点状态和控制约束的最优控制问题的分析与计算

• 批准号: 9803755
• 负责人: Joseph Dunn
• 金额: $ 11.83万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803755&HistoricalAwards=false
• 关键词: Analysis; Computation; Optimal; Control; Problems

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.

[803755] dunn物理系统的状态随着外部施加的变量控制力、电压、温度、利率政策、资源分配策略等的变化而变化，通常用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上进行一些选择时，相关的优化问题自然出现。这些最优控制问题在变分学中有经典的前身，也可以看作是函数空间或有限维序列空间中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点向边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数百或数千个具有相当多约束的变量，并且当以当前接受的数学格式进行转换时，通常是如此糟糕的缩放，以至于最简单和最容易实现的迭代优化方法都有效地失效了。更复杂的牛顿和准牛顿算法确实为接近奇异局部最小值的不良缩放提供了计算对策；然而，牛顿缩放过程是昂贵的，并不能保证良好的非局部收敛性质，而迭代仍然远离局部最小。提出的研究将通过理论和计算评估来解决这些问题，这些标准优化方案在新的非标准数学框架中实施，用于具有点态和控制约束的最优控制问题。替代框架将状态演化方程的局部微分或差分形式替换为非局部积分形式，并将原始变量集中的状态变量替换为满足所有约束条件时与状态一致的新人工变量。最近对混合增广拉格朗日投影方法的实验表明，新公式具有重要的分析和计算优势。本研究的目标是在这种情况下获得更多的算法实现经验，并开发更优化的最优性条件和其他数学工具，以更深入地理解所观察到的行为并改进计算方法。这一领域的改进必须具有直接和显著的实际影响，因为在许多物理环境中出现了大规模的计算挑战最优控制问题。