Simulation, Optimization, and Analysis of High Dimensional Higher Index DAEs

优化应用于方方面面，从航天器的控制到化学过程的优化。优化通常在存在相等和不相等约束的情况下进行。微分代数方程（DAEs）是隐式微分方程组。通常也假定雅可比矩阵对最高阶导数是奇异的。称为指数的整数是衡量DAE与显式常微分方程的差异程度的一种度量。许多问题最初都很自然地被建模为DAE，特别是那些使用计算机生成的数学模型进行分析和模拟的问题。由于原始问题是一个DAE，由于约束的激活，以及当偏微分方程使用线的方法进行数值求解时，在优化中会出现DAE。许多问题还涉及延迟，因为延迟固有地存在于过程中，或者因为控制或反馈循环中的延迟。直接转录法是解决最优控制问题的常用方法。最近的研究表明，当不等式约束最优控制问题的直接转录方法在数值解过程中出现DAE时，通常的DAE积分器理论需要进行实质性的修改。本建议的主要焦点是使用直接转录方法求解不等式约束最优控制问题，包括那些具有延迟的问题。理论将得到发展，理论将应用于数值算法及其在生产代码中的实现，算法将应用于许多应用。优化、延迟、直接转录、数值离散化、状态不等式和平等约束之间的相互作用还远远没有完全理解。本建议将仔细研究这种相互作用。此外，还将开发更通用的DAE仿真算法。设计在某种意义上最优的控制和过程行为是设计高效和有效过程的基础。优化可以包括最小化能耗、在最短时间内生产零件、最大化零件质量或最大化产品产量。一般来说，强大的DAE模拟、分析和优化软件的可用性将导致更准确的模型和性能预测，这反过来又转化为更有效的过程及其更有效的操作，从而产生更好的性能。要研究的特定理论和计算问题是由波音公司的应用程序引起的，并将应用于这些应用程序。具体的应用，每一个都可能有延迟，是实时控制，化学过程，和航空航天飞行器。然而，结果将被公开分发，并将在各种软件包和许多影响广泛行业的类似应用程序中找到用途。高性能和高效地利用复杂过程是现代工程和科学的基本需求。培训本项目领域的年轻科学家，并在他们的研究中建立工业和国际方面，将是本研究项目在未来几年的重大贡献。

Optimization is used in everything from the control of spacecraft to optimizing chemical processes. Optimization usually takes place in the presence of equality and inequality constraints. Differential algebraic equations (DAEs) are implicit systems of differential equations. Usually it is also assumed that the Jacobian with respect to the highest derivative is singular. An integer quantity called the index is one measure of how different a DAE is from being an explicit ordinary differential equation. Many problems are most naturally initially modeled as a DAE particularly those that are analyzed and simulated using computer generated mathematical models. DAEs occur in optimization because the original problem is a DAE, because of the activation of constraints, and when partial differential equations are numerically solved using the method of lines. Many problems also involve delays either because delays are inherently present in the process or because of delays in control or feedback loops. Direct transcription methods are a popular approach for the solution of optimal control problems. Recently it has been shown that the usual theory of DAE integrators needs to be substantially modified when DAEs occur during the numerical solution by direct transcription methods of inequality constrained optimal control problems. The primary focus of this proposal is the numerical solution of inequality constrained optimal control problems including those with delays using direct transcription methods. Theory will be developed, the theory will be applied to numerical algorithms and their implementation in production codes, and the algorithms will be applied to a number of applications. The interaction between optimization, delays, direct transcription, numerical discretizations, and state inequality and equality constraints is far from fully understood. This proposal will carefully examine this interaction. In addition, more general DAE simulation algorithms will be developed. Designing controls and process behavior which are optimal in some sense is fundamental to designing efficient and effective processes. The optimization can include minimizing energy consumption, producing a part in minimum time, maximizing part quality or maximizing product production. In general, the availability of robust DAE simulation, analysis, and optimization software will result in more accurate models and predictions of performance, which in turn translates to more efficient processes and their more efficient operation resulting in better performance. The particular theoretical and computational questions to be examined are motivated by, and will be applied to, applications arising at the Boeing Corporation. The specific applications each of which may have delays, are real time control, chemical processes, and aerospace vehicles. However, the results will be publicly distributed and will find use in a variety of software packages and in a number of similar applications that affect a wide range of industries. High performance and efficient use of complex processes is a basic need in modern engineering and science. The training of young scientists in the areas of this project and the establishment of both industrial and international aspects to their research will be a significant contribution of this research project for years to come.