FRG: Focused Research Collaborative Proposal: Differential Algebraic Inequalities and their Applications in Engineering

重点研究合作建议：微分代数不等式及其在工程中的应用本重点研究项目提出了对微分代数不等式（DAIs）和动态互补问题（dcp）的全面研究，目标是发展广泛的理论，设计和分析有效的算法，并将结果应用于具有实际重要性的工程问题。dai和dcp这两类问题代表了常微分方程（ODE）和微分代数方程（DAE）的重要扩展。提出的研究需要使用最先进的数学规划方法与ODE和DAE方法相结合，以有效地处理DAI/DCP中的不等式和互补条件。后者是ode或DAEs所没有的新特性。由于在实际情况下求解dcp和DAIs的主要方法是数值方法，因此研究将强调公式，计算和数学问题之间的相互作用，例如收敛和近似，包括数值近似和对参数摄动的敏感性。DAIs和dcp为综合处理大量重要的复杂系统应用提供了一个强大的数学框架，这些应用和计算数学家迄今为止只得到了很少的关注。该项目将在应用数学中开启经典微分方程理论与当代数学规划方法相结合的新篇章。该项目的可交付成果将是一套广泛适用的数学和计算工具，这些工具将对几个不同的工程学科产生直接影响，包括：机器人和制造业中产生摩擦接触的约束机械系统，化学和液压过程中的条件建模，以及航空电子设备中遇到的具有可变结构控制的混合系统，智能公路系统和汽车系统。

Focused Research Collaboration Proposal: Differential AlgebraicInequalities and their Applications to EngineeringThis focused research project proposes a comprehensive investigationof Differential Algebraic Inequalities (DAIs) and DynamicComplementarity Problems (DCPs), with the goal of developing anextensive theory, designing and analyzing efficient algorithms, andapplying the results to problems in engineering of practicalimportance. The two problem classes of DAIs and DCPs represent asignificant extension of an ordinary differential equation (ODE) and adifferential algebraic equation (DAE). The proposed study necessitatesthe use of state of the art mathematical programming methods inconjunction with ODE and DAE methods to deal effectively with theinequalities and complementarity conditions in a DAI/DCP. The latterare novel features that are absent in ODEs or DAEs. Since the mainmethods for solving DCPs and DAIs in realistic situations arenumerical, the research will emphasize the interactions amongformulation, computation, and mathematical issues such as convergenceand approximation including numerical approximation and sensitivity toparametric perturbations.DAIs and DCPs provide a powerful mathematical framework for thecomprehensive treatment of a host of important complex systemapplications that have so far received only minimal attention fromapplied and computational mathematicians. The project will open a newchapter in applied mathematics in which classical differentialequation theory is merged with contemporary mathematical programmingmethods. The deliverables of the project will a set of broadlyapplicable mathematical and computational tools that will have adirect impact in several distinct engineering disciplines including:constrained mechanical systems with frictional contact arising inrobotics and manufacturing, conditional modeling in chemical andhydraulic processes, and hybrid systems with variable-structurecontrol encountered in avionics, intelligent highway systems, andautomotive systems.