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

重点研究合作方案：微分代数不等性及其在工程中的应用这一重点研究项目提出了对微分代数不等式(DAI)和动态互补问题(DCP)的全面研究，目的是发展广泛的理论，设计和分析有效的算法，并将结果应用于具有实际意义的工程问题。DAIS和DCPS这两类问题是常微分方程(ODE)和微分代数方程(DAE)的重要推广。所提出的研究需要使用最新的数学规划方法，结合ODE和DAE方法来有效地处理DAI/DCP中的不平等和互补条件。后者是颂歌或代词所没有的新奇特征。由于实际情况中求解DCP和DAI的主要方法是枚举的，因此该研究将侧重于公式、计算和诸如收敛和逼近等数学问题之间的相互作用，包括数值逼近和对参数扰动的敏感性。DAI和DCP为综合处理许多重要的复杂系统应用提供了强大的数学框架，到目前为止，应用数学家和计算数学家对这些应用只给予了很少的关注。该项目将开启应用数学的新篇章，将经典的微分方程式理论与当代的数学规划方法相结合。该项目的成果将是一套广泛适用的数学和计算工具，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、智能公路系统和汽车系统中遇到的具有变结构控制的混合系统。

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.