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这个重点研究项目提出了微分代数不等式（DAI）和动态互补问题（DCP）的全面调查，其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意义的工程问题。 DAI和DCP这两类问题是常微分方程（ODE）和微分代数方程（DAE）的重要扩展。建议的研究需要使用最先进的数学规划方法结合ODE和DAE方法，以有效地处理DAI/DCP中的不等式和互补条件。 后者是ODE或DAE中不存在的新功能。由于在现实情况下解决DCP和DAI的主要方法是数值的，因此研究将强调公式化，计算，DAI和DCP提供了一个强有力的数学框架，用于综合处理许多重要的复杂系统应用，而这些应用迄今为止只得到很少的应用关注和计算数学家。 该项目将开启应用数学的新篇章，其中经典微分方程理论与当代数学编程方法相结合。该项目的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变结构控制混合系统，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.