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.