Focused Research Collaboration Proposal: Differential Algebraic Inequalities and their Applications to Engineering
This focused research project proposes a comprehensive investigation of Differential Algebraic Inequalities (DAIs) and Dynamic Complementarity Problems (DCPs), with the goal of developing an extensive theory, designing and analyzing efficient algorithms, and applying the results to problems in engineering of practical importance. The two problem classes of DAIs and DCPs represent a significant extension of an ordinary differential equation (ODE) and a differential algebraic equation (DAE). The proposed study necessitates the use of state of the art mathematical programming methods in conjunction with ODE and DAE methods to deal effectively with the inequalities and complementarity conditions in a DAI/DCP. The latter are novel features that are absent in ODEs or DAEs. Since the main methods for solving DCPs and DAIs in realistic situations are numerical, the research will emphasize the interactions among formulation, computation, and mathematical issues such as convergence and approximation including numerical approximation and sensitivity to parametric perturbations.
DAIs and DCPs provide a powerful mathematical framework for the comprehensive treatment of a host of important complex system applications that have so far received only minimal attention from applied and computational mathematicians. The project will open a new chapter in applied mathematics in which classical differential equation theory is merged with contemporary mathematical programming methods. The deliverables of the project will a set of broadly applicable mathematical and computational tools that will have a direct impact in several distinct engineering disciplines including: constrained mechanical systems with frictional contact arising in robotics and manufacturing, conditional modeling in chemical and hydraulic processes, and hybrid systems with variable-structure control encountered in avionics, intelligent highway systems, and automotive systems.
