A complementarity system is a dynamical system defined by an ordinary differential equation (ODE) involving the solutions of a finite-dimensional complementarity problem parameterized by the state of the differential equation. Complementarity systems constitute a new mathematical paradigm that finds a wide range of applications in nonsmooth mechanics, robotics, multi-body dynamics, switched circuit systems, economic and traffic systems, and even biological systems. As such, the rigorous study of these systems is warranted. Due to their intrinsically nonsmooth characteristics, such a study defies classical dynamical systems theory and requires novel mathematical analysis, computational and design tools. This proposed project is devoted to the investigation of challenging issues in the analysis and control of complementarity systems. We propose to apply state-of-art techniques from mathematical programming, convex analysis, systems theory, and control theory to tackle these problems. On the analytical side, we will address several fundamental and critical issues of system behavior, e.g., existence and uniqueness of solutions and Zeno property, which are directly related to numerical computation and system analysis. Moreover, we aim at studying controllability and observability and developing control algorithms that will be applied to robot motion planning, multi-body systems subject to unilateral constraints, and constrained dynamic optimization problems.
Dynamical systems modeled by ODEs provide a powerful mathematical and computational framework for the study and understanding of a wide range of time-dependent physical phenomena that naturally occur in many engineering applications. As the application areas expand, researchers and designers are encountering increasingly complex systems subject to various constraints, which arise as a result of the global behavior of the systems and the interactions between them and their complex environment and/or other systems at different levels. A trivial example of such constraints is a falling object before and after hitting the ground. At the instant where the object touches the ground, an impact occurs followed by a rebounce of the object that results in a change of direction of the object's motion. This is a mode transition. The proposed research aims at treating dynamical systems where mode transition is an unknown but important component of the overall system configuration. If successful, the results of the research will let us gain a better understanding of many complex engineering systems with mode changes and will provide a solid foundation for the improved design of such systems that in turn will have a significant impact on many practical fields. The proposed work will also contribute to the advancement of basic sciences and to the education and training of the human workforce.
