Analysis and Control of Complementary Systems

互补系统是由常微分方程（ODE）所定义的一个动力系统，它涉及由微分方程状态参数化的有限维互补问题的解。互补系统构成了一种新的数学范式，在非光滑力学、机器人、多体动力学、开关电路系统、经济和交通系统，甚至生物系统中都有广泛的应用。因此，对这些系统进行严格的研究是必要的。由于其固有的非光滑特性，这种研究违背了经典的动力系统理论，需要新的数学分析、计算和设计工具。这个拟议的项目致力于研究互补系统分析和控制中的挑战性问题。我们建议运用数学规划、凸分析、系统理论和控制理论的最新技术来解决这些问题。在分析方面，我们将解决系统行为的几个基本和关键问题，例如，解的存在性和唯一性以及芝诺性质，这与数值计算和系统分析直接相关。此外，我们的目标是研究可控性和可观察性，并开发控制算法，将应用于机器人运动规划，受单边约束的多体系统，以及约束动态优化问题。由ode建模的动力系统为研究和理解在许多工程应用中自然发生的广泛的随时间变化的物理现象提供了强大的数学和计算框架。随着应用领域的扩展，研究人员和设计人员正在遇到越来越复杂的系统，这些系统受到各种约束，这些约束是由于系统的全局行为以及它们与复杂环境和/或不同层次的其他系统之间的相互作用而产生的。这种约束的一个简单例子是一个物体在撞击地面之前和之后下落。在物体接触地面的瞬间，撞击发生，随后物体反弹，导致物体运动方向的改变。这是一个模式转换。提出的研究旨在处理动力系统，其中模式转换是一个未知的，但整个系统配置的重要组成部分。如果成功，研究结果将使我们更好地理解许多具有模式变化的复杂工程系统，并将为改进此类系统的设计提供坚实的基础，从而对许多实际领域产生重大影响。拟议的工作还将促进基础科学的进步以及人力资源的教育和培训。

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.