Research in Nonlinear Control Systems Theory: Lyapunov Functions, Stabilization, and Engineering Applications

《非线性控制系统理论研究：李雅普诺夫函数、镇定和工程应用》这项研究将在数学控制理论中提供新的基于李亚普诺夫的稳定性方法，其基础是系统地使用广义微分、最优控制和输入-状态可稳性框架。特别是，该项目将在现有理论无法处理的流形上，为非自治系统和欠驱动机械系统推导和应用新的反馈镇定技术。其他相关的研究方向包括：a)显式构造时变控制-Lyapunov函数，b)不连续反馈分歧点的分析，c)Lyapunov函数的逼近和正则性理论，d)输出稳定系统状态估值器的构造。通过发展强大的新的工程和数学技术，该项目将可能使用一种单一的、系统的、用户友好的方法来分析跨越许多学科的一大类重要的稳定性问题。反馈稳定被用于许多领域，包括工程学(例如，用于军事监视和侦察任务的自主空中和水面飞行器的控制)和系统生物学(例如，分析病毒感染模型中的细胞受体)。这项研究将通过支持毕业研究助理来促进学习，他们将在机械工程控制系统实验室应用和验证项目方法。该项目将在一个传统上吸引许多少数族裔的机构进行，并将特别努力从代表性不足的群体中招聘研究助理。

Abstract DMS-0424011, MalisoffLouisiana State UniversityResearch in Nonlinear Control Systems Theory: Lyapunov Functions, Stabilization, andEngineering ApplicationsThis research will provide new Lyapunov-based stability methods inmathematical control theory, which are based on the systematic use ofgeneralized differentials, optimal control, and the input-to-statestability framework. In particular, the project will derive and apply newfeedback stabilization techniques for nonautonomous systems andunderactuated mechanical systems on manifolds that are not tractable bythe existing theory. Other related lines of research to be pursuedinclude: a) the explicit construction of time-varying control-Lyapunovfunctions, b) the analysis of bifurcation points of discontinuousfeedbacks, c) approximation and regularity theory for Lyapunov functions,and d) the construction of state estimators for output stable systems.By developing powerful new engineering and mathematical techniques, thisproject will make it possible to analyze a large class of importantstability problems across many disciplines using a single, systematic,user-friendly approach. Feedback stabilization is used in many areas,including engineering (e.g., control of autonomous air and surfacevehicles for military surveillance and reconnaissance missions) andsystems biology (e.g., analysis of cell receptors in models of viralinfection). The research will promote learning by supporting graduatestudent research assistants, who will apply and validate the projectmethods in a mechanical engineering control system laboratory. The projectwill be carried out in an institution that traditionally attracts manyminorities, and special efforts will be made to recruit researchassistants from under-represented groups.