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

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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.