This project will develop highly innovative stabilization theory for important classes of nonlinear dynamical systems in mathematical control theory, based on the systematic use of Lyapunov functions, generalized differentials, and the input-to-state stability framework. The ultimate goal is to provide a bold and far-reaching unification of the various existing constructions of time-varying feedback stabilizers and Lyapunov functions, which will apply to very general systems with mixtures of discrete and continuous time scales and outputs, including nonholonomic systems that cannot be globally stabilized by continuous time invariant feedbacks. Such systems are ubiquitous in science. Other topics to be pursued include (a) quantifying the effects of introducing feedback delays into a priori stable closed loop systems, (b) analysis of bifurcation points of discontinuous feedbacks, (c) tracking problems for engineering models with uncertain model parameters, (d) explicit constructions of Lyapunov functions and smooth repulsive feedback stabilizers for systems with measurement uncertainty, and (e) quasi time optimal feedback stabilization. By developing powerful new engineering and mathematical techniques, this work will make it possible to analyze a large class of important stability problems across many disciplines using a single, systematic and user-friendly approach.
Mathematical control theory and optimization provide the theoretical foundations that undergird many modern technologies including aeronautics, biotechnology, communications networks, manufacturing, and models of climate change. Feedback stabilization is used in many of these areas, ranging from electromechanical engineering to biomathematics. This innovative project will strive for breakthroughs rather than incremental improvements. Its methods will lead to feedback stabilizers for significant classes of control systems that are beyond the scope of the known continuous feedback stabilization techniques but which commonly arise in engineering, such as systems with time delays and multiple time scales for which only partial information is available. This project will suggest and explore creative and original feedback concepts in several applications that are of compelling engineering interest, such as chemostats (which model microorganisms competing for limiting nutrients) and micro-electromechanical relays (which are used to open or close connections in electric circuits). The research will promote learning by supporting research assistants, who will apply and validate the methods in a control system laboratory. The work will be carried out at an institution that traditionally attracts many minorities, and special efforts will be made to recruit qualified research assistants from under-represented groups.
