The main objective of the project is the development of new mathematical tools for analysis and design of discontinuous feedback control for finite- and infinite-dimensional dynamical systems. The major directions for research are (i) development of a geometric control theory for nonlinear dynamical systems described by partial differential equations to obtain, in particular, conditions for global controllability in generalized Lie-brackets terms; (ii) design of stabilizing discontinuous feedbacks for such infinite-dimensional control systems and analysis of their robustness properties; (iii) study of dynamical control systems under persistent disturbances and problems of their robust feedback stabilization; (iv) applications of these result to design of nonlinear dynamical observers; (v) development of new optimality conditions for optimal feedback controls in differential games; (vi) application of such conditions to problems of optimal group pursuit; (vii) development of nonsmooth analysis methods for design of nonsmooth control Lyapunov functions and robust discontinuous feedback satisfying required specifications.
It has been recognized during last two decades that high performance of numerous nonlinear control systems arising in engineering applications can be achieved only by using discontinuous feedback controllers. One example of such a controller is given by sliding mode controllers which are used in important electrical-mechanical systems for stabilization and control. This research project will develop a unified mathematical approach to the design of discontinuous feedback controllers and analysis of their robustness and performance characteristics. Other examples of the need for discontinuous feedback controllers are provided by numerous control and stabilization problems for infinite-dimensional systems such as stabilization problems for fluid flows or quantum-mechanical systems arising in new technological developments. This project will develop geometric control theory for such infinite-dimensional systems to address problems of their feedback control and stabilization by using nonsmooth analysis tools. Such new tools can be used to design feedback controllers in numerous cases when traditional engineering linearization techniques don't work. Currently problems of coordinated control for groups of autonomous vehicles are the subject of significant interest for control engineers. New tools of nonsmooth analysis will be used to obtain new algorithms for optimal coordinated control of group of autonomous vehicles, in particular, in optimal group pursuit.
