Nonlinear impulsive and hybrid systems with time delay: theory, methods and applications

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There has been a growing world-wide interest in the study of impulsive and hybrid dynamical systems in recent years. Such systems arise in many applications such as communication networks, drug administration in cancer chemotherapy and diabetes, constrained impact mechanics, switching circuits in power electronics, non-smooth and hybrid control of spacecrafts, and ecosystems management. However, there are many challenging issues. Firstly, many theorems of continuous dynamical systems fail for impulsive and hybrid dynamical systems, especially  when the systems contain time delays. New ideas need to be developed that deal with impacts, jumps, switching, and chattering problems in a robust and easily applicable way. Secondly, the qualitative understanding of their dynamics requires specific methods and cannot be solved with simple adaptations of known techniques for continuous linear or nonlinear dynamical systems. The objectives of the proposed research program are (i) to develop the fundamental theory for impulsive and hybrid dynamical systems with time delay; (ii) to develop some general methods that deal with impacts, jumps, switching, and chattering problems; (iii) to develop stability and robust stability criteria and methods of computing the domain of  attraction; (iv) to develop stabilizing control design methods for impulsive and hybrid dynamical systems with delayed feedback; (v) to develop algorithms for impulsive synchronization of complex networks; (vi) to develop efficient computational algorithms for simulation and control of impulsive and hybrid dynamical systems; (vii) to apply the obtained methods to real world problems. In this program, both fundamental research (philosophy, concepts and theoretical investigation) and technical development (system design and computer simulation) will be conducted. The outcomes will enhance Canada's reputation for leading edge research and facilitate opportunities for international collaboration. It will also provide an excellent opportunity for research training of highly qualified personnel.