Impulsive hybrid systems with delays and applications

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Impulsive hybrid systems have emerged into a new discipline over the past two decades that bridges applied mathematics, control engineering, and computer science. They provide a framework for mathematical modeling of complex reactive systems where physical processes interact with man-made automated environments. Such systems are dynamical systems governed by differential (or difference) equations, which undergo vector field switching and/or state jumps, due to sudden changes in model characteristics. They arise from a wide variety of applications, such as artificial intelligence, communications, space technology, Internet-based health system, and unmanned autonomous vehicles. The differential equations involved may be a set of ordinary differential equations, delay differential equations, integro-differential equations, partial differential equations, stochastic differential equations, or even a mixture of the equations mentioned above. The corresponding difference equations can have a variety of forms as well. In the proposed research program, I will study impulsive hybrid systems that contain state-dependent impulses, state-dependent switching, state-dependent delays, random disturbance, even a combination of two or more of these features. The long-term goal of my research program is to advance the theory of impulsive hybrid systems and develop new ideas and novel methods to deal with state-dependent impulses/switching, and chattering problems, and make possible applications to real world problems. The objectives over the proposed period are to : (i) establish stability and robust stability theory for impulsive hybrid systems with state-dependent impulses and/or distributed delays and/or stochastic disturbances; (ii) develop bifurcation theory for impulsive delay differential equations with state-dependent impulses and those with delayed impulses; and (iii) apply the obtained theory and methods to practical problems, such as formation control of unmanned autonomous vehicles, and endemic models over networks. This research program represents an excellent opportunity for Canada to be at the fore-front of the cutting-edge technology and provide a means to train highly qualified personnel for Canadian universities and industry. It is envisaged that the proposed research will result in many high-quality scientific publications, several well-trained graduate students and undergraduate researchers, and a range of effective solution methods with associated software that will enable a wider range of non-experts to solve practical problems within their own disciplines. These students will obtain adequate analytical and computational skills to conduct independent research, mentor and teach students, and land a career in academia or in certain sectors of industry, such as robotics, artificial intelligence, and public health. The outcomes will greatly enhance Canada's research reputation in the field of impulsive hybrid systems.