Geometric and topological approaches in nonlinear control theory

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Nonlinear control theory is an important branch of control theory, with numerous applications ranging from Nuclear Magnetic Resonance medical imaging systems to airplane guidance and navigation systems. The complexity and the diversity of nonlinear control systems yield substantial challenges for the control theorist, and even basic issues surrounding controllability and stabilizability of general nonlinear control systems are still not completely understood. The research directions we are proposing to investigate are meant to further our understanding of nonlinear control systems by shedding light on: The role of topology in stabilization, the possible relations between local geometric invariants and Small-Time Local Controllability, heat kernels of sub-Riemannian geometries and their asymptotics, as well as properties of the partial differential equations that occur in the stabilization of mechanical systems via energy shaping. The importance of our proposed research stems from the fact that it deals with fundamental issues in nonlinear control theory. We believe that the better understanding of nonlinear control systems which we hope to achieve through our research will also yield practical benefits in all the numerous industries that make use of nonlinear control systems, by guiding the design of improved controllers and stabilizing controllers. We anticipate numerous HQPs to benefit from the training afforded by our proposed research program; in particular, we anticipate the direct involvement of at least 4 MSc, 2 PhD, and 2 PDF HQPs, as well as the involvement of at least 2 summer research undergraduate HQPs every year. These HQPs will gain deep experience in their area of pure and applied mathematics, and will be able to subsequently contribute to the field. Control systems form an essential part of numerous high-technology industries, and an industrialized country such as Canada can only benefit from such training.