On the interplay of controllability and stabilisation

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The proposal is to support the research of the principal investigator and his graduate students in the area of geometric control theory. Control theory is the study of systems whose dynamics can be partially prescribed by the user. The user-controlled dynamics can be designed to achieve a variety of outcomes. Two important properties of a control system connected to any design methodology are controllability and stabilisation. Roughly speaking, controllability is the study of which states of the system can be reached from a given initial state. An example of an application where controllability is important is in satellite control. If one wishes to reorient a satellite in space, it is first useful to understand which orientations of the satellite can be achieved from a given initial orientation. Stabilisation, roughly speaking, is the use of control to make a possibly unstable operating point stable, typically by the use of feedback, i.e., by using measurements of the state to specify the control. A simple example of stabilisation is the cruise control in a car which controls the throttle to maintain a nearly constant speed. The simple examples above suggest the utility of controllability and stabilisation. However, these are also fundamental system attributes, and so understanding these for a system is often a first step towards understanding the system per se. While controllability and stabilisation are well understood for simple classes of systems, they remain poorly understood for many large and interesting classes of systems. In the proposed research, the problems of controllability, stabilisation, and the connections between them will be studied. The systems studied are general nonlinear systems, where the tools of differential geometry are valuable. Indeed, the novel use of these tools will be an essential part of the proposed research. The research will advance the cutting edge of the understanding and application of control theory. The training received by graduate students connected to the proposed research will enable them to contribute to Canada's scientific community and high-tech economy.