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Noncommutative geometric structures on stabilizable infinite-dimensional linear systems

This paper aims at showing that noncommutative geometric structures such as connections and curvatures exist on internally stabilizable infinite-dimensional linear systems and on their stabilizing controllers. To see this new geometry, using the noncommutative geometry developed by Connes, we have to replace the standard differential calculus by the quantized differential calculus and classical vector bundles by projective modules. We give an explicit description of the connections on an internally stabilizable system and on its stabilizing controllers in terms of the projectors of the closed-loop system classically used in robust control. These connections aim at studying the variations of the signals in the closed-loop system in response to a disturbance or a change of the reference. We also compute the curvatures of these connections.