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Section: New Results
Noncommutative geometry approach to infinite-dimensional systems
Participant : Alban Quadrat.
In [112] , [111] , [110] , it was shown how the fractional
representation approach to analysis and synthesis problems developed
by Vidyasagar, Desoer, Callier, Francis, Zames..., could be recast
into a modern algebraic analysis approach based on module theory
(e.g., fractional ideals, algebraic lattices) and the theory of Banach
algebras. This new approach successfully solved open questions in the
literature. Basing ourselves on this new approach, we explain in
[114] , [115] why the non-commutative
geometry developed by Alain Connes is a natural framework for the
study of stabilizing problems of infinite-dimensional systems. Using
the 1-dimensional quantized calculus developed in non-commutative
geometry and results obtained in [112] , [111] , [110] , we show
that every stabilizable system and their stabilizing controllers
naturally admit geometric structures such as connections, curvatures,
Chern classes, ... These results developed in
[114] , [115] are the first steps toward the
use of the natural geometry of the stabilizable systems and their
stabilizing controllers in the study of the important