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 $H_{\infty }$ and $H_2$-problems.