Control theory and series in noncommuting variables

Speaker(s)

Lorenzo Clemente

Affiliation

University of Warsaw

Language of the talk

English

Date

Oct. 9, 2024, 2:15 p.m.

Room

room 5440

Title in Polish

Control theory and series in noncommuting variables

Seminar

Seminar Automata Theory

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We present Michel Fliess' work from the 1970's showing surprising connections between control theory and the theory of series in noncommuting variables.

More precisely, to a dynamical system with inputs S one can associate a generating series in noncommuting variables g(S) : Σ* → ℝ 

in such a way that many properties of S which are of interest in control theory (equivalence, and other problems such as linearity, analyticity, and time-invariance)

can be reduced to decision problems for g(S).

This connection is robust: common operations on dynamical systems (such as addition, product, and derivative)

correspond to natural operations on the generating series.

Fliess' then shows that the generating series of *bilinear systems* (an established subclass of dynamical systems)

coincide with the *rational series* (introduced by Fliess' PhD advisor Marcel-Paul Schützenberger).

This provides yet another characterisation of rational series.

(Moreover, it strongly suggests looking more closely at the relationships between classes of dynamical systems and corresponding classes of generating series.)

We conclude by observing that, since rational series have good algorithmic properties,

equivalence of bilinear systems and the other problems are decidable.

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