Seminar talk, 25 March 2015

From Geometry of Differential Equations
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Speaker: Vladimir Chetverikov

Title: Analysis and synthesis of invertible differential operators in one independent variable

Abstract:
The talk discusses invertible linear differential operators with one independent variable. The problem of description of such operators is important, because it is connected with transformations and the classification of control systems, in particular, with the flatness problem.

Each invertible linear differential operator defines a series of spectral sequences of chain complexes. Studying dimensions of modules of these spectral sequences leads to a correspondence between invertible operators and elementary-geometrical models called [math]\displaystyle{ d }[/math]-schemes of squares. The invertible operator is ambiguously defined by its [math]\displaystyle{ d }[/math]-scheme of squares. The mathematical structure that must be set for its unique definition and an algorithm for the construction of the invertible operator are suggested.

As an example of application of the obtained description of invertible operators there will be computed flatness conditions for systems with two-dimensional control.

In conclusion, we will discuss possible generalizations of offered methods on the cases of partial-differential operators, delay-differential operators and difference operators.

References:
V.N.Chetverikov, Classification and construction of invertible linear differential operators on a one-dimensional manifold, Science and education, BMSTU, 7 (2014), 105-127 (in Russian), local copy