Open AccessInvertible linear differential operators with one independent variable are investigated. 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 represents a square matrix of scalar differential operators. Its product with an operator-column is an operator-column whose order does not exceed the sum of orders of initial operators. The operators-columns, the product with which leads to order fall, i.e. the order of the product is less than sum of orders of factors, are interesting for the description of invertible operators. In this paper the classification of invertible operators is based on dimensions dk,p of intersections of modules Gp and Fk for various k and p, where Gp is the module of all operators-columns of order not above p, and Fk is the module of compositions of the invertible operator with all operators-columns of order not above k. The invertible operators that have identical sets of numbers dk,p form one class.
In the paper the general properties of tables of numbers dk,p for invertible operators are investigated. A correspondence between invertible operators and elementary-geometrical models which in the paper are named by d-schemes of squares is constructed. The invertible operator is ambiguously defined by its d-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 offered.
In the proof of the main result, methods of the theory of chain complexes and their spectral sequences are used. In the paper all necessary concepts of this theory are formulated and the corresponding facts are proved.
Results of the paper can be used for solving problems in which invertible linear differential operators are arisen. Namely, it is necessary to formulate the conditions on the invertible operator which follow from the problem statement, in language of d-schemes of squares. Further the d-scheme, satisfying to these conditions, is searched and the required operator is restored on this scheme.
A discussion of possible generalizations of offered methods on the cases of partial-differential operators, delay-differential operators and difference operators concludes the paper.