Classification and Construction of Generalized Invertible Linear Differential Operators with One Independent Variable

• Main Author: V. N. Chetverikov
• Format: Article
• Language: Russian
• Published: MGTU im. N.È. Baumana 2015-01-01
• Series: Matematika i Matematičeskoe Modelirovanie
• Subjects: invertible linear differential operators; transformations of control systems; spectral sequences of chain complexes
• Online Access: http://mathm.elpub.ru/jour/article/view/23

<p>The paper investigates invertible linear differential operators. Description of such operators is an important problem, because it is related to transformations of control systems. Namely, a C-transformation is an invertible transformation when the variables of one system are expressed in terms of variables and derivatives of dependent variables with respect to independent variables of second system. The lack of a useful description of C-transformations does not allow developing the theory of their application. C-Transformations of linear systems are represented as invertible linear differential operators. In the case of nonlinear systems, linearizations of C-transformations are interpreted as invertible linear differential operators. Therefore, the study of invertible linear differential operators should be considered as the first step to the description of C-transformations of both linear and nonlinear systems.<br />This paper is the second work devoted to the description of invertible linear differential operators with one independent variable and their generalizations. In the first work, a table of integers was associated to each invertible linear differential operator. These tables were described in terms of elementary geometry. Thus some elementary-geometrical model was assigned an invertible operator. This model was called a d-scheme. Invertible linear differential operators are classified by d-schemes.<br />An invertible operator is not uniquely determined by its d-scheme. It was shown in previous work how to construct some invertible differential operators for a given d-scheme and what mathematical structures still should be given for this construction. However, a description of all invertible operators with a given d-scheme was not there obtained.<br />In this work, a complete description of all invertible linear differential operators with a given d-scheme is obtained. In addition, this result and the results of the first article are generalized to invertible mappings of filtered modules generated by one differentiation. In particular, the linearizations of C-transformations and mappings determined by unimodular matrices are such generalized invertible operators.<br />The results of this paper can be used to describe C-transformations of control systems and to classify such systems.</p>