Cluzeau, Thomas (INRIA Sophia Antipolis), Quadrat, Alban (INRIA Sophia Antipolis)

Using Morphism Computations for Factoring and Decomposing General Linear Functional Systems

Scheduled for presentation during the Mini-Symposium "Multidimensional systems: Algebraic and behavioral approaches" (ThA11), Thursday, July 27, 2006, 10:25−10:50, Room 104a

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on May 19, 2024

Abstract

Within a constructive homological algebra approach, we study the factorization and decomposition problems for general linear functional systems and, in particular, for multidimensional linear systems appearing in control theory. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M', where M (resp., M') is a module intrinsically associated with the linear functional system R y=0 (resp., R' z=0). These morphisms define applications sending solutions of the system R' z=0 to the ones of R y=0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module M is equivalent to the existence of a non-trivial factorization R=R_2 R_1 of the system matrix R. The corresponding system can then be integrated in cascade. Under certain conditions, we also show that the system R y=0 is equivalent to a system R' z=0, where R' is a block-triangular matrix. We show that the existence of projectors of the ring of endomorphisms of the module M allows us to reduce the integration of the system R y=0 to the integration of two independent systems R_1 y_1=0 and R_2 y_2=0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system R y=0, i.e., they allow us to compute an equivalent system R' z=0, where R' is a block-diagonal matrix. Many applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in a package Morphisms based on the library OreModules.