Factoring and decomposing a class of linear functional systems
نویسندگان
چکیده
Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, over-determined, under-determined) systems. 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 ′, whereM (resp.,M ′) is a module intrinsically associated with the linear functional system Ry = 0 (resp., R′ z = 0). These morphisms define applications sending solutions of the system R′ z = 0 to solutions of Ry = 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 = R2R1 of the system matrix R. The corresponding system can then be integrated “in cascade”. Under certain conditions, we also show that the system Ry = 0 is equivalent to a system R′ z = 0, where R′ is a block-triangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system Ry = 0 to the integration of two independent systems R1 y1 = 0 and R2 y2 = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system Ry = 0, i.e., they allow us to compute an equivalent system R′ z = 0, where R′ is a block-diagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in a Maple package Morphisms based on the library OreModules.
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تاریخ انتشار 2007