Using morphism computations for factoring and decomposing linear functional systems
Within a constructive homological algebra approach, we study the factorization and decomposition problems for general linear functional systems (determined, under-determined, over-determined). 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 of the same size as $R$. 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 of the same size as $R$. Many 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.