Using morphism computations for factoring and decomposing general linear functional systems

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 = R2 R1 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 R1 y1 = 0 and R2 y2 = 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. Keywords—Linear functional systems, factorization and decomposition problems, morphisms, equivalences of systems, Galois symmetries, r-pure autonomous observables, controllability, quadratic first integrals of motion, quadratic conservation laws, constructive homological algebra, module theory, symbolic computation.