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Factoring and decomposing a class of linear functional systems. (English) Zbl 1131.15011

The authors study, within a constructive homological algebra approach, the factorization and decomposition problems for a class of linear functional systems. Using the concept of Ore algebras of functional operators, the first part of the paper aims at computing effectively morphisms from a left \(D\)-module \(M\), finitely presented by a matrix \(R\) with entries in a certain Ore algebra \(D\), to a left \(D\)-module \(M'\) presented by a matrix \(R'\). These morphisms define applications sending solutions of the system \(R'z=0\) to solutions of \(Ry=0\).
The authors explicitly characterize the kernel, coimage, image and cokernel of a morphism from \(M\) to \(M'\) and deduce an heuristic method to check the equivalence of the corresponding systems \(Ry=0\) and \(R'z=0\). They prove that the existence of a non-injective endomorphism of the left \(D\)-module \(M\), corresponds to a factorization of the form \(R=R_1R_2\), where \(R_1\) and \(R_2\) are two matrices with entries in \(D\). As a consequence, the integration of the system \(Ry=0\) is reduced to a cascade of integrations. Under certain conditions, the authors 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 \(R\).
In the fourth section of the paper, the authors show how to effectively compute some idempotents of the endomorphism ring of the module \(M\) and prove that they allow to decompose the system \(Ry=0\) into two decoupled systems \(S_1y_1=0\) and \(S_2y_2=0\), where \(S_1\) and \(S_2\) are two matrices with entries in \(D\). Furthermore, the authors prove that, under certain conditions on the idempotents, the system \(Ry=0\) is equivalent to a block diagonal system \(R'z=0\).
All along the paper, the authors illustrate their results by considering some applications coming from mathematical physics (e.g. computation of quadratic first integrals of motion and quadratic conservation laws, testing the equivalence of linear systems of partial differential equations (PDEs) appearing in mathematical physics, factoring, decomposing and computing Galois transformations of the classical linear systems of PDEs appearing in elasticity theory, electromagnetism, hydrodynamics) and in control theory (factorization, decomposition and computation of Galois transformations of classical linear functional systems, parametrizations, decoupling the autonomous and the controllable subsystems).

MSC:

15A23 Factorization of matrices
15A06 Linear equations (linear algebraic aspects)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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