Symmetries, parametrizations and potentials of multidimensional linear systems

Within the algebraic analysis approach to linear systems theory, the purpose of this paper is to study how left Dhomomorphisms between two finitely presented left D-modules associated with two linear systems induce natural transformations on the autonomous elements of the two systems and on the potentials of the parametrizations of the parametrizable subsystems. Extension of these results are also considered for linear systems inducing a chain of successive parametrizations. I. HOMOMORPHISMS OF LINEAR SYSTEMS Let D be a ring of functional operators (e.g., ordinary or partial differential operators, time-delay operators, shift operators) and R ∈ Dq×p (resp., R′ ∈ Dq×p ) a q × p (resp., q′×p′) matrix. We consider the left D-module finitely presented by R (resp., R′), namely, M = D1×p/(D1×q R) (resp., M ′ = D1×p ′ /(D1×q ′ R′)). A left D-homomorphism f (or simply morphism) from M to M ′ is a left D-linear map f : M −→ M ′. The abelian group of all morphisms from M to M ′ is denoted homD(M,M ′). If M = M ′, f ∈ homD(M,M ′) is called a left D-endomorphism of M . We denote by endD(M) = homD(M,M) the ring of all endomorphisms of M also called the endomorphism ring. Lemma 1.1 ([3], Corollary 2.1): With the previous notations, let us consider the finite presentations of M and M ′ D1×q .R −→ D1×p π −→M −→ 0, D1×q ′ .R′ −→ D1×p π ′ −→M ′ −→ 0, (1) where (.R)(λ) = λR for all λ ∈ D1×q and similarly for R′, namely, (1) are exact sequences, i.e., π (resp., π′) is surjective and kerπ = D1×q R (resp., kerπ′ = D1×q ′ R′). 1) The existence of a left D-morphism f : M −→M ′ is equivalent to the existence of two matrices P ∈ Dp×p ′ , Q ∈ Dq×q ′ satisfying the commutation relation: RP = QR′. Then, we have the commutative exact diagram D1×q .R −→ D1×p π −→ M −→ 0 ↓ .Q ↓ .P ↓ f D1×q ′ .R′ −→ D1×p π ′ −→ M ′ −→ 0, Dedicated to Professor Ulrich Oberst on the occasion of his 70th birthday. where f(π(λ)) = π′(λP ) for all λ ∈ D1×p. 2) If we denote by R′ 2 ∈ D ′ 2×q ′ a matrix satisfying kerD(.R) , {λ ∈ D1×q ′ | λR = 0} = D1×q ′ 2 R′ 2, then P and Q are defined up to homotopy, i.e., { P = P + Z1R, Q = Q+RZ1 + Z2R 2, where Z1 ∈ Dp×q ′ and Z2 ∈ Dq×q ′ 2 are two arbitrary matrices, satisfy the same relation RP = QR′, and f(π(λ)) = π′(λP ) for all λ ∈ D1×p. See [3] for algorithms which compute the matrices P and Q when D is a commutative polynomial ring over a computable field or a noncommutative polynomial rings for which Buchberger’s algorithm terminates for any admissible term order. These algorithms are implemented in the package OREMORPHISMS ([4]) for classes of Ore algebras ([1]). In the particular case where R′ = R, from Lemma 1.1, we obtain that the existence of a left D-endomorphism f of M is equivalent to the existence of two matrices P ∈ Dp×p and Q ∈ Dq×q satisfying the following commutation relation: