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**ABSTRACT:**Given an ideal I in A, the polynomial ring in n-indeterminates, the affine variety of I is the set of common zeros in Cn of all the polynomials that belong to I, and the Hilbert Nullstellensatz states that there is a bijective correspondence between these affine varieties and radical ideals of A. If, on the other hand, one thinks of a polynomial as a (constant coefficient) partial differential operator, then instead of its zeros in Cn, one can consider its zeros, i.e., its homogeneous solutions, in various function and distribution spaces. An advantage of this point of view is that one can then consider not only the zeros of ideals of A, but also the zeros of submodules of free modules over A (i.e., of systems of PDEs). The question then arises as to what is the analogue here of the Hilbert Nullstellensatz. The answer clearly depends on the function–distribution space in which solutions of PDEs are being located, and this paper considers the case of the classical spaces. This question is related to the more general question of embedding a partial differential system in a (two-sided) complex with minimal homology. This paper also explains how these questions are related to some questions in control theory.Advances in Applied Mathematics - ADVAN APPL MATH. 01/1999; 23(4):360-374. -
##### Conference Proceeding: Multidimensional Constant Linear Systems.

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**ABSTRACT:**A continuous resp. discrete r-dimensional (r=1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Computer Aided Systems Theory - EUROCAST'91, A Selection of Papers from the Second International Workshop on Computer Aided Systems Theory, Krems, Austria, April 15-19, 1991, Proceedings; 01/1991 · 0.99 Impact Factor - [show abstract] [hide abstract]

**ABSTRACT:**This paper develops a theory of feedback stabilization for SISO transfer functions over a general integral domain which extends the well-known coprime factorization approach to stabilization. Necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained. These conditions are then refined in the special cases of unique factorization domains (UFDs), Noetherian rings, and rings of fractions. It is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring. This interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant. The set of transfer functions is topologized so as to restrict to the graph topology of Vidyasagar [IEEE Trans. Automatic Control, ACo29 (1984), pp. 403-418], when the ring is a Bezout domain. It is shown that stability of a feedback system is robust in this topology when the ring is a UFD. This theory is then applied to the problem of stabilization of multidimensional systems. The above stabilizability criterion is interpreted geometrically in terms of affine varieties in C" when the stability region is the complement of a compact polynomially convex domain F. This criterion restricts to the well-known result for two-dimensional systems when F is the unit polydisc; it also allows the resolution of an open problem of Guiver [Multidimensional Systems Theory, D. Reidel, 1985]. Finally it is shown that while feedback stabilizability is robust, it is not, however, a generic property.SIAM Journal on Control 02/1992; 30(002):11-30.

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