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 Grrbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author). given uniquely by the equations P(L)(K.(v))=v, P(L)(Iq(u))=Q(L)(u), K(v)IG=H(u)IG=O. I call F G the canonical state space of the system S=((uy);Py=Qu}. With the same technique I prove in theorem 5.71 that for arbitrary large injective cogenerators A over F[s]=F[sl,---,Sr] the IO-system SeA m+p with given IO-structure (u,y) can be uniquely characterized by a pair (Prg,H) of matrices where Prg~F[s]l'P,l:=l(D(1))l+---+l(D(p))[, is the reduced Grt~bner matrix of S and H~ F(s)P'm its transfer matrix satisfying prgH~F[s]l'n.In other words, (prg,H) is a complete system of invariants 8 ULRICH OBERST for ScA m÷p with given 1P-structure (u,y). The questions of §6, but not the results were again inspired by [GRE]. The starting point is Willems' observation [WILl that in the one-dimensional discrete case any system S¢(FN)rn÷P=(F{t})m+P admits a IO-structure with proper transfer matrix H(s). This means that H(s) is a power series in t:=s -1, i.e. H(s)~(F(s)f]F{t}) p'm. Moreover it is then well-known that the transfer operator H from (7) is given by convolution with H, i.e.~I(u)=H*u where u=(Y. nUl(n)tn;j=l,'-,m)~F{t}...

No abstract

No abstract

We give an elementary and constructive, purely algebraic proof for the existence of noetherian differential operators for primary submodules of finite-dimensional free modules over polynomial algebras. By means of these operators the submodule can be described by differential conditions on the associated characteristic variety. This important result and the terminology are due to V. P. Palamodov. However, his, L. Ehrenpreis' and later J.-E. Bjork's proofs of the existence theorem use complicated algebraic and analytic techniques and are not constructive as far as we see. The idea to characterize primary ideals by their associated differential operators is due to W. Grobner. But M. Noether's Fundamentalsatz is based on similar ideas and is obviously the origin of Palamodov's terminology. ᮊ 2000 Academic Press