Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference equations with constant coefficients

Abstract

In this paper we investigate the solvability of inhomogeneous linear systems of partial differential-difference equations with constant coefficients and also the corresponding duality problem in how far the solutions of the corresponding homogeneous systems determine the equations. For ordinary delay-differential (DD) equations these behavioral problems were investigated in a seminal paper by Glüsing-Lürssen (SIAM J Control Optim 35:480–499, 1997) and in later papers by Habets, Glüsing-Lürssen, Vettori and Zampieri. In these papers the delay-differential operators are considered as distributions with compact support which act on smooth functions or on arbitrary distributions via convolution. The entire analytic Laplace transforms of the distributions with compact support play an important part in the quoted papers. In our approach the partial differential-difference operators belong to various topological operator rings $$A$$ of holomorphic functions on subsets of $$\mathbb C ^n$$ and are thus studied in the frequency domain, the arguments of these (operator) functions being interpreted as generalized frequencies. We show that the topological duals $$A^{\prime }$$ of these operator rings with the canonical action of $$A$$ on $$A^{\prime }$$ have strong elimination and duality properties for $$_A A^{\prime }$$ -behaviors and admit concrete representations as spaces of analytic functions of systems theoretic interest. In particular, we study systems with generalized frequencies in the vicinity of suitable compact sets. An application to elimination for systems of periodic signals is given. We also solve an open problem of the quoted authors for DD-equations with incommensurate delays and analytic signals. Module theoretic methods in context with DD-equations have also been used by other authors, for instance by Fliess, Mounier, Rocha and Willems.