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

Partial differential-difference equations are the multidimensional generalization of ordinary delay-differential equations. We investigate behaviors of analytic signals governed by equations of this type, i.e., solution modules of linear systems with constant coefficients of such equations, and especially the problems of elimination and duality. The first concerns the question whether the images of behaviors are again behaviors and in particular the existence of solutions of inhomogeneous linear systems which satisfy the obvious necessary compatibility conditions. Duality refers to the determination of the module of equations by the behavior. Our theory is presently restricted to analytic signals because the proofs make substantial use of the Stein algebra of multivariate entire functions and of Stein modules over it, but the extension to smooth or distributional signals is of course an important task for the future. We especially prove the validity of elimination for delay-differential equations with incommensurate delays and thus solve, for analytic signals, an open problem stated by Gluesing-Luerssen, Vettori and Zampieri. Duality is expressed and derived by means of the polar theorem for locally convex spaces in duality. GluesingLuerssen’s rather complete and seminal behavioral theory of delay-differential equations with commensurate delays relies on the fact that the appropriate ring of operators is a Bezout domain and especially coherent. Coherence of the relevant rings of operators in the more general situations is important, but has not yet been proven. Further contributors to the module theoretic or behavioral approach to delay-differential equations are Fliess, Habets, Mounier, Rocha, Willems et al.. AMS-classification: 93B25, 93C05, 93C23, 93C35