An algebraic approach to multidimensional behaviors

The main aim of this thesis is the study of systems described by linear constant coefficient partial differential equations (PDEs) using commutative algebra. In order to do so we use an elegant mathematical approach called "the behavioral" framework. In this approach a system is determined by its behavior which is the set of trajectories that satisfy the laws ( i.e. equations) of the system. Fundamental questions of systems and control theory of PDEs are posed and solved in this context, e.g. feedback control, regular interconnection, partial interconnection, decomposition into controllable and autonomous part, spectral factorization, etc. In order to tackle these problems we will develop and use a powerful natural duality between these systems and finitely generated modules over the polynomial ring of differential operators. Hence, we can to translate many analytic properties of the system into algebraic properties of the corresponding module. This is very useful since it allows the use of the potent machine of computational algebra to provide constructive solutions.