A constructive solution to interconnection and decomposition problems with multidimensional behaviors

By a multidimensional behavior, we mean the solution space of a linear constant-coefficient system of partial difference or differential equations. Within the behavioral framework, a natural concept of interconnection has been introduced by J. C. Willems. The regular interconnection problem can be formulated as follows: Given a behavior (the plant), find---if possible---another behavior (a controller) such that their interconnection is not redundant and is equal to a certain desired behavior. We give a constructive solution to this problem, which is in terms of polynomial matrix equations that can be solved using the theory of Grobner bases. We also study a dual problem concerning the existence of direct sum decompositions of multidimensional behaviors. Finally, we present a unified framework for studying both problems as special cases of constructing relative complements in the lattice of behaviors.