Regular implementation in the space of compactly supported functions

This article extends results on regular implementability in [P. Rocha, Canonical controllers and regular implementation of nd behaviors, in: Proceedings of the 16th IFAC World Congress, 2005] and [H.L. Trentelman, D. Napp Avelli, On the regular implementability of nD systems, Systems Control Lett. 56 (4) (2007) 265-271] to the case when the signal space is not an injective cogenerator, for instance, the space D of compactly supported smooth functions on Rn. In this case the bijective correspondence between behaviors and modules fails to hold; also projections and sums of behaviors need not in general be behaviors. A more general version of implementability is introduced and necessary and sufficient conditions are established for the implementation and regular implementation of a given desired behavior.