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Abstract: A continuous multi-dimensional behavior is defined to be the solution space of a linear system of constant-coefficient partial differential equations. After revisiting the concepts of controllability and extendibility for systems of this type, the following main result is obtained: behaviors that are both controllable and extendable are characterized by a concatenation property that naturally generalizes the notion of strong controllability, originally introduced for two-dimensional discrete behaviors by Willems and the second author.

We characterize the discrete 2D systems with kernel representation that admit a state/driving-variable (SDV) representation. This characterization is based on the possibility of decomposing a behaviour ß as the sum of its controllable part with a suitable autonomous part (controllable-autonomous decomposition). We show that ß has a SDV representation if and only if it allows for a controllable-autonomous decomposition where the autonomous part is SDV representable. This means that ß has a kernel representation matrix which can be decomposed as the product of two 2D L-polynomial matrices such that the left factor is factor left prime and the right factor is square and properly invertible.