On the decomposition of differential behaviors into the direct sum of irreducible components

In a recent pair of contributions [2, 3], starting from a set of results about the relationship between the direct sum decomposition of (linear, time-invariant, differential) behaviors and the solvability of certain two-sided Bézout equations, it was shown that every autonomous behavior can be expressed as a direct sum of irreducible components. In this contribution, we further explore this issue, by naturally introducing an irreducibility notion for controllable behaviors and by proving that every controllable behavior can be naturally expressed as the direct sum of a finite number of irreducible controllable behaviors. As a result, we can extend the whole direct sum (irreducible) decomposition analysis to the general case of (linear, time-invariant, differential) behaviors. Key-Words:Autonomous/controllable behaviors, (internally) skew-prime matrices, direct sum decomposition, irreducible autonomous behaviors, irreducible controllable behaviors.