Two-dimensional behavior decompositions with finite-dimensional intersection: a complete characterization

In this paper, the following problem is addressed: given a two-dimensional complete behavior B and one of its sub-behaviors B_B, under what conditions a third complete behavior B_A can be found, such that B = B_A + B_B and the intersection of B_A and B_B is finite-dimensional autonomous? This constitutes a complete generalization of the decomposition theorem, as it represents a decomposition with “minimal intersection”, in which one of the two terms has been a priori fixed. The analysis carried on here completes the preliminary results reported in [1] and completely generalizes the direct sum decomposition problem presented in [2].