The Riccati PDE´s Associated with Invariant Distributions and Minimal Factorization of Systems

The paper does three things. First invariant distributions play an important role in a variety of problems in systems theory (cf. [I]). We show that such invariant distributions correspond to a Riccati type partial differential equations. These concrete PDE\´s appear to be a useful way for analysis and computation in specific problems. Secondly, we use this to prove that invariant distributions are rare and to classify them in some cases. Thirdly we study "minimal" factorizations of systems. We show that minimal factorizations are nongeneric, indicate a consequence for control, and compute them explicitly for a class of special problems, thereby obtaining a generalization and strengthening of a theorem of Boyd and Chua.