Left-coprimeness condition for the reachability in finite time of pseudo-rational systems of order zero with an application to difference delay systems

The paper proves that a pseudo-rational system Σ_Q is X_{Q,1}-reachable in finite time T for every T > −l(det Q) if and only if it is left-coprime (∃R,S∈M(R−) with Q∗R+P∗S=δ0 Id). The “if” part is established constructively and the time bound then follows from Theorem 4.1; the “only if” part uses a functional-analytic argument (approximation plus open mapping theorem) to build S and then R, yielding the Bezout identity Q∗R+P∗S=δ0 Id . By contrast, the model’s (i)⇒(ii) step incorrectly claims surjectivity of the convolution map u↦P∗u on M(R−)^m→M(R−)^d from reachability and then invokes a splitting argument to conclude P∗S=δ0 Id, which is far stronger than needed and not justified by the reachability hypothesis; moreover it conflates equality after truncation π with equality of distributions. The model’s time-bound attempt l(S)+l(Q)≥l(det Q) relies on an invalid claim that the l of a sum equals the minimum l of the summands, ignoring cancellations in distributions. The paper’s route avoids these pitfalls and is correct, relying on Theorem 4.1 for the minimal-time bound .