Two invariants for weak exponential stability of linear time-varying differential behaviors

In the paper [H. Bourlès, B. Marinescu, U. Oberst, Weak exponential stability (w.e.s.) of linear time-varying (LTV) differential behaviors, Linear Algebra Appl. 486 (2015) 523-571] we studied the problem of the title. If a finitely generated torsion module over an appropriate ring of differential operators and its associated autonomous system are regular singular the system is never w.e.s. In contrast we computed a square complex matrix for each irregular singular module and showed that the system is w.e.s. resp. not stable if all eigenvalues of the matrix have positive real parts resp. if at least one eigenvalue has negative real part. In this supplement of the quoted paper we show that the spectrum of the matrix and the decay exponent are isomorphy invariants of the module. The proofs make essential use of results exposed in [P. Maisonobe, C. Sabbah, D-module cohérents et holonomes, Hermann, Paris, 1993]. We also complement the main w.e.s. result of our quoted paper by the case where at least one eigenvalue of the matrix is purely imaginary.