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Two invariants for weak exponential stability of linear time-varying differential behaviors. (English) Zbl 1337.93080
Summary: In the paper [H. Bourlès et al., Linear Algebra Appl. 486, 523–571 (2015; Zbl 1321.93053)] 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 (ed.) and C. Sabbah (ed.), $$D$$-modules cohérents et holonomes. Cours d’été du CIMPA ’Éléments de la théorie des systèmes différentiels’, août et septembre 1990, Nice, France. Paris: Hermann (1993; Zbl 0824.00033)]. 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.
##### MSC:
 93D20 Asymptotic stability in control theory 93C15 Control/observation systems governed by ordinary differential equations 93B25 Algebraic methods 34D20 Stability of solutions to ordinary differential equations
OreModules
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##### References:
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