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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
Software:
OreModules
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References:
[1] Berger, T.; Ilchmann, A.; Wirth, F. R., Zero dynamics and stabilization for analytic linear systems, Acta Appl. Math., 138, 17-57, (2015) · Zbl 1316.93093
[2] Bourlès, H.; Marinescu, B., Linear time-varying systems, (2011), Springer Berlin · Zbl 1159.93015
[3] Bourlès, H.; Marinescu, B.; Oberst, U., Weak exponential stability of linear time-varying differential behaviors, Linear Algebra Appl., 486, 523-571, (2015) · Zbl 1321.93053
[4] Chyzak, F.; Quadrat, A.; Robertz, D., Oremodules: a symbolic package for the study of multidimensional linear systems, (Chiasson, J.; Loiseau, J.-J., Applications of Time-Delay Systems, Lecture Notes in Control and Information Sciences, vol. 352, (2007), Springer), 233-264 · Zbl 1248.93006
[5] Hill, A. T.; Ilchman, A., Exponential stability of time-varying linear systems, IMA J. Numer. Anal., 31, 865-885, (2011) · Zbl 1226.65065
[6] Hinrichsen, D.; Pritchard, A. J., Mathematical systems theory I, (2005), Springer Berlin
[7] Komatsu, H., An introduction to the theory of hyperfunctions, (Hyperfunctions and Pseudo-Differential Equations, Lecture Notes in Mathematics, vol. 287, (1973), Springer Berlin), 3-40
[8] Maisonobe, P.; Sabbah, C., \(\mathcal{D}\)-modules cohérents et holonomes, (1993), Hermann Paris
[9] Oberst, U., Stabilizing compensators for linear time-varying differential systems, Internat. J. Control, (2015)
[10] van der Put, M.; Singer, M. F., Galois theory of linear differential equations, (2003), Springer Berlin · Zbl 1036.12008
[11] Ramis, J.-P., Théorèmes d’indices Gevrey pour LES équations différentielles ordinaires, Mem. Amer. Math. Soc., vol. 296, (1984) · Zbl 0555.47020
[12] Robertz, D., Recent progress in an algebraic analysis approach to linear systems, Multidimens. Syst. Signal Process., 26, 349-388, (2015) · Zbl 1348.93078
[13] Rugh, W. J., Linear system theory, (1996), Prentice Hall Upper Saddle River, NJ · Zbl 0892.93002
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