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On stability of linear time-varying infinite-dimensional discrete-time systems. (English) Zbl 0543.93057

Author’s abstract: ”The asymptotic behaviour of linear time-varying infinite-dimensional discrete-time systems is considered. The introduced notions are: weak power equistability, power equistability, uniform power equistability, l\({}^ p\)-equistability, uniform \(l^ p\)-equistability and \(l^{p(x)}\)-equistability. It is shown that they are identical. A generalization of the concept of spectral radius of a single operator is also proposed. It is proven that any time-varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one.”
Reviewer: R.Curtain

MSC:

93D20 Asymptotic stability in control theory
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
93C99 Model systems in control theory
93C25 Control/observation systems in abstract spaces
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[1] Bourbaki, J. N., (Théorie Spectrale (1967), Hermann: Hermann Huntington, NY), Chapters 1-2
[2] Brockett, R. W., (Finite Dimensional Linear Systems (1970), Wiley: Wiley Paris) · Zbl 0216.27401
[3] Curtain, R. F.; Pritchard, A. J., (Infinite Dimensional Linear Systems Theory (1978), Springer: Springer New York) · Zbl 0389.93001
[4] D’Angelo, H., (Linear Time-Varying Systems: Analysis and Synthesis (1970), Allyn and Bacon: Allyn and Bacon Berlin) · Zbl 0202.08502
[5] Datko, R., An extension of a theorem of A.M. Lyapunov to semi-groups of operators, J. Math. Anal. Appl., 24, 290-295 (1968) · Zbl 0186.45802
[6] Datko, R., Extending a theorem of A.M. Lyapunov to Hilbert space, J. Math. Anal. Appl., 32, 610-616 (1970) · Zbl 0211.16802
[7] Dunford, N.; Schwartz, J. T., (Linear Operators (1966), Wiley-Interscience: Wiley-Interscience Boston), Part I
[8] Elsgol’c, L. E., (Introduction to the Theory of Differential Equations with Deviating Argument (1964), Nauka: Nauka New York), (in Russian)
[9] Fuhrmann, P. A., On weak and strong reachability and controllability of infinite dimensional linear systems, J. Optim. Theory Appl., 9, 77-89 (1972) · Zbl 0215.30203
[10] Fuhrmann, P. A., On observability and stability in infinite-dimensional linear systems, J. Optim. Theory Appl., 12, 173-181 (1973) · Zbl 0247.93005
[11] Hager, W. W.; Horowitz, L. L., Convergence and stability properties of the discrete Riccati operator equation and the associated optimal control and filtering problems, SIAM J. Control Optim., 14, 295-312 (1976) · Zbl 0328.93028
[12] Hahn, W., (Theory and Application of Liapunov’s Direct Method (1963), Prentice Hall: Prentice Hall Moscow) · Zbl 0119.07403
[13] Halanay, A.; Wexler, D., (Teoria Calitaliva a Sistemelor cu Impulsuri (1968), Editura Academiei R.S.R: Editura Academiei R.S.R NJ) · Zbl 0176.05202
[14] Hale, J. K., (Theory of Functional Differential Equations (1977), Springer. New York: Springer. New York Bucareşti)
[15] Kalman, R. E.; Bertram, J. E., Control system analysis and design via the ‘second method’ of Lyapunov. 11: Discrete-time system, ASME J. Basic Engineering Ser. D, 394-400 (1960)
[16] Kamen, E. W.; Green, W. L., Asymptotic stability of linear difference equations defined over a commutative Banach algebra, J. Math. Anal. Appl., 75, 584-601 (1980) · Zbl 0471.39006
[17] Kubrusly, S., Mean square stability for discrete bounded linear systems in Hilbert space (1983), Laboralório de Computação Cientifica, Report
[18] Perron, O., Über Stabilität und asymptotisches Verhalten der Lösungen eines Systems endlicher Differenzengleichungen, J. Reine Angew. Math., 161, 41-61 (1929) · JFM 55.0869.02
[19] Pritchard, A. J.; Zabczyk, J., Stability and stabilizability of infinite dimensional systems, SIAM Review, 30, 25-52 (1981) · Zbl 0452.93029
[20] Przyłuski, K. M., Infinite dimensional discrete-time equations as models for linear systems with time delay. Preprints of the, (2nd IFAC Symp. on Distributed Parameter Systems. 2nd IFAC Symp. on Distributed Parameter Systems, Warwick (1977))
[21] Przyłuski, K. M., The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space, Appl. Math. Optim., 6, 97-112 (1980) · Zbl 0435.47017
[22] Przyłuski, M., Stability of linear infinite dimensional systems revisited, (Memorandum 1982-17 (1982), Eindhoven Univ. of Technology, Dept. of Math.: Eindhoven Univ. of Technology, Dept. of Math. Rio de Janeiro) · Zbl 0517.93046
[23] Li, Ta, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63, 99-141 (1934) · JFM 60.0397.03
[24] Zabczyk, J., Remarks on the control of discrete-time distributed parameter systems, SIAM J. Control, 12, 721-735 (1974) · Zbl 0254.93027
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