Summary
Two criteria, one postulating the existence of a not necessarily positive definite Lyapunov function and the other postulating conditional input-output stability, are given, ensuring that a system of linear differential equations has an exponential dichotomy.
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References
B. D. O. Anderson,External and internal stability of linear systems — a new connection, IEEE Trans. Automatic Control,17 (1972), pp. 107–111.
B. D. O. Anderson -J. B. Moore,New results in linear system stability, SIAM J. Control,7 (1969), pp. 398–414.
R. W.Brockett,Finite dimensional linear systems, New York, 1970.
W. A.Coppel,Dichotomies in stability theory, Berlin, 1978.
P.Hartman,Ordinary differential equations, New York, 1964.
N. N.Krasovskii,Stability of motion (translated from the Russian by J. L. Brenner), Stanford, 1963.
J. L. Massera -J. J. Schäffer,Linear differential equations and functional analysis, I., Ann. of Math.,67 (1958), pp. 517–573.
J. L.Massera - J. J.Schäffer,Linear differential equations and function spaces, New York, 1966.
M. Megan,On exponential stability of linear control systems in Hilbert spaces, Anal. Univ. Timişoara,14 (1976), pp. 125–130.
M. Megan,On the input-output stability of linear controllable systems, Canad. Math. Bull.,21 (1978), pp. 187–195.
K. J. Palmer,The structurally stable linear systems on the half-line are those with exponential dichotomies, J. Differ. Equs.,33 (1973), pp. 16–25.
O. Perron,Die Stabilitätsfrage bei Differentialgleichungen, Math. Z.,32 (1930), pp. 703–728.
L. M. Silverman -B. D. O. Anderson,Controllability, observability and stability of linear systems, SIAM J. Control,6 (1968), pp. 121–130.
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Thanks are due to Mr. W. A. Coppel for suggesting the problems and helpful advice.
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Palmer, K.J. Two linear systems criteria for exponential dichotomy. Annali di Matematica pura ed applicata 124, 199–216 (1980). https://doi.org/10.1007/BF01795393
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DOI: https://doi.org/10.1007/BF01795393