On the improperness sets of families of linear differential systems |
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Authors: | E. A. Barabanov |
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Affiliation: | 1.Institute of Mathematics,National Academy of Sciences,Minsk,Belarus |
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Abstract: | We consider families of linear differential systems continuously depending on a real parameter with continuous (or piecewise continuous) coefficients on the half-line. The improperness set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are Lyapunov improper. We show that a subset of the real axis is the improperness set of some family if and only if it is a G δσ -set. The result remains valid for families in which the matrices of the systems are bounded on the half-line. Almost the same result holds for families in which the parameter occurs only as a factor multiplying the system matrix: their improperness sets are the G δσ -sets not containing zero. For families of the last kind with bounded coefficient matrix, we show that their improperness set is an arbitrary open subset of the real line. |
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