Abstract
We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying the matrix of the system. The asymptotic stability set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are asymptotically stable. We prove that a set on the real axis is the asymptotic stability set of such a family if and only if it is an F σδ -set lying entirely on an open ray with origin at zero. In addition, for any set of this kind, the coefficient matrix of a family whose asymptotic stability set coincides with this set can be chosen to be infinitely differentiable and uniformly bounded on the time half-line.
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Barabanov, E.A., Structure of Stability Sets and Asymptotic Stability Sets of Families of Linear Differential Systems with Parameter Multiplying the Derivative. I, Differ. Uravn., 2010, vol. 46, no. 5, pp. 611–625.
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Original Russian Text © E.A. Barabanov, 2010, published in Differentsial’nye Uravneniya, 2010, Vol. 46, No. 6, pp. 791–800.
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Barabanov, E.A. Structure of stability and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: II. Diff Equat 46, 798–807 (2010). https://doi.org/10.1134/S0012266110060042
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DOI: https://doi.org/10.1134/S0012266110060042