The necessary and sufficient conditions for the stability of linear systems with an arbitrary delay |
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Authors: | AA Zevin |
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Institution: | 1. School of Basic Sciences, Changchun University of Technology, Changchun, 130012, PR China;2. College of Mathematics, Jilin University, Changchun, 130012, PR China;1. Dpto. de Aeronáutica, Facultad de Ciencias Exactas, Físicas y Naturales, Universidad Nacional de Córdoba, Avenida Vélez Sarsfield 1611–5000 Córdoba, Argentina;2. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina;3. Dpto. Física Aplicada, ETSI Aeronáuticos, Universidad Politécnica de Madrid, 28040 Madrid, Spain |
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Abstract: | A system of linear differential equations with a Hurwitz matrix A and a variable delay is considered. The system is assumed to be stable if it is stable for any delay function τ(t) ≤ h. The necessary and sufficient condition for stability, expressed using the eigenvalues of the matrix A and the quantity h, is found. It is established that the function τ(t), corresponding to the critical value of h, is constant or piecewise-linear depending on to which eigenvalue of matrix A (complex or real respectively) it corresponds. In the first case, the critical values of h in systems with a variable and constant delay are identical and, in the second case, they differ very slightly. |
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