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Determining observations for stability and bifurcation on a finite time interval in variational control systems with a parameter. (Russian. English summary) Zbl 1359.93339

Summary: Stability and bifurcation on a finite time interval for a thermovisco-elastoplastic contact problem are considered. To describe such a type of contact Coulomb’s law for dry friction, which is written as a variational inequality is used. The contact problem is presented as a parameter dependent variational system. Phase spaces for the system are given by scales of Hilbert spaces. Determining observation operators for bifurcation of the system and output convergence are introduced. The frequency theorem is applied in order to describe stability and the bifurcation which is understood as a loss of stability on a finite time interval. Frequency-domain conditions for the existence of determining observations and for absolute dichotomy of a variational equation are given. The connection between the frequency-domain condition and the completeness defect of the observation operator is considered.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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