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Popov theories and qualitative behavior of dynamic and control systems. (English) Zbl 1293.93597

Summary: This paper is an attempt to summarize some problems that may be solved within the framework of the Popov and Popov-like theories. Besides classical absolute stability problem, problems as forced oscillations, systems with multiple equilibria, dissipativity (ultimate boundedness) are considered. The Popov theory and results are viewed as emerging in three stages: the first one is concerned with the discovery of the frequency domain inequality connected with integral equations, the second one corresponds to the generalization of Yakubovich Kalman lemma to the multivariable case as well as to hyper stability theory formulation, and the third stage incorporates results on various qualitative behavior for systems with multiple equilibria (mutability-dichotomy and gradient like behavior).

MSC:

93D10 Popov-type stability of feedback systems
93C35 Multivariable systems, multidimensional control systems
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