ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Factorization of the Characteristic Polynomial of the Equilibrium State for an Аutonomous System Having an Attracting Invariant Manifold


Alexander V. Bratishchev

Don State Technical University, Professor of Applied Mathematics Department ,
Rostov-on-Don, Gagarin sq.,1
Professor, doctor fiz.-mat. nauk


Let an autonomous n-th order system have m variable parameters. In this paper, the Erugin method is used to select such parameter realizations that the obtained system has a predetermined (n-m)-dimensional invariant manifold which is the Kolesnikov stable. It is proved that the characteristic polynomial corresponding to the equilibrium state of this system can be represented as the product of explicitly computed polynomials of powers m and n-m. The similar results may be obtained when the autonomous system without parameters already has the Kolesnikov stable invariant manifold. The result obtained is used in the problem of inverted pendulum, where the nonlinear control stabilizing the pendulum in the upper position has been synthesized by the method of analytical design of aggregated regulators.



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