TY - JOUR

T1 - On problems of Aizerman and Kalman

AU - Leonov, G.A.

AU - Kuznetsov, N.V.

AU - Bragin, V.O.

PY - 2010

Y1 - 2010

N2 - The problems of the stability of nonlinear control systems posed by Aizerman and Kalman have stimulated the development of methods for detecting hidden periodic oscillations in multidimensional dynamical systems. In the 1950s, Pliss developed an analytical method for detecting periodic oscillations in third-order systems satisfying the generalized Routh-Hurwitz conditions. It has turned out that this generalized method of Pliss can be regarded as a special version of the describing function method in the critical case. Being combined with computational procedures based on applied bifurcation theory, this method makes it possible to obtain new classes of systems for which the conjectures of Aizerman and Kalman are false. The known approaches to constructing counterexamples to Aizerman's and Kalman's conjectures proposed by Fitts, Barabanov, and Llibre are reviewed. A new effective analytical-numerical method for constructing such counterexamples is presented. The method is based on combining the classical theo

AB - The problems of the stability of nonlinear control systems posed by Aizerman and Kalman have stimulated the development of methods for detecting hidden periodic oscillations in multidimensional dynamical systems. In the 1950s, Pliss developed an analytical method for detecting periodic oscillations in third-order systems satisfying the generalized Routh-Hurwitz conditions. It has turned out that this generalized method of Pliss can be regarded as a special version of the describing function method in the critical case. Being combined with computational procedures based on applied bifurcation theory, this method makes it possible to obtain new classes of systems for which the conjectures of Aizerman and Kalman are false. The known approaches to constructing counterexamples to Aizerman's and Kalman's conjectures proposed by Fitts, Barabanov, and Llibre are reviewed. A new effective analytical-numerical method for constructing such counterexamples is presented. The method is based on combining the classical theo

KW - absolute stability

KW - Aizerman conjecture

KW - hidden oscillations

KW - Kalman conjecture

KW - periodic solutions

U2 - 10.3103/S1063454110030052

DO - 10.3103/S1063454110030052

M3 - статья

VL - 43

SP - 148

EP - 162

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -