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

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

Stability of One Class of Differential-functional Equations in the Most Simple Critical Case


N. V. Kuznetsov

Saint-Petersburg State University, Department of mathematics and mechanics
Russia, 198904, Saint-Petersburg, Petrodvoretz, Bibliotechnaya pl., 2


Consider a system of functional differential equations      dz/dt=Az+bf, s=c'z, f=Ns, where A is a constant (m × m) - matrix, all the eigenvalues of which, except for that to be zero, lay in the left semiplane, b and c are constant m-dimensional columns, N is a nonlinear operator describing a dynamic pulse modulator. These equations describe a wide class of nonlinear pulse systems with different types of pulse modulation (amplitude, pulse-width, frequency, and so on). In this case s(t) is a signal at the modulator input, f(t) is a signal at the its output. By means of the analysis of Lyapunov functions in terms of the quartic forms and the averaging method [1] a new frequency criterion of stability of functional differential equations, describing the dynamic of nonlinear pulse systems in the case of one zero root of a characteristic equation with monotonic differentiable nonlinear characteristic, is obtained. Reference 1. Gelig A.Kh., Churilov A.N. Stability and Oscillations of Nonlinear Pulse-Modulated Systems. Birkhauser, Boston, 1998, 362 p.

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