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


A. Kh. Gelig

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


A system of differential-functional equations  dz/dt = Az+bf,  s = c'z,  f = Ns  is considered. Here  A  is a constant  m×m-matrix, its  n – 1 eigenvalues are in the left half-plane, one eigenvalue is zero,  b  and  c  are constant  m-columns,  N  – nonlinear operator, wich describes pulse modulator dynamics. These equations describe a broad class of nonlinear impulse systems [1] with various kinds of modulation (amplitude, frequency, width etc.). Here  s(t)  is modulator's input,  f(t)  is modulator's output. By using the averaging method [1], by constructing the Lyapunov function as a fourth degree form and with help of V.A.Yakubovich's frequence theorem we received new frequency conditions of global stability for solution  x = 0. Reference. 1. Gelig A.Kh., Churilov A.N. Stability and Oscillations of Nonlinear Pulse-Modulated Systems. Birkhäuser, Boston, 1998, 362 p.

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