Asymptotic stability of nonlinear pulse systems. (Q556739)

The paper considers a nonlinear pulse system described by the functional differential equation \[ \dot x=g(x)+b(x)\xi,\quad\sigma =c^{*}(x)x,\quad\xi =\mathcal M\sigma , \tag{1} \] where \(g(x)\), \(b(x)\), \(c(x)\) are continuous \(m\)-dimensional vector functions, \(\xi (t)\) is a signal at the output, \(\sigma (t)\) is a signal at the input. In (1) \(\mathcal M\) is a nonlinear operator describing the work of a pulse component so that for each continuous function \(\sigma (t)\) on \([0,+\infty )\) both the function \(\xi (t)\) and the sequence \(\{t_n\}\) (\(n=0,1,2,\dots\); \(t_0=0\)) satisfy certain properties. Along with system (1) the ``equivalent'' continuous nonlinear system \[ \dot x=g(x)+b(x)\varphi (\sigma), \quad\sigma =c^{*}(x)x, \tag{2} \] which is obtained from (1) by replacing a pulse element by its static characteristic, is considered. It is shown that for sufficiently large values of the pulse frequence, asymptotic stability of the equilibrium state of the pulse system (1) follows from the stability by the first approximation of the ``equivalent'' system (2).