Periodic solutions of one class of functional differential equations

Abstract

We consider a nonlinear pulse system

$\dot x = Ax + bf,\sigma = c*x + \psi ,$

where A ∈ ℝ^{m × m} is a constant Hurwitz matrix, b ∈ ℝ^{m × 1} and c × ℝ^{m × 1}, ψ is a nonzero constant, σ and f are input and output signals of the modulator, which generates momentary pulses described by the delta functions f(t)= Σ ^∞ _{n = 0} λ _n δ(t − t _n ), where

$\lambda _n = \left\{ \begin{gathered} \operatorname{sgn} \sigma (t_n - 0),if\sigma (t_n - 0) \ne 0; \hfill \\ 0,if\sigma (t_n - 0) = 0; \hfill \\ \end{gathered} \right. $

t _{n + 1} = t _n + τ, n = 0, 1, …; τ is the smallest positive root of the equation

$\left| {\int\limits_0^\tau {\sigma (t_n + \lambda )e^{ - \varepsilon (\tau - \lambda )} d\lambda } } \right| = \Delta ,$

where Δ and ɛ are positive constants.

In the phase space we construct a domain in which the operator of displacement along the trajectories is continuous and use it to derive sufficient conditions for the existence of a periodic solution with one pulse per period.