Stable periodic modes in control systems with monotone nonlinearities (Q923026)

We consider systems described by equations of the form \[ (1)\quad dz/dt=Az+bf(t,x),\quad x=(c,z), \] where \(f(t,x)\) is a continuous function T-periodic in t. Here, \(z,b,c\) are vectors in \({\mathbb{R}}^ N\), and A is a \(N\times N\) Hurwitz matrix. The existence of stable T-periodic modes in systems (1) is a classical problem of control theory. We derive existence conditions of stable periodic modes in cases when Eq. (1) is known to have several periodic modes. The results are proved using a new approach which relies, on the one hand, on the methods of \textit{M. A. Krasnosel'skij} and the first author [Sov. Math., Dokl. 17, 128-132 (1976); translation from Dokl. Akad. Nauk SSSR 226, 506-509 (1976; Zbl 0341.45013)] and the first author [Autom. Remote Control 47, 451-458 (1986); translation from Avtom. Telemekh. 1986, No.4, 16-23 (1986; Zbl 0604.93050)] and, on the other, on a new class of Lyapunov functions.