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Forced oscillations in control systems under near-resonance conditions. (English. Russian original) Zbl 0932.93038
Autom. Remote Control 56, No. 11, Pt. 1, 1575-1584 (1995); translation from Avtom. Telemekh. 1995, No. 11, 87-98 (1995).
The author considers a single-loop control system whose dynamics is described by the differential equation \[ L(d/dt)x = M(d/dt)(f(x) + u(t)) \] where \(L(p)\) and \(M(p)\) are polynomials with constant coefficients, \(\deg L>\deg M \geq 0\), \(f(x)\) is a smooth function and the external action \(u(t)\) is assumed to be \(T\)-periodic. The resonance conditions under which \(L(p)\) has two primitive roots \(\lambda_{1,2} = \pm (2k \pi i)/T\) and the other roots lie left of the imaginary axis are assumed to be satisfied. The new method of multi-sheeted vector guiding functions developed by the author is applied to prove the existence of \(T\)-periodic trajectories for a given system.

93C15 Control/observation systems governed by ordinary differential equations
34C25 Periodic solutions to ordinary differential equations