# zbMATH — the first resource for mathematics

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.

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