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Separation of motions in multirate discontinuous systems with time delay. (English. Russian original) Zbl 0932.93040
Autom. Remote Control 58, No. 7, Pt. 2, 1263-1275 (1997); translation from Autom. Telemekh. 1997, No. 7, 240-254 (1997).
The author considers a system of the form $\mu dz/dt = g(z,s,x,u), \quad ds/dt = h_1(z,s,x,u), \quad dx/dt = h_2(z,s,x,u)$ where $$z \in R^m$$, $$s \in R$$, $$x \in R^n$$, $$u(s) = - \text{sign}[s(t-1)]$$, $$g$$,$$h_1$$ and $$h_2$$ are smooth functions and $$\mu$$ is a small parameter. Sufficient conditions are given under which for sufficiently small $$\mu$$ there exists an orbitally asymptotically stable periodic solution. The algorithm of the construction of an asymptotics of a periodic solution is suggested. An example illustrating how the obtained results can be used for the separation of motions in singularly perturbed discontinuous systems with delay is given.

##### MSC:
 93C23 Control/observation systems governed by functional-differential equations 34C25 Periodic solutions to ordinary differential equations 34K13 Periodic solutions to functional-differential equations 34D15 Singular perturbations of ordinary differential equations 93D20 Asymptotic stability in control theory 93D15 Stabilization of systems by feedback