Designing a Control for a Multidimensional System of OrdinaryDifferential Equations with Relay Hysteresis and Perturbation

We consider a multidimensional controlled system with a constant matrix, a significant nonlinearity of the two-position relay type with hysteresis as a control, and a continuous periodic perturbation function. The system matrix has simple real nonzero eigenvalues, among which one can be positive. Conditions for the system parameters, including the nonlinearity ones, are established under which there exists a single two-point oscillatory periodic solution with a period comparable to the period of the perturbation function in the case of a special type of the feedback vector. The asymptotic stability of the solution has been proven using the phase plane method. The results obtained are illustrated by examples for three-dimensional systems.