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

We consider a multidimensional controlled system with a constant matrix, a significantnonlinearity of the two-position relay type with hysteresis as a control, and a continuous periodicperturbation function. The system matrix has simple real nonzero eigenvalues, among which onecan be positive. Conditions for the system parameters, including the nonlinearity ones, areestablished under which there exists a single two-point oscillatory periodic solution with a periodcomparable to the period of the perturbation function in the case of a special type of the feedbackvector. 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.