TY - JOUR

T1 - Criterion for the Existence of Two-Point Oscillatory Solution of a Perturbed System with a Relay

AU - Yevstafyeva, V.V.

PY - 2023/8/1

Y1 - 2023/8/1

N2 - Abstract: We study an n -dimensional system of ordinary differential equations with a constant matrix in the linear part, a discontinuous hysteresis-type nonlinearity, and a continuous bounded perturbation function in the nonlinear part. The nonlinearity is described by a characteristic of the on-off nonideal relay. The matrix of the system has real simple nonzero eigenvalues. We study oscillatory solutions with two switching points in the phase space of the system and an arbitrary period of return to each of these points. We consider the system in the original and canonical forms. The Cauchy problem is solved with initial and boundary conditions at the switching points. For the canonical system with nonzero vector feedback, the vector of units in the case of nonlinearity, and a perturbation function of general form, we prove a criterion for the existence and uniqueness of a solution with an arbitrary return period. Moreover, in the case of a periodic perturbation function, a necessary and sufficient condition for the existence of a unique periodic solution with a given period is obtained. We present an example of the existence of a solution for a three-dimensional system.

AB - Abstract: We study an n -dimensional system of ordinary differential equations with a constant matrix in the linear part, a discontinuous hysteresis-type nonlinearity, and a continuous bounded perturbation function in the nonlinear part. The nonlinearity is described by a characteristic of the on-off nonideal relay. The matrix of the system has real simple nonzero eigenvalues. We study oscillatory solutions with two switching points in the phase space of the system and an arbitrary period of return to each of these points. We consider the system in the original and canonical forms. The Cauchy problem is solved with initial and boundary conditions at the switching points. For the canonical system with nonzero vector feedback, the vector of units in the case of nonlinearity, and a perturbation function of general form, we prove a criterion for the existence and uniqueness of a solution with an arbitrary return period. Moreover, in the case of a periodic perturbation function, a necessary and sufficient condition for the existence of a unique periodic solution with a given period is obtained. We present an example of the existence of a solution for a three-dimensional system.

KW - bounded oscillatory solution

KW - continuous bounded perturbation function

KW - discontinuous hysteresis nonlinearity

KW - periodic solution

KW - relay system with hysteresis

KW - switching hyperplanes

KW - switching points

KW - system of ordinary differential equations

UR - https://www.mendeley.com/catalogue/ea1577e1-2d5a-37ee-b007-81791267145d/

U2 - 10.1134/s0001434623070222

DO - 10.1134/s0001434623070222

M3 - Article

VL - 114

SP - 212

EP - 222

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 2

ER -