TY - JOUR
AU - Aisagaliev, S. А.
AU - Aisagalieva, S. S.
PY - 2018/08/27
Y2 - 2024/10/13
TI - Improper integrals for stability theory of multidimensional phase systems
JF - Journal of Mathematics, Mechanics and Computer Science
JA - JMMCS
VL - 97
IS - 1
SE - Mathematics
DO - 10.26577/jmmcs-2018-1-483
UR - https://bm.kaznu.kz/index.php/kaznu/article/view/483
SP - 38-53
AB - <p>A class of ordinary differential equations describing the dynamics of multidimensional phase systems<br>with a countable equilibrium position with periodic nonlinear functions from a given set is<br>considered. Such uncertainty of right side of the differential equation gives rise to the nonuniqueness<br>of the solution, which leads to the study of the properties of solutions of equations with<br>differential inclusions. A completely new approach to the study of the properties of the solution of<br>dynamical systems with a countable equilibrium position with incomplete information on nonlinearities<br>is proposed. By a nonsingular transformation, the original system is reduced to a special<br>kind, consisting of two parts. The first part of the differential equations is solvable with respect to<br>the components of the periodic function and the second part does not contain nonlinear functions.<br>The properties of solutions are studied, estimates for the solutions of the original system and the<br>transformed system are obtained and their boundedness is proved. Identities with respect to the<br>components of the nonlinear function are obtained and their relation to the phase variables is<br>established. The properties of quadratic forms with respect to phase variables and derivatives are<br>studied. The estimates of improper integrals along the solution of the system are obtained for two<br>cases: when the values of the integrals of the components of the nonlinear function in the period<br>are zero; When the values of the integrals in the period are different from zero. These results can<br>be used to obtain conditions for global asymptotic stability of multidimensional phase systems.</p>
ER -