@article{Aisagaliev_Ayazbayeva_2019, title={Absolute stability of multidimensional regulated systems. Aizerman problem}, volume={101}, url={https://bm.kaznu.kz/index.php/kaznu/article/view/619}, DOI={10.26577/JMMCS-2019-1-619}, abstractNote={<p>We consider one class of ordinary differential equations describing the dynamics of multidimensional&nbsp;controlled systems with a single equilibrium state with nonlinearities from a given set. Such&nbsp;uncertainty of the nonlinear function generates a non-uniqueness of the solution, which leads&nbsp;to the study of the properties of solutions of equations with differential inclusions. A new method&nbsp;for studying the absolute stability of the equilibrium state of controlled systems with many<br>continuous nonlinearities with incomplete information about them is proposed. By non-singular&nbsp;transformation, the original system is reduced to a special form, which allows using information&nbsp;about the properties of nonlinearities. We study properties of the solutions, obtain estimates for&nbsp;the solutions of the original system and the transformed system, and prove their boundedness.&nbsp;The identities with respect to the components of the nonlinear function are obtained and their<br>connection with the phase variables is established. Estimates of improper integrals along the&nbsp;solution of the system are obtained and they are used to obtain conditions for absolute stability.<br>The class of multidimensional nonlinear controlled systems for which the problem of Aizerman has&nbsp;a solution is highlighted. For this class of regulated systems, necessary and sufficient conditions&nbsp;for absolute stability are obtained.</p>}, number={1}, journal={Journal of Mathematics, Mechanics and Computer Science}, author={Aisagaliev S. А. and Ayazbayeva, A. M.}, year={2019}, month={Apr.}, pages={29–47} }