@article{Aisagaliev_Ayazbayeva_2018, title={Improper integrals for stability theory of multidimensional regulated systems}, volume={95}, url={https://bm.kaznu.kz/index.php/kaznu/article/view/464}, DOI={10.26577/jmmcs-2017-3-464}, abstractNote={<p>A class of ordinary differential equations described the dynamics of nonlinear regulated systems<br>the right-hand part of which contains the nonlinear functions of the given set is considered. The<br>uncertainty of the right-hand side arises the non-uniqueness of the solution, that leads to the<br>necessity to study the group properties of solutions of the system. One such property is the absolute<br>stability of the trivial solution, i.e. properties at which all decisions coming from any starting point<br>for any non-linear functions of the given set tend over time to an equilibrium position. A completely<br>new method for the study of absolute stability of nonlinear regulated systems without involving<br>any Lyapunov functions and frequency theorems is proposed by evaluating improper integrals<br>along the solutions of the system. The motion equations of the system is led to a special form<br>by non-singular transformation, which allows to represent the integrand improper integrals as the<br>sum of two terms. The first term is a quadratic form reduced to the diagonal form, and the second<br>term is the total differential function on time. The representation of the integrand, ultimately,<br>leads to easily verifiable criteria for absolute stability.</p>}, number={3}, journal={Journal of Mathematics, Mechanics and Computer Science}, author={Aisagaliev S. А. and Ayazbayeva, A. M.}, year={2018}, month={Aug.}, pages={3–20} }