On construction of a field of forces along given trajectories in the presence of random perturbations
Кездейсоқ түрткi болғанда берiлген траекториялар бойынша күштер өрiсiн тұрғызу туралы
О построении поля сил по заданным траекториям при наличии случайных возмущений
Tleubergenov M.I. Vassilina G.K. Tuzelbaeva G.A.
2021E.A.Buketov Karaganda State University Publish House
Bulletin of the Karaganda University. Mathematics Series
2021#101Issue 198 - 103 pp.
In this paper, a force field is constructed along a given integral manifold in the presence of random perturbing forces. In this case, two types of integral manifolds are considered separately: 1) trajectories that depend on generalized coordinates and do not depend on generalized velocities, and 2) trajectories that depend on both generalized coordinates and generalized velocities. The construction of the force field is carried out in the class of second-order stochastic Ito differential equations. It is assumed that the functions in the right-hand sides of the equation must be continuous in time and satisfy the Lipschitz condition in generalized coordinates and generalized velocities. Also this functions satisfy the condition for linear growth in generalized coordinates and generalized velocities.These assumptions ensure the existence and uniqueness up to stochastic equivalence of the solution to the Cauchy problem of the constructed equations in the phase space, which is a strictly Markov process continuous with probability 1. To solve the two posed problems, stochastic differential equations of perturbed motion with respect to the integral manifold are constructed. Moreover, in the case when the trajectories depend on generalized coordinates and do not depend on generalized velocities, the second order equations of perturbed motion are constructed, and in the case when the trajectories depend on both generalized coordinates and generalized velocities, the first order equations of perturbed motion are constructed. And further, in both cases by Erugin’s method necessary and sufficient conditions for solving the posed problems are derived.
integral manifold , inverse problems , stability , stochastic differential equations
Text of the article Перейти на текст статьи
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Al-Farabi Kazakh National University, Almaty, Kazakhstan
Almaty University of Power Engineering and Telecommunications named after G.Daukeev, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling
Al-Farabi Kazakh National University
Almaty University of Power Engineering and Telecommunications named after G.Daukeev
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026