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One of the inverse problems of dynamics in the presence of random perturbations is considered. This is the problem of the simultaneous construction of a set of first-order Ito stochastic differential equations with a given integral manifold, and a set of comparison functions. The given manifold is stable in probability with respect to these comparison functions.

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.

Given second-order Ito stochastic equations, we construct almost surely equivalent stochastic equations of the Lagrangian structure. We establish the conditions for the direct and indirect analytical representations of the Lagrangian in the presence of random perturbations. The results obtained are illustrated by examples. [ABSTRACT FROM AUTHOR]

In the class of ordinary differential equations the following modification of the inverse problem of differential systems was previously considered: to construct both a set of systems of differential equations and a set of comparison functions for the given program motion. In this article, the modification of the inverse problem is considered in Stochastic case. In this problem it is assumed that random perturbations are from the class of processes with independent increments. By the given program of motion, two sets are constructed: the set of first - order Ito stochastic differential equations and the set of comparison functions. It is proved that there is a stability in probability of the given program motion with respect to the constructed comparison functions. To solve the problem Lyapunov functions method is used. Using Lyapunov’s second method makes it is possible to weaken the conditions imposed on the components of the constructed comparison functions, in contrast to the application of the Lyapunov characteristic numbers method for solving the inverse problem in the class of ordinary differential equations...

We consider inverse problems for differential systems in the presence of random perturbations. By the method of additional variables, we establish sufficient conditions for the representation of stochastic differential equations of the second order in the form of stochastic Lagrange equations, as well as for the representation of stochastic differential equations of the first order in the form of stochastic canonical equations. The obtained results are illustrated by specific examples. [ABSTRACT FROM AUTHOR]

We consider the reconstruction problem in the class of stochastic differential equations of Itô type on the basis of given motion properties that depend only on part of the variables. We determine the set of controls providing necessary and sufficient conditions for the existence of a given integral manifold. [ABSTRACT FROM AUTHOR]

The separation method is used obtain sufficient conditions for the solvability of the main (according to Galiullin’s classification) inverse problem in the class of first-order Itô stochastic differential systems with random perturbations from the class of Wiener processes and diffusion degenerate with respect to a part of variables. [ABSTRACT FROM AUTHOR]