# Abstract

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.

## 1 Introduction

The theory of inverse problems of differential systems and general methods of their solving are quite fully developed in the class of ordinary differential equations [1,2,3, 4,5,6, 7,8,9, 10,11].

Note that one of the important requirements in the theory of inverse problems of differential systems, which is associated with the system’s intransigence to perturbations, is the requirement of stability of the given properties of motion [5]. Therefore, the solution to the problem of stability of program motion is essential for the further development of the theory of inverse problems of differential systems and the theory of constructing systems of program motion.

In the theory of stability, possible perturbed motions of a material system are compared with unperturbed motion in relation to the corresponding values of the given kinematic indicators of motion for each moment of time. In the established formulations of stability problems, the comparison functions

A. S. Galiullin posed the following inverse problem of dynamics, namely, the problem of constructing a set of ordinary differential equations possessing a given integral manifold, as well as the problem of constructing a set of comparison functions with respect to which the stability of the given integral manifold takes place [1,2,3].

It is required to construct the set of equations of motion for the material system

The posed problem of constructing the set of ordinary differential equations and a set of comparison functions was solved in [1,2,3] by the Lyapunov characteristic numbers method. The solution of this problem determines a set of kinematic indicators of motion, in relation to which the given properties of the motion of the system under consideration are stable.

The set of equations of motion of the system, for which the given motion (1.2) is one of the possible, is constructed in [1,2,3] in the following form

The set of sought components of the vector functions

Ito stochastic differential equations describe models of mechanical systems that are important in the application, which take into account the effect of external random forces, for example, the motion of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces [13] or fluctuation drift of a heavy gyroscope in a gimbal [14] and many others. An example showing the importance of taking into account random perturbations is the inverse problem of the dynamics of a spacecraft flight. For example, the aerodynamic moments of a spacecraft always have random components [13] due to fluctuations in the density of the planet’s atmosphere. And, random changes in the moments of inertia cause thermoelastic vibrations of stabilizing rods and vibrations of liquids in banks, antennas, and solar panels. Analysis of the influence of random perturbations is so important that ignoring these perturbations of the spacecraft can significantly reduce its service life [15]. Inverse problems in the class of stochastic differential systems were considered in [16,17,18]. And, the stability in probability of the given program motion is investigated by the Lyapunov functions method in [19,20]. Let us consider the problem posed earlier in the class of ordinary differential equations [1,2,3] under the additional assumption of the presence of random perturbations.

## 2 Problem statement

Let the program motion

Here,

Following [2,16], the equation of perturbed motion of the material system, for which the given motion (2.1) is possible, is represented as

**Definition 1.** [23] A function

**Definition 2.** [24] The program manifold (2.1) of the equation (2.2) is called

### 2.1 The set of vector functions
Q
(
y
)
that do not explicitly depend on time

**Theorem 1.** *Let in the neighborhood*
*there exist a Lyapunov function*
*of the integral manifold*
*with the properties*

**Proof.** Let us consider the difference

### 2.2 The set of vector functions
Q
(
y
,
t
)
depending on
y
and
t

**Theorem 2.** *Let in the neighborhood*
*there exist a Lyapunov function*
*of the integral manifold*
*with the properties* (2.4), (2.5). *Then*, *the program motion*
*of system* (2.3) *is asymptotically*
*stable in probability with respect to an arbitrary continuous in*
*and*
*dimensional vector function*
*satisfying the condition*

*Here*,

**Proof.** The existence of the function

Consequently,

### 2.3 Program motion
λ
(
y
,
t
)
=
0
and a set of comparison vector functions
Q
(
λ
,
t
)

Let us define the program motion in the following form:

Suppose that it takes place

The set of equations of perturbed motion can be represented as

Consider the continuous

**Theorem 3.** *Suppose that in neighborhood* (2.9) *of the integral manifold* (2.8) *there exists a Lyapunov function*
*with properties* (2.4) *and*

*Then, the integral manifold*
*(2.8) is asymptotically stable in probability with respect to an arbitrary*
*dimensional vector function*
*which is continuous with respect to*
*and*
*and satisfying condition (2.11) for*

The proof of Theorem 3 is carried out similarly to the proof of Theorem 2.

### 2.4 A set of
n
-dimensional vector functions of the form
C
(
t
)
λ

Assume that the perturbed motion equation (2.3) in the first approximation has the form

Assume that

(i) the matrix

(ii) the matrix

Then, taking into account properties (i), (ii) and Theorem 2, the following theorem holds.

**Theorem 4.** *Let*
*and*
*be continuous matrices such that conditions* (*i*) *and* (*ii*) *are satisfied*.

Then, the motion

**Remark.** In Theorem 4, the weaker condition (2.15) is required instead of condition (1.4).

**Funding information:** This research was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP08955847).

**Conflict of interest:** Authors state no conflict of interest.

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**Received:**2020-11-25

**Accepted:**2020-12-09

**Published Online:**2021-05-03

© 2021 Marat I. Tleubergenov and Gulmira K. Vassilina, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.