CRITERION OF ABSOLUTE STABILITY OF SOLUTIONS OF STOCHASTIC DIFFUSION DYNAMIC INFORMATION SYSTEMS OF AUTOMATIC REGULATION WITH EXTERNAL DISTURBANCES

. The conditions of asymptotic stability of the trivial solution of stochastic diffusion dynamic systems of automatic regulation with external disturbances are obtained. In work is to find criteria for the absolute stability of solutions of stochastic diffusion dynamic systems of automatic regulation (SAR) with external disturbances. The problem of determining sufficient algebraic conditions for the stability of stochastic d iffusion dynamic SARs with aftereffect and taking into account external disturbances is relevant, since its solution makes it possible to study complex stochastic systems using matrix calculations.


INTRODUCTION
In order to research and study real-life random processes, a whole constellation of scientistsmathematicians study stochastic models, which are described by differential, differential functional, differentialdifference equations with disturbances of various kinds, the parameters of which are random functions of time [1, 2, ].
The main indicator of high-quality physical implementation of the process is its stability [4,5]. As a result, the academician O.M. Lyapunov at the end of the 19th and the beginning of the 20th century created a new scientific trend in the theory of motion -the theory of stochastic stability, which has gained wide development [6,7,8].
The purpose of this work is to find criteria for the absolute stability of solutions of stochastic diffusion dynamic systems of automatic regulation (SAR) with external disturbances. The problem of determining sufficient algebraic conditions for the stability of stochastic diffusion dynamic SARs with aftereffect and taking into account external disturbances is relevant, since its solution makes it possible to study complex stochastic systems using matrix calculations. МЕТОДИ ТА СИСТЕМИ ОПТИКО-ЕЛЕКТРОННОЇ І ЦИФРОВОЇ ОБРОБКИ ЗОБРАЖЕНЬ ТА СИГНАЛІВ 6

CONDITIONS FOR THE ASYMPTOTIC STABILITY OF THE TRIVIAL SOLUTION OF STOCHASTIC DIFFUSION DYNAMIC SARS WITH EXTERNAL DISTURBANCES
Consider the Lyapunov-Krasovsky stochastic functional [1]: where the symmetric positive definite matrix H is a solution of Sylvester's matrix equations [1]: The ( )inequality [1] is valid for: According to the substitution of Ito [2], the stochastic differential of the functional ( )at the solutions of NDSRAR (1) -(2) has the form: According to the definition of the stochastic differential, equality (11) should be understood as an integral does not exist with probability 1 [2].
Next, we will calculate the mathematical expectation on the left and right sides of the corresponding integral equality (11), taking into account the equality of zero from the Wiener-Ito integral [2] for ∀ ∈ [0, ]: As a result, we get: Under condition (3), we have ( ) ( ) − 2 ( ) > 0, then equality (12) will turn into inequality: The last inequality can be written in vector-matrix form, namely: is negative definite, and the block symmetric matrix is ̃u nderdefinite [1]. We will denote further ̃≤ 0 . Matrix { 1 ( 1 ( ))}( + ) + { 2 2 ( 2 ( ))} is negative definite if and only if the matrix is a solution of the Sylvester matrix equation (7), in which there are mathematical expectations: The matrix ̃( 14) is ill-defined only in the case of ill-defined matrix-blocks standing on its main diagonal, namely, the matrix is ill-defined if and only if the number < 0, the number − 1 < 0, which is equivalent to the condition: Thus, the under definiteness of the block matrix ̃( 14) under the conditions < 01, (7) and (17) also requires the underdefiniteness of the following matrix requirement (17) means the density of the matrix { 1 ( 1 ( ))} + , which characterizes the exponential stability of the matrix A.
We write down the condition of positive definiteness of the Lyapunov function (6) on the linear characteristic of the Hurwitz angle ( ) = : Hence the positive definiteness of the matrix follows Multiplying this inequality on the left by , and on the right by , we obtain an equivalent inequality Thus, the condition of negative definiteness of mathematical expectation requires negative definiteness of the matrix (condition (7)) and the fulfillment of the inequality Let's open this determinant by the last column Let's get the statement.  (7); 3) the matrix inequality (13)

Then:
I) The trivial solution of system (4) is asymptotically stable according to Lyapunov. II) In inequality (25), S is defined as follows [3] Verification of conditions (1) -4) is effective when using modern computer technologies.

CONCLUSION
The conditions of asymptotic stability of the trivial solution of stochastic diffusion dynamic systems of automatic regulation with external disturbances are obtained. In work is to find criteria for the absolute stability of solutions of stochastic diffusion dynamic systems of automatic regulation (SAR) with external disturbances. The problem of determining sufficient algebraic conditions for the stability of stochastic diffusion dynamic SARs with aftereffect and taking into account external disturbances is relevant, since its solution makes it possible to study complex stochastic systems using matrix calculations.