ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Combination of the Method of Steps and an Expansion of the State Space for Analyzing Linear Stochastic Systems with Various Forms of Delays and Random Inputs in the Form of Additive and Multiplicative White Noises

### Author(s):

Igor Egorovich Poloskov

Perm State University, Department of Higher Mathematics,
Head of the Department,
Associate Professor, Doctor of Physics and Mathematics
Russia, 614990, Perm, Bukirev St., 15

Igor.Poloskov@psu.ru

### Abstract:

We represent the theoretical method and technique for stochastic analysis of systems of linear differential equations which have two forms of finite delays (concentrated and distributed) and are perturbed by additive and multiplicative white noises. The research problem is solved on the basis of constructing ordinary differential equations for the first moment functions of the system state vector. As a tool for such a construction, we consider a scheme that integrates the classical method of steps and an expansion of the state space, and uses the technique of multidimensional matrices. By using this scheme we construct a chain of systems of stochastic differential equations without delays, and then the equations for the moments required. The program implementing the proposed scheme was performed in the environment of the computer algebra package Mathematica and was used to analyze a model system. The final part of the work presents both the results of calculations and some details of the program algorithm.

### Keywords

• additive and multiplicative noises
• linear dynamic system
• lumped and distributed delays
• moment functions
• state vector
• stochastic analysis

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