АСИМПТОТИЧНІ ВЛАСТИВОСТІ ІМПУЛЬСНОГО ПРОЦЕСУ ЗБУРЕНЬ В УМОВАХ ПУАССОНОВОЇ АПРОКСИМАЦІЇ З ТОЧКОЮ РІВНОВАГИ КРИТЕРІЮ ЯКОСТІ

• Date: 2020
• Main Authors: Chabanyuk, Y.M., Nіkіtіn, А.V., Khimka, U.T., Nikitina, T.R.
• Format: Article
• Language: English
• Published: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Subjects: випадкова еволюція; пуассонова апроксимація; марковські переключення
• Online Access: https://jais.net.ua/index.php/files/article/view/501
• Journal Title: Problems of Control and Informatics

For a system of stochastic differential equations with Markov switchings and impulse perturbation under Poisson approximation scheme and under the conditions of the existence of a single equilibrium point of the quality criterion, the limit generators for the impulse process and the dynamic system are constructed. The complexity of the proposed evolutionary model lies in its three properties. Firstly, the system is under conditions of an external random impact, which is modeled using the Markov switching process. Processes with independent increments, which also depend on the Markov switching process, have certain characteristics between the moments of its restoration, and at the moments of restoration these characteristics change. Therefore, the so-called «gluing» of trajectories of processes with independent increments occurs. Secondly, the model contains a Poisson approximation scheme, which is a generalization of the classical averaging scheme and is determined by normalization depending on a small parameter. In the classical approximation scheme in the limit process, we do not see large jumps in the system. The maximum that we get is the shift of the deterministic trajectory. But in the Poisson approximation scheme, which was invented by Korolyuk and Limnios in the 2005 monograph, this problem is eliminated, that is, in the limit there will be both a deterministic shift and large jumps. And thirdly, the system has a control function, which is determined using the Robins-Monroe stochastic approximation procedure. This procedure solves the problem of finding the equilibrium point of the regression function and consists in finding the only solution to the equation with respect to control. Assuming the existence of a single control on each interval, we solve a two-level problem. The article examines the questions of how the behavior of the limiting process depends on the prelimit normalization of the stochastic system in an ergodic Markov environment.