Про проблему стабілізації керованих стохастичних дифереціально-функціональних систем із скінченним запізненням

Abstract

Works [1–4] analyze a problem of stabilization of systems, which are defined by a stochastic differential-functional equations with impulsive Markov disturbances with a constant or finite delay, in presence of a transitional process and a delay at the same time. This work considers a more generalized problem of stabilization of the control stochastic differential-functional systems with finite delay and mutually independent Wiener processes. The delay is constructed on the space of the Skorokhod of right continuous functions with left limits [1]. This systems must be asymptotically stable by the probability and provide preassigned optimivity of a transient process. The control be selected is built on the principle of inverse communication, obtained as a Markov process [3, 4]. The problem of optimal stabilization is considered in the context of the given quality criteria, builds on Bellman's dynamic programming principles. The first part of the work analyzes the properties of Markov processes. The corresponding lemma is formulated as a result. In the second part obtained the infinitesimal operator of the corresponding Markov process, is formulated and the basic theorem of stabilization is proved. The proof algorithm is based on using the Ito-formula. Examples of use are given. In the third part an optimal stabilization algorithm has been demonstrated to investigate a linear systems. For the case of linear systems, the stabilization theorem is formulated. The results of the scientific research were obtained for use in technical systems. The results obtained and the arguments given are valid in the determined case as well. This work is part one of the first scientific research, the second part will contain more examples and use the method of successive approximations.