Оптимізація функціоналів в умовах невизначеності для стохастичних диференціальних рівнянь Іто-Скорохода у гільбертових просторах

Abstract

In the article for the stochastic differential equations of Ito-Skorokhod, problems of optimization of functionals under conditions of uncertainty in Hilbert spaces are investigated. Purpose of the article is to investigate some properties of stochastic differential equations in Hilbert spaces. These objects arise in diverse areas of applied mathematics as models for various natural phenomena, in particular, the evolution of complex systems with infinitely many degrees of freedom. For instance, one may think of the liquid fuel motion in the tank of a spacecraft. Spacecraft constructors should take into account this motion, for it influences heavily the path of a spacecraft. Also, optimization of the motion is an issue of principal importance. It is not trivial to carry over the results concerning stochastic differential equations in finite-dimensional spaces to the infinite dimensional case. We give some statements, in which the existence, uniqueness is proved and the explicit form μ-optimal controls for such equations is constructed, in particular, μ-optimal control is found as a linear inverse relationship.