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Abstracts

XX conference

Program Control with Probability One for Stochactic Systems

Karachanskaya E.V.

Pacific National University, Khabarovsk, Russia, karachanskaya@mail.khstu.ru

Usually, a program moving is considered as moving on a given

manifold. The term "stochastic optimization"\, was actual for

stochastic system. Terms "program moving"\, and "program

control"\, for stochastic system didn't exist there. There

exists a function, which conserves with probability one a

constant value for all solutions of stochastic differential

equations system, and it is called a first integral of SDE

system \cite{D_78,D_02,11_KchUpr}. Then we can set a program

control problem with probability one and solve it

\cite{11_KchUpr,08_ChContrEn}.

\verb"Definition 1." Let us call \textit{a Program Control with

Probability One} (PCP1) as a control in stochastic system,

which with probability one provides an insensitivity of this

system to random perturbations.

\verb"Definition 2." Let us ${\bf x}(t; {\bf x}_{o},{\bf

s};\omega)$ be a solution of a SDE system:

\begin{equation}\label{UprPuas2}

\begin{array}{c}

d {\bf x}(t)= \Bigl[ P(t;{\bf x}(t)) + Q(t;{\bf x}(t)) \cdot {\bf

s}(t;{\bf x}(t)) \Bigr] dt + B(t;{\bf x}(t)) d {\bf

w}(t) %+ \\

+\displaystyle\int G(t;{\bf

x}(t);\gamma)\nu(dt;d\gamma),

\end{array}

\end{equation}

where ${\bf w}(t)$ is a $m$-dimensional Wiener process;

$\nu(t;\triangle \gamma)$ is a non-centered Poisson measure. A

non-random function is a first integral of SDE system

\eqref{UprPuas2} with initial condition ${\bf x}(t;{\bf

x}_{o})\bigr|_{t=0}={\bf x}_{o}$. \textit{ A Program Moving of

a stochastic system} we will call a solution ${\bf x}(t; {\bf

x}_{o},{\bf s};\omega)$, which with a some PCP1 ${\bf

s}(t;{\bf x})$ allows this system to remain on the given

integral manifold $ u\bigl(t;{\bf x}(t;{\bf x}_{o})\bigr)=

u(0;{\bf x}_{o})$ with probability one for any $t$.

\begin{thebibliography}{100}

\bibitem{D_78} \textit{Doobko V. A.} A first integral for a

stochastic differential equations system (Preprint / Inst.

Math. Ac.Sci. USSR) -- Kiev, 1978. 22 p.

\bibitem{D_02} \textit{Doobko V. A.} \textit{Open evolving

systems} // I inter. sci.-appl. conf.

"Open evolving systems"\, (2002), Kiev, 2002. P. 14--31.

\bibitem{11_KchUpr} \textit{Karachanskaya E. V.} Construction

of program control with probability one for a dynamical

system with poisson perturbations // \textit{Bulletin of PNU} No 2 (21), 2011. P. 51-60.

\bibitem{08_ChContrEn} \textit{Chalykh E.} Constructing the set

of program controls with probability 1 for one class of

stochastic systems // \textit{Automation and Remote

Control} \textbf{70}, No 8, 2009. P. 1364--1375.

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