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\input zb-ioport
\iteman{io-port 04070787}
\itemau{Evstigneev, I.V.}
\itemti{Controlled random field on oriented graphs.}
\itemso{Teor. Veroyatn. Primen. 33, No.3, 465-479 (1988).}
\itemab
Let ($\Omega$,${\cal F},P)$ be a probability space and T a finite set endowed with the structure of an oriented graph. Suppose that a $\sigma$- algebra ${\cal F}\sb t\subseteq {\cal F}$ and a measurable space $(A\sb t,{\cal A}\sb t)$ (where $A\sb t$ is a linear space) correspond to all $t\in T$. The set of functions $z:=\{z\sb t$; $t\in T$, $\Omega \to A\sb t\}$ is said to be a control if for all $t\in T$ the function $z\sb t$ is ${\cal F}\sb t$-measurable. Let Z be a convex subset of all controls, and let a real valued functional F: $Z\to {\bbfR}$ (supposed to be convex from above) the defined on this set. Let us consider the mapping $T\times Z\ni (t,z)\mapsto G\sb t(.,z)$, where $G\sb t(.,z)$ is an $m\sb t$- dimensional random vector. (Here $m\sb t\in {\bbfN}$ and $z\mapsto G\sb t(.,z)$ is also supposed to be convex from above.) The problem is to find a control $z\in Z$ for which F is maximal on Z under the condition $$ (1)\quad G\sb t(.,z)\ge 0\quad almost\quad surely\quad for\quad all\quad t\in T. $$ If $T:=\{1,2,...,N\}$ and ${\cal F}\sb t$ (the $\sigma$- algebra of events observable up to t) do not decrease with t then the traditional problems of stochastic optimization are obtained. The given model, however, may also be interpreted in such a way that the control is realized by afinte number of participants $t\in T$, and the decision $z\sb t$ by any of them depends on the occurrence or non-occurrence of events from ${\cal F}\sb t$. The participants follow a common goal described by the functional F obeying the restriction give in (1). Suppose that for all $t\in T$ there exists K(t)$\subseteq T$ (t$\in K(t))$ (K(t) is the set of participants directly dependent on t) and $$ (2)\quad {\cal F}\sb s\supseteq {\cal F}\sb t,\quad if\quad s\in K(t) $$ holds. Then the problem above to find the maximum may be interpreted as a problem to control the random field $\{z\sb t\}$ given on the graph T. Condition (2) means that randomness effecting participant t also effects particpants depending on t. Under conditions connecting condition (1) with the structure of the graph T, a necessary and sufficient condition of optimality (an optimality principle) will be stated as a theorem. (The form of this result is analogous to the maximum principle of Pontryagin.) As an application of this result a statement is proven on the stability of optimal controls The existence of an optimal control is also shown in a special case.
\itemrv{J.T\'oth}
\itemcc{}
\itemut{oriented graph; stochastic optimization; random field; maximum principle; stability of optimal controls; existence of an optimal control}
\itemli{}
\end