Introduction

Abstract

The theory of controlled random sequences is a well developed area of applied mathematics. One can form a very good idea of this theory from the monographs [36, 74]. Different controlled stochastic processes with continuous time were investigated in the books [1, 60, 91, 101, 131, 136, 145, 208]. Broadly speaking, the problem consists in the following. Let π be some control method, that is, a control strategy. We assume that there is pre-set real-valued functional $\tilde{R}(\pi )$, and one must minimize it:

$$\tilde{R}(\pi ) \to \mathop{{\inf }}\limits_{\pi } .$$

(0.1)

The strict mathematical model can be built in two ways: (1) the stochastic basis is fixed, and any strategy determines the specific stochastic process; (2) the probability space and the stochastic process are given, and any strategy π determines the probability measure P^π. In both cases the initial distribution is assumed to be known. As a rule the second approach is employed, in so doing one should take $\tilde{R}(\pi ) \to R({{P}^{\pi }})$. However, it will be shown in Chapter 1 that both versions are equivalent in some sense. In the present work we shall also use the second way. Very often we shall say ‘stochastic process’ instead of ‘random sequence’; in this case the time is always assumed to be discrete.