Control of Random Sequences in Problems with Constraints

We consider the optimal control problem for a stochastic sequence (not necessarily Markov) in which besides the main criterion there are a number of other functionals of the sample paths; their expectations must satisfy a certain system of inequalities. The life time of the process is assumed to be finite and all basic spaces are Borel spaces. The paper studies some properties of the space of strategy measures (for example, we prove that the selectors correspond to extreme points of this space). The optimal control problem is reformulated in terms of the theory of abstract linear programming which allows us to obtain necessary and sufficient conditions of optimality. Moreover, the paper proves the existence of an optimal strategy and its form is established (a finite mixture of selectors); at the end some exactly solved examples are cited.