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Optimal stopping areas for discrete time linear quadratic control problem

The problem of determining the optimal stopping areas for a discrete time linear stochastic controlled system is investigated in this paper. Sometimes we have no information how long the system will be controlled. In this case we have a complex problem: the system should be controlled and stopped at the appropriate moment. Thus at each moment we make a decision about continuation of control and when the decision is positive, we need to determine the optimal control for stochastic system. To solve this problem the dynamic programming and the optimal stopping rules for stochastic processes were employed. The paper presents the method of determining the optimal stopping areas — sets of states where the system should be stopped. A numerical example is included and illustrates the behavior of these sets.