In this paper we dealt with a new formulation of the classic total expected value - discounted cost (TEV-DC) control problem, over an infinite time horizon, where a set-valued control policy and a set of initial states must be found. The control problem is applied to a periodic discrete-time dynamic system, affected by stochastic disturbances. The solution is obtained by manipulating the solution of an analogous, but point-valued, TEV-DC control problem.