In this paper we consider the problem of controlling a multi–inventory system in the presence of uncertain demand. The demand is unknown but bounded in an assigned compact set. The control input is assumed to be also constrained in a compact set. We consider an integral cost function of the buffer level and we face the problem of minimizing the worst case cost. We show that the optimal cost of a suitably constructed auxiliary problem with no uncertainties is always an upper bound for the original problem. In the special case of minimum–time control, this upper bound is tight, namely its optimal cost is equal to the worst case cost for the original system. Furthermore the result is constructive, since the optimal control law can be explicitly computed.