Optimal control for resource allocation in discrete event systems

Supervisory control for discrete event systems (DESs) belongs essentially to the logic level for control problems in DESs. Its corresponding control task is hard. In this paper, we study a new optimal control problem in DESs. The performance measure is to maximize the maximal discounted total reward among all possible strings (i.e., paths) of the controlled system. The condition we need for this is only that the performance measure is well defined. We then divide the problem into three sub-cases where the optimal values are respectively finite, positive infinite and negative infinite. We then show the optimality equation in the case with a finite optimal value. Also, we characterize the optimality equation together with its solutions and characterize the structure of the set of all optimal policies. All the results are still true when the performance measure is to maximize the minimal discounted total reward among all possible strings of the controlled system. Finally, we apply these equations and solutions to a resource allocation system. The system may be deadlocked and in order to avoid the deadlock we can either prohibit occurrence of some events or resolve the deadlock. It is shown that from the view of the maximal discounted total cost, it is better to resolve the deadlock if and only if the cost for resolving the deadlock is less than the threshold value.