For the control of discrete event systems, the notion of directed control refines that of supervisory control. A directed controller is one that selects at most one controllable event to be enabled at any state (without disabling any uncontrollable event), which is in fact how a discrete event control is implemented. In contrast, a supervisory controller computes a maximal allowable set of controllable events at each state, leaving undecided exactly which one is to be enabled.

We model discrete event systems using the automaton formalism. Under directed control, our first goal is to achieve logical correctness of the controlled system behavior as specified by safety and nonblocking. Subsequently we address the best performance issue by providing an optimization based framework. The optimization task is to direct a system in such a way that regardless of the history of evolution, it accomplishes a pending task in a minimal cost.

In a state-based setting, we formulate and study the existence and synthesis problems with the above objectives. We first show that the existence and the synthesis of a safe and nonblocking directed controller are both solvable in polynomial complexity. Then we present a novel approach with polynomial complexity for the synthesis (and the existence) of an optimal director, thus providing a complete solution to the problems in study.