Optimal control for discrete event systems with arbitrary control pattern

In this paper, we present a new model for optimal control of discrete event systems (DESs) with an arbitrary control pattern. Here, a discrete event system is defined as a collection of event sets that depend on strings. When the system generates a string, the next event that may occur should be in the corresponding event set. In the optimal control model, there are rewards for choosing control inputs at strings and the sets of available control inputs also depend on strings. The performance measure is to find a policy under the condition where the discounted total reward among strings from the initial state is maximized. By applying ideas from Markov decision processes, we divide the problem into three sub-cases where the optimal value is respectively finite, positive infinite and negative infinite. For the case with finite optimal values, the optimality equation is shown and further characterized with its solutions. We also characterize the structure of the set of all optimal policies. Moreover, we discuss invariance and closeness of several languages. We present a new supervisory control problem of DESs with the control pattern being dependent on strings. We study the problem in both the event feedback control and the state feedback control by generalizing concepts of invariant and closed languages/predicates. Finally, we apply the above model and results to a job-matching problem.