Optimal control of discrete event systems

We are interested in the problem of designing control software for large-scale systems having discrete event-driven dynamics, in situations where the performance is specified by numerical measures. The paradigm of Supervisory Control Theory, developed for Discrete Event Systems (DES) constrained by legality specifications (0, $\infty$-cost structure), is useful and we show how it can be extended to this more general class of specifications. We assume the DES is represented by a formal language consisting of strings contained in the Kleene closure of an alphabet. This language has two kinds of costs on its usage of resources. The design objective is to find sublanguages that minimize these costs. Both deterministic and stochastic languages are looked at. We present modelling methods and examples to motivate interesting ways of using our problem formulation. An existence theory is developed. Amongst other things, the theory establishes the existence of minimally restrictive solutions. Various related paradigms in stochastic control, dynamic programming and finite vertex directed graphs are discussed. For DES modelled by finite state machine we establish computability and present algorithms of polynomial complexity to compute optimal sublanguages. Our findings can collectively be considered as a theory of optimal control for DES, though it differs from the classical theory in interesting ways.