Numerical Methods for Control and Optimization

One of the main techniques presented in Chapters 7 and 8 is to reduce the complexity of singularly perturbed systems by studying the corresponding limit systems that are easier to handle than the original problems. The optimal or nearly optimal controls of the limit problems can be used to construct nearly optimal controls of the original systems. Although the limit systems are substantially simpler than the original pre-limit ones, very often closed-form solutions are still difficult to obtain, except in special cases. For example, in the context of stochastic manufacturing systems, a closed-form solution for optimal production planning is obtained for a system with one-machine and one-part-type by Akella and Kumar [2] for a discounted cost problem, and Zhang and Yin [251] for a finite horizon counterpart. Such closed-form solutions do not seem possible for more general manufacturing systems such as flowshops and jobshops (see Sethi and Zhang [192]). For many applications, one has to resort to a viable alternative – numerical methods.