Extensions of the Maximum Principle

In the preceding seven chapters we explored the basic principles of optimal control theory applied to problems typically arising in economics and management science. In particular we illustrated how the interplay of analytical and numerical methods can be used to obtain insight when fighting drugs, corruption, and terror. Before concluding our tour we should briefly sketch several important extensions of “standard-type” optimal control theory.

Usually, in optimal control theory, systems evolve continuously in time, but in reality sudden discontinuous changes may happen at certain time points. Those changes may pertain not only to the model's parameter values but also to sudden modifications of the model's functional forms. In Sect. 8.1 we give a short introduction into the important and growing field of multi-stage models. Section 8.2 provides a brief introduction to the fundamentals of differential games. The Nash and the Stackelberg concepts are introduced. Examples from the fields of corruption and terrorism are provided to illustrate the flavor of differential games.

The homogeneity of economic agents is admittedly a fiction. There are several avenues to introduce heterogeneity. One of them has been discussed in Chap. 6, namely multi-compartment models. Partial differential equations (PDEs) are another powerful tool for distinguishing individuals according to various characteristics evolving over time.1 In Sect. 8.3 we present some basic facts about distributed parameter control.

Finally, in Sect. 8.4 we mention a number of other important topics in optimal control modeling, including stochastic models, impulse control, delayed systems, and nonsmooth systems.