Higher-Dimensional Models

The theory of optimal control is wonderful. Now, how does one actually apply it? In particular, how does one compute the solutions numerically, including for higher-dimensional systems (since solving these systems can substantially be harder than solving one-dimensional models)? This third part of the book sheds light on the practical side of applying optimal control theory.

By “practical side” we still mean mathematical calculations, not political or organizational issues associated with persuading decision-makers to implement a solution or even statistical issues associated with data collection and model parameterization. Too often, however, texts on optimal control tell the reader whatto do but not howor even whenand whyto do it.

The “practical side” of how to solve optimal control problems involves writing computer programs because few models have analytic solutions. The “when and why” questions concern how to organize all of the components of an overall analysis. Given a particular mathematical formulation, what does one do first? What second? What does one do when one analytic strategy hits a dead end? We illustrate how to answer such questions with three innovative examples that are worked in depth.1 We offer one example each from the three main application areas discussed in this book: drug control, corruption, and counter-terror.

This chapter presents the model formulations and meta-analytic strategies employed to analyze them. Points in the analysis that require nontrivial computations are flagged, with a reference forward to the following chapter on numerical methods. In that subsequent Chap. 7, the discussion walks the reader step by step through the details of how to do computations of that sort, including references to the computer code available on the companion Web site for performing these numerical calculations.