CHAPTER Review of the Theory of Optimal Control The previous chapter introduced the basic tools of dynamic optimization in discrete time We will now review a number of basic results in dynamic optimization in continuous time–particularly the so-called optimal control approach Both dynamic optimization in discrete time and in continuous time are useful tools for macroeconomics and other areas of dynamic economic analysis One approach is not superior to another; instead, certain problems become simpler in discrete time while, certain others are naturally formulated in continuous time Continuous time optimization introduces a number of new mathematical issues The main reason is that even with a finite horizon, the maximization is with respect to an infinite-dimensional object (in fact an entire function, y : [t0 , t1 ] → R) This requires us to review some basic ideas from the calculus of variations and from the theory of optimal control Nevertheless, most of the tools and ideas that are necessary for this book are straightforward We start with the finite-horizon problem and the simplest treatment (which is much more similar to calculus of variations than optimal control) to provide the basic ideas We will then move to the more powerful theorems from the theory of optimal control as developed by Pontryagin and co-authors The canonical problem we are interested in can be written as Z t1 f (t, x (t) , y (t)) dt max W (x (t) , y (t)) ≡ x(t),y(t) subject to x˙ (t) = g (t, x (t) , y (t)) and y (t) ∈ Y (t) for all t, x (0) = x0 , 313