Optimal Control Problems With Multiple Turnpikes

This book studies the turnpike phenomenon arising in optimal control theory. Its goal is to analyze the turnpike phenomenon for various classes of optimal control problems possessing a finite number of turnpikes. The book considers both discrete-time and continuous-time optimal control problems, including problems in metric spaces. The term "turnpike phenomenon" was first coined by P Samuelson in 1948. In this book, each turnpike is a stationary trajectory (a point).The most important feature of this book is the study of a large, general class of optimal control problems in metric spaces with multiple turnpikes, probably for the first time in the literature, to the best of the author's knowledge. It provides a full description of the structure of approximate optimal solutions. Roughly speaking, it is shown that, for each approximate optimal trajectory-control pair (x, u) defined on an interval [0, T], where the length T is sufficiently large, there are mutually disjoint subintervals of [0, T] such that their number does not exceed the number of turnpikes; the Lebesgue measure of the complement of their union does not exceed a constant that does not depend on T; and, for each subinterval, the function x maps it to a small neighborhood of a turnpike that is the same for all points of the subinterval.