The structure of approximate solutions of autonomous discrete-time
optimal control problems and individual turnpike results for optimal
control problems without convexity (concavity) assumptions are examined
in this book. In particular, the book focuses on the properties of
approximate solutions which are independent of the length of the
interval, for all sufficiently large intervals; these results apply to
the so-called turnpike property of the optimal control problems. By
encompassing the so-called turnpike property the approximate solutions
of the problems are determined primarily by the objective function and
are fundamentally independent of the choice of interval and endpoint
conditions, except in regions close to the endpoints. This book also
explores the turnpike phenomenon for two large classes of autonomous
optimal control problems. It is illustrated that the turnpike phenomenon
is stable for an optimal control problem if the corresponding infinite
horizon optimal control problem possesses an asymptotic turnpike
property. If an optimal control problem belonging to the first class
possesses the turnpike property, then the turnpike is a singleton (unit
set). The stability of the turnpike property under small perturbations
of an objective function and of a constraint map is established. For the
second class of problems where the turnpike phenomenon is not
necessarily a singleton the stability of the turnpike property under
small perturbations of an objective function is established. Containing
solutions of difficult problems in optimal control and presenting new
approaches, techniques and methods this book is of interest for
mathematicians working in optimal control and the calculus of
variations. It also can be useful in preparation courses for graduate
students.
