2022 | Buch

# Turnpike Phenomenon and Symmetric Optimization Problems

verfasst von: Alexander J. Zaslavski

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

2022 | Buch

verfasst von: Alexander J. Zaslavski

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

Written by a leading expert in turnpike phenomenon, this book is devoted to the study of symmetric optimization, variational and optimal control problems in infinite dimensional spaces and turnpike properties of their approximate solutions. The book presents a systematic and comprehensive study of general classes of problems in optimization, calculus of variations, and optimal control with symmetric structures from the viewpoint of the turnpike phenomenon. The author establishes generic existence and well-posedness results for optimization problems and individual (not generic) turnpike results for variational and optimal control problems. Rich in impressive theoretical results, the author presents applications to crystallography and discrete dispersive dynamical systems which have prototypes in economic growth theory.

This book will be useful for researchers interested in optimal control, calculus of variations turnpike theory and their applications, such as mathematicians, mathematical economists, and researchers in crystallography, to name just a few.

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Abstract

In this chapter we consider optimization problems on complete metric spaces without compactness assumptions, optimization problems arising in crystallography and symmetric optimization problems in abstract spaces. We also discuss turnpike properties in the calculus of variations. To have the turnpike property means, roughly speaking, that the approximate solutions of the problems are determined mainly by the integrand and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.

Abstract

In this chapter we study several classes of symmetric optimization problems which are identified with the corresponding spaces of objective functions, equipped with appropriate complete metrics. Using the Baire category approach, for any of these classes, we show the existence of subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution. These results are obtained as realizations of a general variational principle which is established in this chapter. We extend these results for certain classes of symmetric optimization problems using a porosity notion. We identify a class of symmetric minimization problems with a certain complete metric space of functions, study the set of all functions for which the corresponding minimization problem has a solution, and show that the complement of this set is not only of the first category but also a σ-porous set.

Abstract

In this chapter we study classes of symmetric parametric optimization problems which are identified with the corresponding spaces of functions, equipped with appropriate complete metrics. Using the Baire category approach, for any of these classes we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every function from this intersection the corresponding symmetric parametric optimization problems possess solutions for any parameter. These results are obtained as realizations of a general variational principle which is established in this chapter. The results of this chapter are new.

Abstract

The study of infinite dimensional optimal control has been a rapidly growing area of research. In this chapter we present preliminaries which we need in order to study turnpike properties of infinite dimensional optimal control problems. We discuss Banach space valued functions, unbounded operators, C
_{0} semigroups, evolution equations, and admissible control operators.

Abstract

In this chapter we study the turnpike properties for symmetric variational problems in Banach spaces. To have the turnpike property means, roughly speaking, that the approximate solutions of the problems are determined mainly by the integrand and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. It is shown that for symmetric variational problems the turnpike is a minimizer of the integrand. The results of this chapter are new.

Abstract

In this chapter we study the turnpike phenomenon for continuous-time optimal control problems in infinite dimensional spaces that have a certain symmetric property. To have the turnpike property means, roughly speaking, that the approximate solutions of the problems are determined mainly by the integrand and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. It is shown that for our class of problems the turnpike is a minimizer of the integrand. We also study the structure of approximate solutions on large intervals in the regions close to the right end points. The results of this chapter are new. They will be obtained for two large classes of problems that will be treated simultaneously.

Abstract

In this chapter we study symmetric optimization problems arising in crystallography, which are identified with the corresponding space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach and the porosity notion, we show that a typical (generic) problem has exactly two different solutions and is well-posed.

Abstract

In this chapter we study turnpike properties for a discrete dispersive dynamical system generated by set-valued mappings, which was introduced by A. M. Rubinov in 1980. This dispersive dynamical system has a prototype in mathematical economics. In particular, it is an abstract extension of the classical von Neumann–Gale model. Our dynamical system is described by a compact metric space of states and a transition operator, which is set-valued. Our goal is to study the asymptotic behavior of the trajectories of this dynamical system.