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2013 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Nonconvex Optimal Control and Variational Problems

verfasst von: Alexander J. Zaslavski

Verlag: Springer New York

Buchreihe : Springer Optimization and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Nonconvex Optimal Control and Variational Problems is an important contribution to the existing literature in the field and is devoted to the presentation of progress made in the last 15 years of research in the area of optimal control and the calculus of variations. This volume contains a number of results concerning well-posedness of optimal control and variational problems, nonoccurrence of the Lavrentiev phenomenon for optimal control and variational problems, and turnpike properties of approximate solutions of variational problems.

Chapter 1 contains an introduction as well as examples of select topics. Chapters 2-5 consider the well-posedness condition using fine tools of general topology and porosity. Chapters 6-8 are devoted to the nonoccurrence of the Lavrentiev phenomenon and contain original results. Chapter 9 focuses on infinite-dimensional linear control problems, and Chapter 10 deals with “good” functions and explores new understandings on the questions of optimality and variational problems. Finally, Chapters 11-12 are centered around the turnpike property, a particular area of expertise for the author.

This volume is intended for mathematicians, engineers, and scientists interested in the calculus of variations, optimal control, optimization, and applied functional analysis, as well as both undergraduate and graduate students specializing in those areas. The text devoted to Turnpike properties may be of particular interest to the economics community.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Let \(-\infty < T_{1} < T_{2} < \infty \), \(A \subset [T_{1},T_{2}] \times {R}^{n}\) be a closed subset of the t x-space R n+1 and let A(t) denote its sections, that is
$$\displaystyle{A(t) =\{ x \in {R}^{n} : (t,x) \in A\},\quad t \in [T_{ 1},T_{2}].}$$
For every (t,x)∈A let U(t,x) be a given subset of the u-space R m , \(x = (x_{1},\ldots x_{n})\), \(u = (u_{1},\ldots u_{m})\).
Alexander J. Zaslavski
Chapter 2. Well-posedness of Optimal Control Problems Without Convexity Assumptions
Abstract
In this chapter we prove generic existence results for classes of optimal control problems in which constraint maps are also subject to variations as well as the cost functions. These results were obtained in [87, 90]. More precisely, we establish generic existence results for classes of optimal control problems (with the same system of differential equations, the same boundary conditions and without convexity assumptions) which are identified with the corresponding complete metric spaces of pairs (f, U) (where f is an integrand satisfying a certain growth condition and U is a constraint map) endowed with some natural topology. We will show that for a generic pair (f, U) the corresponding optimal control problem has a unique solution.
Alexander J. Zaslavski
Chapter 3. Well-posedness and Porosity in Nonconvex Optimal control
Abstract
In[86, 88] we considered a class of optimal control problems which is identified with the corresponding complete metric space of integrands, say \(\mathcal{F}\). We did not impose any convexity assumptions. The main result in[86, 88] establishes that for a generic integrand \(f \in \mathcal{F}\) the corresponding optimal control problem is well posed. In this chapter based on[89] we study the set of all integrands \(f \in \mathcal{F}\) for which the corresponding optimal control problem is well posed. We show that the complement of this set is not only of the first category but also of a σ-porous set.
Alexander J. Zaslavski
Chapter 4. Well-posedness of Nonconvex Variational Problems
Abstract
In this chapter based on [92,93] we study variational problems in which the values at the end points are also subject to variations. Using the Baire category approach and the porosity notion we show that most variational problems are well posed.
Alexander J. Zaslavski
Chapter 5. Generic Well-posedness Result for a Class of Optimal Control Problems
Abstract
In this chapter we prove a generic existence and uniqueness result for a class of optimal control problems in which the right-hand side of differential equations is also subject to variations as well as the integrands.
Alexander J. Zaslavski
Chapter 6. Nonoccurrence of the Lavrentiev Phenomenon for Variational Problems
Abstract
In this chapter we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex nonautonomous constrained variational problems. A state variable belongs to a convex subset H of a Banach space X with nonempty interior. Integrands belong to a complete metric space of functions \(\mathcal{M}_{B}\) which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. This space will be described below. In [97] we considered a class of nonconstrained variational problems with integrands belonging to a subset \(\mathcal{L}_{B} \subset \mathcal{M}_{B}\) and showed that for \(f \in \mathcal{L}_{B}\) the following property holds:
Alexander J. Zaslavski
Chapter 7. Nonoccurrence of the Lavrentiev Phenomenon in Optimal Control
Abstract
In this chapter we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex optimal control problems which is identified with the corresponding complete metric space of integrands \(\mathcal{M}\) which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. We establish that for most elements of \(\mathcal{M}\) (in the sense of Baire category) the infimum on the full admissible class of trajectory-control pairs is equal to the infimum on a subclass of trajectory-control pairs whose controls are bounded by a certain constant.
Alexander J. Zaslavski
Chapter 8. Generic Nonoccurrence of the Lavrentiev Phenomenon
Abstract
In this chapter we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex optimal control problems. We show that for most problems (in the sense of Baire category) the infimum on the full admissible class of trajectory-control pairs is equal to the infimum on a subclass of trajectory-control pairs with bounded controls.
Alexander J. Zaslavski
Chapter 9. Infinite-Dimensional Linear Control Problems
Abstract
In this chapter we show nonoccurrence of gap for two large classes of infinite-dimensional linear control systems in a Hilbert space with nonconvex integrands. These classes are identified with the corresponding complete metric spaces of integrands which satisfy a growth condition common in the literature. For most elements of the first space of integrands (in the sense of Baire category) we establish the existence of a minimizing sequence of trajectory-control pairs with bounded controls. We also establish that for most elements of the second space (in the sense of Baire category) the infimum on the full admissible class of trajectory-control pairs is equal to the infimum on a subclass of trajectory-control pairs whose controls are bounded by a certain constant.
Alexander J. Zaslavski
Chapter 10. Uniform Boundedness of Approximate Solutions of Variational Problems
Abstract
In this chapter, given an \(x_{0} \in {R}^{n}\) we study the infinite horizon problem of minimizing the expression \(\int _{0}^{T}f(t,x(t),x^{\prime}(t))dt\) as T grows to infinity where \(x : [0,\infty ) \rightarrow {R}^{n}\) satisfies the initial condition x(0) = x 0. We analyze the existence and properties of approximate solutions for every prescribed initial value x 0.
Alexander J. Zaslavski
Chapter 11. The Turnpike Property for Approximate Solutions of Variational Problems
Abstract
In this chapter we study the structure of approximate solutions of variational problems with continuous integrands \(f : [0,\infty ) \times {R}^{n} \times {R}^{n} \rightarrow {R}^{1}\) which belong to a complete metric space of functions \(\mathfrak{M}\). We do not impose any convexity assumption and establish the existence of an everywhere dense G δ -set \(\mathcal{F}\subset \mathfrak{M}\) such that each integrand in \(\mathcal{F}\) has the turnpike property.
Alexander J. Zaslavski
Chapter 12. A Turnpike Result for Discrete-Time Optimal Control Systems
Abstract
In this chapter we study a turnpike property of approximate solutions for a general class of discrete-time control systems without discounting and with a compact metric space of states. This class of control systems is identified with a complete metric space of objective functions. We show that for a generic objective function approximate solutions of the corresponding control system possess the turnpike property.
Alexander J. Zaslavski
Backmatter
Metadaten
Titel
Nonconvex Optimal Control and Variational Problems
verfasst von
Alexander J. Zaslavski
Copyright-Jahr
2013
Verlag
Springer New York
Electronic ISBN
978-1-4614-7378-7
Print ISBN
978-1-4614-7377-0
DOI
https://doi.org/10.1007/978-1-4614-7378-7

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