nach oben

2000 | Buch

Kapitel lesen Erstes Kapitel lesen

Optimality Conditions: Abnormal and Degenerate Problems

verfasst von: Aram V. Arutyunov

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

This book is devoted to one of the main questions of the theory of extremal prob lems, namely, to necessary and sufficient extremality conditions. It is intended mostly for mathematicians and also for all those who are interested in optimiza tion problems. The book may be useful for advanced students, post-graduated students, and researchers. The book consists of four chapters. In Chap. 1 we study the abstract minimization problem with constraints, which is often called the mathemati cal programming problem. Chapter 2 is devoted to one of the most important classes of extremal problems, the optimal control problem. In the third chapter we study one of the main objects of the calculus of variations, the integral quadratic form. In the concluding, fourth, chapter we study local properties of smooth nonlinear mappings in a neighborhood of an abnormal point. The problems which are studied in this book (of course, in addition to their extremal nature) are united by our main interest being in the study of the so called abnormal or degenerate problems. This is the main distinction of the present book from a large number of books devoted to theory of extremal problems, among which there are many excellent textbooks, and books such as, e.g., [13, 38, 59, 78, 82, 86, 101, 112, 119], to mention a few.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Extremal Problems with Constraints

Abstract

Let two vector spaces X and Y, a mapping F : X → Y, a scalar-valued function f ₀ : X → R ¹, and a convex cone C ⋸ Y be given. We consider the problem

$${f_0}(x) \to \min |x:F(x) \in C $$

(1.1)

Aram V. Arutyunov

Chapter 2. Optimal Control Problem. Pontryagin maximum Principle

Abstract

We study the following general optimal control problem:

$$\dot x = f(x,u,t)\quad t \in [{t_1},{t_2}],{t_1} < {t_{2,}}$$

(1.1)

$$u = ({u_1},{u_2}),\quad {u_2}(t) \in {U_2}(t)\forall t,$$

(1.2)

$$R(x,{u_1},t) \leqslant 0$$

(1.3)

$$G(x,t) \leqslant 0,$$

(1.4)

$${K_1}(p) \leqslant 0,\quad {K_2}(p) = 0,$$

(1.5)

$$p = ({x_1},{x_2},{t_1},{t_2}),\quad {x_1} = x({t_1}),{x_2} = x({t_2})$$

(1.6)

$$J = J(p,u) = {K_0}(p) + \int\limits_{{t_1}}^{{t_2}} {{f^0}} (x,u,t)dt \to \min .$$

(1.7)

Aram V. Arutyunov

Chapter 3. Degenerate Quadratic Forms of the Calculus of Variations

Abstract

When solving problems of the classical calculus of variations and examining the solution of the Euler equation obtained via second-order conditions, there arises the problem of verifying the positive semi-definitness of the integral quadxatic form. This form looks as follows:

$$U(x) = \int_0^1 {\left\langle {A(t)\dot x(t),\dot x(t)} \right\rangle \quad + \left\langle {B(t)x(t),x(t)} \right\rangle \quad + 2\left\langle {C(t)\dot x(t),x(t)} \right\rangle dt\quad + \left\langle {\Omega (x(0),x(1)),(x(0),x(1))} \right\rangle } $$

(1.1)

Aram V. Arutyunov

Chapter 4. Study of Mappings in a Neighborhood of an Abnormal Point

Abstract

Let a vector space X, a Banach space Y, and a smooth mapping F: X × Y → Z, where Z = R ^k is the k-dimensional arithmetical space, be given¹. Also, let points x ₀ ∈ X, y ₀ ∈ Y, and z ₀ ∈ Z satisfying the condition F(x ₀, y ₀) = z ₀ be given. For an arbitrary (y, z) lying in a neighborhood of the point (y ₀, z ₀) we consider the equation

$$F(x,y) = z$$

(1.1)

in the unknown x. If the point (x ₀, y ₀) is normal, i.e., if

$$\operatorname{Im} \frac{{\partial F}} {{\partial x}}({x_0},{y_0}) = Z \Leftrightarrow \frac{{\partial F}} {{\partial x}}({x_0},{y_0})X = Z$$

(1.2)

then, according to the classical implicit function theorem (see, e.g., [13]), in a neighborhood of the point (y ₀, z ₀) a mooth mapping ø = ø y, z) that yields a solution to Eq. (1.1) is defined, i.e.,

$$ F(\phi (y,z),y) \equiv z,\quad \phi ({y_0},{z_0}) = {x_0} $$

Aram V. Arutyunov

Backmatter

Titel: Optimality Conditions: Abnormal and Degenerate Problems
verfasst von: Aram V. Arutyunov
Verlag: Springer Netherlands
Electronic ISBN: 978-94-015-9438-7
Print ISBN: 978-90-481-5596-5
DOI: https://doi.org/10.1007/978-94-015-9438-7

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Extremal Problems with Constraints

Chapter 2. Optimal Control Problem. Pontryagin maximum Principle

Chapter 3. Degenerate Quadratic Forms of the Calculus of Variations

Chapter 4. Study of Mappings in a Neighborhood of an Abnormal Point

Backmatter