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## Über dieses Buch

A. Blaquière: Quelques aspects géométriques des processus optimaux.- C. Castaing: Quelques problèmes de mesurabilité liés à la théorie des commandes.- L. Cesari: Existence theorems for Lagrange and Pontryagin problems of the calculus of variations and optimal control of more-dimensional extensions in Sobolev space.- H. Halkin: Optimal control as programming in infinite dimensional spaces.- C. Olech: The range of integrals of a certain class vector-valued functions.- E. Rothe: Weak topology and calculus of variations.- E.O. Roxin: Problems about the set of attainability.

## Inhaltsverzeichnis

### Quelques Aspects Geometriques des Processus Optimaux

Nous considèrerons un système dynamique dont l'état, au temps t, est représenté par le point ⫘ = (x1, x2, & xn) dans un espace Euclidien à n-dimensions, En. Nous supposerons que le comportement du système, c'est-à-dire I'évolution au cours du temps de ses variables d'état, dépend du choix de certaines règies dans un ensemble règles donné. Pour le moment il ne sera pas nécessaire de préciser les propriétés de cet ensemble de règies; par contre, nous ferons quelques hypothèses en ce qui concerne le comportement du système.
Austin Blaquiere

### Quelques Problemes de Mesurabilite Lies a la Theorie des Commandes

DEFINITION 1 . - Une multi-application (i.e. application multivoque ) Γ d'une espace topologique T dans un espace topologique E est dite semi-continue inférieurement si l'ensemble
$$\Gamma ^ - \Omega = \left\{ {{\text{t}} \in {\text{T/}}\Gamma \,\,\left( {\text{t}} \right) \cap \,\,\,\,\,\,\,\,\Omega \ne \varphi} \right\}$$
est ouvert dans T pour tout ouvert Ω dans E .
Ch. Castaing

### Existence Theorems for Lagrange and Pontryagin Problems of The Calculus of Variations and Optimal Control. More Dimensional Extensions in Sobolev Spaces

Let A be a closed subset of the tx-space E1 × En, t∈E1, x = (x1,…xn)∈En and for each (t, x)∈A, let U(t,x) be a closed subset of the u-space Em, u = (u1,…,um). We do not exclude that A coincides with the whole tx-space and that U coincides with the whole u-space. Let M denote the set of all (t, x, u) with (t, x)∈A. u∈U(t, x). Let f(t, x, u) = (f0, f) = (f0, f1,…, fn) be a continuous vector function from M into En+1. Let Bbe a closed subset of points (t1x1,t2,x2)of E2n+2, x1 = (x1 1,…x1 n, x2 = (x2 1,…x2 n.We shall consider the class of all pairs x(t), u(t), t1 ≤t≤t2, of vector functions x(t), u(t) satisfying the following conditions :
(a)
x(t) is absolutely continuous (AC) in [t1, t2];

(b)
u(t) is measurable in [t1, t2];

(c)
(t,x(t))∈A for every t∈[t1, t2];

(d)
u(t)∈U(t, x(t)) almost everywhere (a.e.) in [t1, t2];

(e)
f0 (t,x(t), u(t)) is L-integrable in[t1, t2];

(f)
dx/dt / f(t, x(t), u(t)) a.ein [t1, t2];

(g)
(t 1,x(t 1), t 2, x(t 2))∈B.

Lamberto Cesari

### Optimal Control as Programming in Infinite Dimensional Spaces

Introduction. The terms “Programming” and “Mathematical Programming” refer here to the usual constrained maximization problem of the type: given a function φ (x) = (φ A (x),…, φk (x)) from En (the Euclidean n-dimensional space) into Ek, find a point x̂ ϵ En such that φ A (x̂) is maximized subject to equality and/or inequality constraints on φ2(x), …, φk(x̂). This problem has received considerable attention and has led to interesting results ranging from the initial work of Lagrange to the more recent studies of Kuhn and Tucker. Calculus of Variations and Optimal Control are also concerned with constrained maximization problems but over given sets of continuous curves instead of finite dimensional Euclidean spaces as in Programming. Programming has always extended a strong influ ence on Calculus of Variations and one of the motivating forces behind the creation of Functional Analysis was to build a bridge between these two fields. The methods of Functional Analysis have been used to derive Euler-Lagrange equation but we do not know any previous successful attempts to apply those methods to the deri vation of the Weierstrass-E test and the Multiplier Rule for the pro blem of Bolza. This is indeed the purpose of the present paper.
In Section I we shall study a mathematical programming problem in infinite dimensional spaces. In Section II we prove that the standard optimal control problem (a generalization of the problem of Bolza) can be casted into a problem of the type studied in Section I, and by applying to this problem the results of Section I we obtain a generalization of the Maximum Principle of Pontryagin which is itself a generalization of the classical Weierstrass-E test and of the Multiplier Rule for the problem of Bolza (including the abnormal case).
Hubert Halkin

### The range of integrals of a certain class of vector-valued functions

Introduction. Let J be a compact interval of R1, say [0, 1]. Consider the metric space (M, ρ), where $${\text{M}}\,{\text{ = }}\,{\text{M}}_{{\text{Rn}}} ({\text{J}})$$ is the space of Lebesgue measurable functions of J into Rn and the metric function ρ is given by
$$\rho \left(\text{f,}\,\text{g} \right) = \mathop{\inf}\limits_{\alpha > 0} \left(\alpha + \mu \left(\text{s:}\mid \,\text{f(s)}\mid\, >\,\alpha\right) \right),\,\, \text{f, g} \in \text{M},$$
(0,1)
where | | stands for a norm in Rn and μ denotes the lebesgue measure in R1. We identify two functions of M if they differ on a set of measure zero.
Czelaw Olech

### Weak Topology and Calculus of Variations

1. Introduction. It might be said that the use of weak compactness in the calculus of variations is implicitly contained in the classical method of proving existence theorems by selecting weakly convergent subsequences. In recent years the use of weak topology in the calculus of variations has become a good deal more explicit and systematic. It is based on the following facts:
I. In any topological space E a real valued function f defined on a compact subset A takes a (absolute) minimum in some point of A if it is lower semi-continuous, and also a maximum if it is continuous.
E. H. Rothe

### Problems About the Set of Attainability

1. Control systems. In this series of lectures we will consider “control systems” described by differential equations of the type
$${\text{dx/dt}}\,\,{\text{ = }}\,{\text{x'}}\,\,{\text{ = }}\,{\text{f}}\left( {{\text{t,}}\,{\text{x,}}\,{\text{u}}} \right),$$
(I.1.1)
where t is a real variable representing the time, x is an n-vector determining the instantaneous state of the (physical) system, and u is an m-vector corresponding to the instantaneous action of a certain control mechanism. This control action is supposed to be adjustable as a function of time u(t), or of the state u(x), or of both: u = u(t, x). Once the control action is prescribed, equation (I.1.1) is a differential equation governing the evolution of the system. A typical problem of control theory, is how to choose the control law u(t, x), in order to achieve a certain goal (for example to reach a given point in minimum time).
In these lectures we will consider only problems in which it is assumed that the control action u(t, x), even if not determined from the beginning, can be adjusted in an exact manner. In other words, we will not be concerned about “stochastic systems”, where some “random functions” (which are neither exactly known nor adjustable) influence the evolution of the system.
E. O. Roxin
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