By M.I. Zelikin, S.A. Vakhrameev

ISBN-10: 3642086039

ISBN-13: 9783642086038

The one monograph at the subject, this booklet matters geometric equipment within the idea of differential equations with quadratic right-hand facets, heavily with regards to the calculus of adaptations and optimum keep an eye on concept. in line with the author’s lectures, the ebook is addressed to undergraduate and graduate scholars, and clinical researchers.

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Read or Download Control Theory and Optimization I: Homogeneous Spaces and the Riccati Equation in the Calculus of Variations (Encyclopaedia of Mathematical Sciences) PDF

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Extra info for Control Theory and Optimization I: Homogeneous Spaces and the Riccati Equation in the Calculus of Variations (Encyclopaedia of Mathematical Sciences)

Example text

In dealing with uncertain outcomes, von Neumann and Morgenstern8 suggested to construct a real valued function u: R1 + Rl so that one alternative, represented by random variable X, is preferred to the other alternative, represented by random variable Y iff Eu(X) ) Eu(Y). Such a real valued function is known as a utility function for the preference over uncertain outcomes. 1 As noted earlier, much research has been devoted to studying the existence conditions of such a utility function. conditions must be extremely strict.

The basic idea of the nondominated set is to narrow the set of all available alternatives down to a set which contains the optimal choice by eliminating the inferior alternatives that are dominated by at least one alternative in the set. 5. (ii) (iii) N2 (o, s 2 ) • N3 (o, mv) if 0 is a class of normally distributed random variables. In the following figure we summarize relationships among the four dominance concepts. ___x_a Y___. 2 References 1. Fiahburn, P. , Utility Theory for Decision Making, John Wiley and Sons, New York, New York, 1970.

Some Relationships among Different Dominance Concepts In this section we investigate relationships among the four domi- nance concepts discussed in the previous sections. 1. 1 (i) X s 1 Y iff X u1 Y (ii) X s 2 Y iff X Proof. u2 Y See Radar and Russe~ 10 • 26 Hanoch and Levy, 9 and Bawa. 1 erence in the infinite dimensional space. alence. arge unless 0 is small and/or with some special structures. 2 .. _f.. x\t) • Pr[Y ~ tiX • t]. Then Ex[Fylx(X)] • FYIX(t)dFx(t) ~ ß is a necessary condition for X ß Y.

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Control Theory and Optimization I: Homogeneous Spaces and the Riccati Equation in the Calculus of Variations (Encyclopaedia of Mathematical Sciences) by M.I. Zelikin, S.A. Vakhrameev


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