By Alberto Bressan, Benedetto Piccoli

ISBN-10: 1601330022

ISBN-13: 9781601330024

This e-book presents an advent to the mathematical conception of nonlinear regulate structures. It includes many subject matters which are often scattered between various texts. The e-book additionally provides a few subject matters of present examine, that have been by no means earlier than integrated in a textbook. This quantity will function a terrific textbook for graduate scholars. it really is self-contained, with numerous appendices masking a large mathematical history. scholars should be aided by means of its lucid exposition. greater than a hundred figures and a hundred routines were inserted, supporting the readers to appreciate the main geometric rules and construct their instinct. For technological know-how or engineering scholars, this publication offers a richly illustrated evaluate of the elemental options and leads to the speculation of linear and nonlinear keep watch over. extra mathematically orientated scholars can use this article as an invaluable advent, ahead of tackling extra complicated, learn orientated monographs

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**Extra resources for Introduction to the mathematical theory of control**

**Example text**

Here, all concavity concepts are taken with respect to i" . 49 6. LocaL versus gLobaL optimaLity and related remarks There are three well-known results which immediately enter one's mind if quasi concavity or semistrict quasiconcavity is mentioned in connection with optimization: (1) The feasible set defined by a set of inequalities of the form g(X) ;;. 0, where the components ,91(-) are quasiconcave, is convex. (2) The set of optimal points in a (scalar) maximization problem is convex if the objective function is quasiconcave.

To complete the proof, suppose that J{x) i- J{y). ) Choose any aE(O,1}. Then, i[J{x)] l= i[J{y)], by definition because J{x) i- J{y); i[J{xay)] l;;. i[(J{x)aJ{y)], because fis concave; i[J{x)aJ{y)] l;;. i[J{x)]ai[J{y)], by (2) and (3) of Lemma 1; i[J{x)]ai[J{y)] l= i[J{y)]ai[J{y)], since i[J{x)] l= i[J{y)]; i[J{y)]ai[J{y)] l= i[J{y)], which is immediate. Hence, i[J{xay)] l;;. i[J{y)], which means that J{xay) i;;. J{Y), by definition. The proof is complete. 0 46 Corollary. In view of the above proof it is immediately clear that concave and quasi concave can be replaced by *-concave and *-quasiconcave in the above theorem, where * is any of the folLoVlling concepts: a (ae(O, 1) predefined), rationally (all a E (0,1 )nQ) , midpoint (a=~) .

Moreover, there stiLL remains the possibility of reducing problem (6) to a set of consecutive maxmin problems if the individual functions are quasi concave 41 and semistrictLy quasiconcave. JL(-» := (h1(,u1(-»'··· ,~(,uk(-)W is used in probLem (6) instead of 11(-), 'Nhere the hi'S are strictLy monotone increasing. S-shaped functions do have this property. PracticaL experiences reported by LeberLing [19], Zimmermann and Zysno [32], and others (see Werners [30)) show that a decision-maker's behaviour towards fuzzy probLems usuaLLy is of such an "S-shaped" nature.

### Introduction to the mathematical theory of control by Alberto Bressan, Benedetto Piccoli

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