Type of Publication:  Dissertation 
Publication status:  Published 
URI (citable link):  http://nbnresolving.de/urn:nbn:de:bsz:35221b4w1bae5w20e2 
Author:  Hamadneh, Tareq 
Year of publication:  2018 
Summary: 
This thesis considers bounding functions for multivariate polynomials and rational functions over boxes and simplices. It also considers the synthesis of polynomial Lyapunov functions for obtaining the stability of control systems. Bounding the range of functions is an important issue in many areas of mathematics and its applications like global optimization, computer aided geometric design, robust control etc. The expansion of a given polynomial into Bernstein polynomials provides bounds for the range of this polynomial over the given domain. The Bernstein expansion is used due to the tightness of the enclosure and its rate of convergence to the true range. Bounds for the range of a rational function can easily obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. The main achievements of this thesis are as follows: Firstly, we show that the 'enclosure' bounds converge monotonically and linearly to the range of the (polynomial and rational) function if the degree of the Bernstein expansion is elevated. If the region is subdivided then the convergence is quadratic with respect to the width of the subregions. The inclusion isotonicity and the sharpness property of the related enclosure function are investigated for polynomials and rational functions given over boxes or simplices. We provide a representation for computing the Bernstein coefficients and the enclosure bound of polynomials over a simplex. % Additionally, we study the face values of the simplicial Bernstein form. Algebraic identities certifying the positivity of polynomials and rational functions over boxes and simplices are given, i.e., certificates of positivity in the (tensorial and simplicial) Bernstein basis. Subdivision of a simplex is a widely applied scheme, wherein a starting region is successively subdivided into subregions. It follows that the barycentric Bernstein form is inclusion isotone and the certificates of positivity are local. The Lyapunov stability of systems is addressed over boxes and simplices. We consider the stability by developing algorithms to search for Lyapunov functions that demonstrate stability of control systems. We then extend these algorithms for obtaining the stability of the feedback controller design. Subsequently, the control synthesis problem is reduced to finite number of evaluations of a polynomial within a real bound in the space of parameters representing controls and Lyapunov functions. These results provide theoretical foundations for analysis and design of polynomial control systems, i.e., control systems with polynomial vector fields. Minimization of polynomials and rational functions over a given box or simplex is considered, too. Finally, several new methods are presented for the construction of affine lower bounding functions for polynomials and rational functions over boxes and simplices. The convergence properties of these bounds are shown.

Examination date (for dissertations):  Jan 8, 2018 
Dissertation note:  Doctoral dissertation, University of Konstanz 
MSC Classification:  Numerical Analysis; Control Theory 
Subject (DDC):  510 Mathematics 
Keywords:  Bernstein Basis, Numerical Optimization, Control Theory 
Link to License:  In Copyright 
HAMADNEH, Tareq, 2018. Bounding Polynomials and Rational Functions in the Tensorial and Simplicial Bernstein Forms [Dissertation]. Konstanz: University of Konstanz
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