Bounding Polynomials and Rational Functions in the Tensorial and Simplicial Bernstein Forms

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

Convergence and Inclusion Isotonicity of the Tensorial Rational Bernstein Form

A method is investigated by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials in Bernstein polynomials. Convergence of the bounds to the range with respect to degree elevation of the Bernstein expansion, to the width of the box and to subdivision are proven and the inclusion isotonicity of the related enclosure function is shown.publishe

Convergence of the Simplicial Rational Bernstein Form

Abstract. Bernstein polynomials on a simplex V are considered. The expan-sion of a given polynomial p into these polynomials provides bounds for the range of p over V. Bounds for the range of a rational function over V can easily obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. In this paper it is shown that these bounds con-verge monotonically and linearly to the range of the rational function if the degree of the Bernstein expansion is elevated. If V is subdivided then the con-vergence is quadratic with respect to the maximum of the diameters of the subsimplices

Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

In this paper, we provide tight linear lower bounding functions for multivariate polynomials given over boxes. These functions are obtained by the expansion of polynomials into Bernstein basis and using the linear least squares function. Convergence properties for the absolute difference between the given polynomials and their lower bounds are shown with respect to raising the degree and the width of boxes and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate continuous rational function based on the Bernstein control points, the convex hull of a non-positive polynomial s, and degree elevation. Numerical comparisons with the well-known Bernstein constant lower bounding function are given. Finally, with these affine functions, the positivity of polynomials and rational functions can be certified by computing the Bernstein coefficients of their linear lower bounds