Investigation of the global positivity of polynomials via linear optimization

In this paper the robust positivity of multivariate polynomials under coefficient perturbation is investigated. This robust positivity of multivariate polynomials can be used for polynomial systems in order to determine the robust asymptotic stability of the system. It is assumed that the polynomials under investigation are linearly dependent on some parameters. The aim in this article is to determine the parameter perturbation region as a hypersphere, for which the polynomial is globally positive. The theorem of Ehlich and Zeller and linear optimization are used together to achieve this aim. The theorem of Ehlich and Zeller enables to give conditions in the parameter space for global positivity. These conditions are linear inequalities. By means of these inequalities a linear optimization problem is defined and solved for the relevant perturbation region which is a hypersphere. Two nontrivial examples conclude the paper and show the effectiveness of the presented method.