On boundary implications of stability and positivity properties of multidimensional systems

Multidimensional generalizations of various 1-D results on the robustness of Hurwitz, Schur, and positivity properties of polynomials and rational functions are considered. More specifically, the convexity property of the stable region in the coefficient space of multivariable polynomials is studied. Multidimensional generalizations of Kharitonov-type results are reviewed, and further extensions, including that of the 1-D edge theorem, are discussed. Interval positivity properties of multivariate rational functions are characterized in terms of ratios of a finite number of Kharitonov-type polynomials constructed from the extreme values of the intervals of perturbation.