Polytopes of Nonnegative Polynomials

Algebraic, recursive, and finite tests are proposed to verify if a convex polytope in the parameter space contains only axis or circle nonnegative polynomials. An extraordinary numerical simplicity of the tests is a consequence of the fact that the nonnegativity regions in the parameter space are shown to be convex, and it suffices to check only the vertex polynomials of the polytope. The tests are applied to robustness analysis of absolute stability of nonlinear continuous and discrete systems, optimality of LQ regulators, positive realness of rational functions and matrices, and positivity of polynomial matrices appearing in stability criteria for 2-D polynomials.