In many problems in control, optimal and robust control, one has to solve global optimization problems of the form: P : f* = minx { f(x) : x ∈ K}, or, equivalently, f* = max{λ : f - λ ≥ 0 on K}, where f is a polynomial (or even a semi-algebraic function) and K is a basic semi-algebraic set. One may even need solve the "robust" version min{f(x) : x ∈ K; h(x; u) ≥ 0, ∀u ∈ U} where U is a set of parameters. For instance, some static output feedback problems can be cast as polynomial optimization problems whose feasible set K is dened by a polynomial matrix inequality (PMI). And robust stability regions of linear systems can be modeled as parametrized polynomial matrix inequalities (PMIs) where parameters u account or uncertainties and (decision) variables x are the controller coefficients.

Therefore, to solve such problems one needs tractable characterizations of polynomials (and even semi-algebraic functions) which are nonnegative on a set, a topic of independent interest and of primary importance because it also has implications in many other areas.

We will review two kinds of tractable characterizations of polynomials which are non- negative on a basic closed semi-algebraic set K ⊂ Rn. The rst type of characterization is when knowledge on K is through its dening polynomials, i.e., K = {x : gj(x) ≥ 0; j = 1, . . . ,m}, in which case some powerful certicates of positivity can be stated in terms of some sums of squares (SOS)-weighted representation. For instance, this allows to dene a hierarchy fo semidenite relaxations which yields a monotone sequence of lower bounds converging to f* (and in fact, nite convergence is generic). There is also another way of looking at nonnegativity where now knowledge on K is through moments of a measure whose support is K. In this case, checking whether a polynomial is nonnegative on K reduces to solving a sequence of generalized eigenvalue problems associated with a count- able (nested) family of real symmetric matrices of increasing size. When applied to P, this results in a monotone sequence of upper bounds converging to the global minimum, which complements the previous sequence of upper bounds. These two (dual) characterizations provide convex inner (resp. outer) approximations (by spectrahedra) of the convex cone of polynomials nonnegative on K.