In many problems in control, optimal and robust control, one has to solve global optimization problems of the form: $\mathbf{P}:f^\ast=\min_{\mathbf x}\{f(\mathbf x):\mathbf x\in\mathbf K\}$, or, equivalently, $f^\ast=\max\{\lambda:f-\lambda\geq0\text{ on }\mathbf K\}$, where $f$ is a polynomial (or even a semi-algebraic function) and $\mathbf K$ is a basic semi-algebraic set. One may even need solve the "robust" version $\min\{f(\mathbf x):\mathbf x\in\mathbf K;h(\mathbf x,\mathbf u)\geq0,\forall \mathbf u\in\mathbf U\}$ where $\mathbf U$ is a set of parameters. For instance, some static output feedback problems can be cast as polynomial optimization problems whose feasible set $\mathbf K$ is defined by a polynomial matrix inequality (PMI). And robust stability regions of linear systems can be modeled as parametrized polynomial matrix inequalities (PMIs) where parameters $\mathbf u$ account for 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 $\mathbf K\subset\mathbb R^n$. The first type of characterization is when knowledge on $\mathbf K$ is through its defining polynomials, i.e., $\mathbf K=\{\mathbf x:g_j(\mathbf x)\geq 0, j =1,\dots, m\}$, in which case some powerful certificates of positivity can be stated in terms of some sums of squares (SOS)-weighted representation. For instance, this allows to define a hierarchy fo semidefinite relaxations which yields a monotone sequence of lower bounds converging to $f^\ast$ (and in fact, finite convergence is generic). There is also another way of looking at nonnegativity where now knowledge on $\mathbf K$ is through moments of a measure whose support is $\mathbf K$. In this case, checking whether a polynomial is nonnegative on $\mathbf K$ 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 $\mathbf P$, 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 $\mathbf K$.

We investigate various properties of the sublevel set $\{x : g(x) \leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$ are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(- g)$ on the whole space $\mathbb{R}^n$ (a non-Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of its coefficients.

In fact, whenever $h$ is non-negative, the functional $\int \phi(g)h dx$ is a convex function of $g$ for a large class of functions $\phi:\mathbb{R}_{+} \to \mathbb{R}$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently, to approximate the associated non-Gaussian integral). We also show that finding the sublevel set $\{x : g(x) \leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non-Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.