Positive Forms

Abstract

This chapter addresses the problem of studying positivity of a form, i.e. a polynomial whose terms have all the same degree. This is a key issue which has many implications in systems and control theory. A basic tool for the representation of forms, which is known in the literature as Gram matrix or SMR, is introduced. The main idea is to represent a form of a generic degree through a quadratic form, by introducing suitable base vectors and corresponding coefficient matrices, which are called power vectors and SMR matrices, respectively. It is shown that a positive semidefinite SMR matrix exists if and only if the form is an SOS form. This allows one to establish whether a form is SOS via an LMI feasibility test, which is a convex optimization problem. Hence, sufficient conditions for positivity of forms can be formulated in terms of LMIs. Then, the SMR framework is extended to address the case of matrix forms. Another contribution of this chapter is to show how some problems involving positivity of polynomials over special sets, such as ellipsoids or the simplex, can be cast in terms of unconstrained positivity of a form. Finally, it is shown how the power vectors belonging to assigned linear spaces can be determined via linear algebra operations. This is instrumental to the extraction of solutions in the robustness problems addressed in subsequent chapters.