Quantitative Local Analysis of Nonlinear Systems - University of ...

e represented as a sum of squares of finitely many polynomials, i.e., it is a sum-of-squares
(SOS) polynomial, then it trivially follows that the polynomial is globally nonnegative. An
appealing property of sum-of-squares polynomials is that checking whether a polynomial
is sum-of-squares can be formulated as a SDP feasibility problem. Consequently, in the
following sections, the strategy for dealing with global polynomial nonnegativity constraints
will be replacing nonnegativity constraints by sum-of-squares conditions.
We now review useful facts about SOS polynomials. Formally, a polynomial p in x ∈ R n
is said to be SOS if, for p 1 , . . . , p M ∈ R[x] it can be decomposed in the form
p(x) =
M∑
p i (x) 2 . (2.3)
i=1
Obviously, a SOS polynomial is globally nonnegative. Therefore, the set Σ[x] of SOS polynomials
in x (of some fixed degree 3 ), defined as
{
Σ[x] := s ∈ R[x] : ∃M < ∞, p 1 , . . . , p M ∈ R[x] such that s(x) =
}
M∑
p i (x) 2 ,
i=1
is a subset of the set of globally nonnegative polynomials. In fact, Σ[x] is a strict subset
of the set of globally nonnegative polynomials except for univariate polynomials, quadratic
polynomials and quartic polynomials in two variables [53].
Let p be a polynomial in x of degree m and z(x) be a vector of monomials in x up
to degree min{m e ∈ Z : m/2 ≤ m e }. Then, for some symmetric matrix Q, p can be
decomposed as
p(x) = z(x) T Qz(x). (2.4)
Based on this fact, the following theorem provides a characterization of SOS polynomials.
3 We do not specify the degree of polynomials in Σ[x] in notation unless it leads to confusion and use
Σ[x] to denote the set of sum-of-squares polynomials in x ∈ R n of some fixed degree to be inferred from the
context.
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