Polynomial Matrix Inequality and Semidefinite Representation

Consider a convex set S = {x ∈ D : G(x) o 0} where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D ⊆ R is a domain on which G(x) is defined, and G(x) o 0 means G(x) is positive semidefinite. The set S is called semidefinite representable if it equals the projection of a higher dimensional set which is defined by a linear matrix inequality (LMI). This paper studies sufficient conditions guaranteeing semidefinite representability of S. We prove that S is semidefinite representable in the following cases: (i) D = R, G(x) is a matrix polynomial and matrix sos-concave; (ii) D is compact convex, G(x) is a matrix polynomial and strictly matrix concave on D; (iii) G(x) is a matrix rational function and q-module matrix concave on D. Explicit constructions of semidefinite representations are given. Some examples are illustrated.