Convex Invertible Cones, Nevalinna-Pick Interpolation and the Set of Lyapunov Solutions

For a real matrix A whose spectrum avoids the imaginary axis, it is shown that the three following, seemingly independent problems, are in fact equivalent. • Characterizing the set of real symmetric solutions of the algebraic Lyapunov inclusion associated with A. • The image of all Nevalinna-Pick interpolations associated with the spectrum of A. • The structure of the Convex Invertible Cone generated by A. The analogous result for the case where the matrix A is complex and the characterization is of the set of Hermitian solutions to the algebraic Lyapunov inclusion, is addressed as well. The inertia of matrix A whose spectrum avoids the imaginary axis will be called regular. For such a matrix we explore three seemingly independent problems. Due to space limitations, we mostly consider the case where A is real. (a) All Real Symmetric Lyapunov Solutions. Denoting by P (P) the set of Hermitian positive (semi)-definite matrices, let us define the set S(A) of all real symmetric solutions to a Lyapunov inclusion, associated with A, S(A) := {S : (SA+AS) ∈ P}. In this context, there are (at least) two classical questions associated with a given pair of matrices A,B. First, under what conditions S(A) ⋂ S(B) is not empty and second, under what conditions S(A) ⊆ S(B). The first problem was explored in [2] and references therein. In practice, one uses the LMI approach to find a matrix S which belongs to the intersection S(A) ⊆ S(B). Here we focus our attention on the second problem. Relatively little is known on the structure of the set S(A). Clearly, S(αA) = S(A) = S(A−1), (1) for all α > 0. In fact, S(A) is an open convex cone of non-singular real symmetric matrices, for more details see [2, sections III, IV]. Next, recall that the case of equality S(A) = S(B) was characterized by R. Loewy. Theorem 1 :[5]. Let A and B be a pair of matrices with regular inertia. Then S(A) = S(B), if and only if for some α > 0 either B = αA or B = αA−1. In the sequel we shall use H(A), the complex analogous of the set S(A). Namely, for A which is not necessarily real, but with regular inertia, we denote by H(A) the set of all Hermitian solutions to the Lyapunov inclusion, H(A) := {H : (HA+A∗H) ∈ P}. We shall also find it convenient to define a set H̃(A) which strictly contains H(A), H̃(A) := {H : HA+A∗H := Q ∈ P, A∗, Q controllable}. Clearly, all matrices in H̃(A) are non-singular and share the same inertia.