Sufficient conditions for Schur and Hurwitz diagonal stability of complex interval matrices

Let C=A+jB be a complex interval matrix, where A,B⊂Rn×n are real interval matrices, and j2=−1; denote by D+ the set of positive definite diagonal n×n matrices. The Schur [respectively Hurwitz] diagonal stability of C, abbreviated as SDS [respectively HDS], is defined in terms of the Stein [respectively Lyapunov] inequality associated with a generic matrix C∈C; the inequality must have a common solution Q∈D+ for all C∈C. The paper develops two types of results for the SDS [respectively HDS] analysis, which exploit different properties of C. Each type of results presents (i) a sufficient condition relying on the solvability of a matrix norm inequality (SDS analysis) or a matrix measure inequality (HDS analysis); (ii) a construction procedure for a concrete matrix that satisfies the inequality at (i), and, concomitantly, yields a common solution Q∈D+. For some particular cases, both necessary and sufficient conditions can be derived. The Schur and Hurwitz approaches are developed in parallel, for concision reasons, as well as for emphasizing the similarities. Numerical examples illustrate the applicability of the theoretical results.