On a theorem of Redheffer concerning diagonal stability

An important problem in system theory concerns determining whether or not a given LTI system (x) over dot = Ax is diagonally stable. More precisely, this problem is concerned with determining conditions on a matrix A such that there exists a diagonal matrix D with positive diagonal entries (i.e. a positive diagonal matrix), satisfying A(T)D + DA = -Q < 0, Q = Q(T) > 0. While this problem has attracted much attention over the past half century, two results of note stand out: (i) a result based on Theorems of the Alternative derived by Barker et al.; and (ii) algebraic conditions derived by Redheffer. This paper is concerned with the second of these conditions. Our principal contribution is to show that Redheffer's result can be obtained from the Kalman-Yacubovich-Popov lemma. We then show that this method of proof leads to natural generalisations of Redheffer's result and we use these results to derive new conditions for diagonal and Hurwitz stability for special classes of matrices. (c) 2009 Elsevier Inc. All rights reserved.