Available online at www.sciencedirect.com Systems & Control Letters 52 (2004) 283–288 www.elsevier.com/locate/sysconle H ∞ and BIBO stabilization of delay systems of neutral type Jonathan R. Partington a ; ∗ , Catherine Bonnet b a School of Mathematics, University of Leeds, Leeds LS2 9JT, UK b INRIA Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay cedex, France Received 7 March 2003; received in revised form 22 August 2003; accepted 22 September 2003 Abstract Frequency-domain tests for the H∞ and BIBO stability of large classes of delay systems of neutral type are derived. The results are applied to discuss the stabilizability of such systems by nite-dimensional controllers. c  2004 Elsevier B.V. All rights reserved. Keywords: Time-delay system; Robust stabilization; Neutral system; H∞; BIBO stability; Stabilizability 1. Introduction We shall study the stability analysis of delay sys- tems from an input–output point of view (the delay system being specied by a transfer function), and concentrate on delay systems with innitely many poles. Then, as is well known and can be found in the book of Bellman and Cooke [1], the poles occur in chains of three types: 1. Chains of retarded type, where the poles (s n ) satisfy Re s n → -∞, and thus there are only nitely many poles in any right half-plane. 2. Chains of neutral type, where the poles lie in a band centred on the imaginary axis; this is the most delicate case, and the object of our present study. 3. Chains of advanced type, where the poles (s n ) sat- isfy Re s n →∞. * Corresponding author. E-mail addresses: j.r.partington@leeds.ac.uk (J.R. Partington), catherine.bonnet@inria.fr (C. Bonnet). We are interested in questions of stability and stabilizability. The notions on which we shall con- centrate are (a) H ∞ stability, that is, nite L 2 (0; ∞) gain, where the transfer function lies in H ∞ (C + ), the space of functions analytic and bounded in the right half-plane C + ; and (b) BIBO stability, that is, nite L ∞ (0; ∞) gain, where bounded inputs produce bounded outputs. The rst notion is weaker than the second, and usually easier to analyse. We shall concentrate on what is generally regarded as the most delicate case, that of delay systems of neutral type. For results on stability and stabilization of delay systems of retarded and advanced type, we refer the reader to [3,5,6,10,12], for example. We refer to poles in the closed right half-plane C + as unstable poles, and those in the open left half-plane C - as stable poles. However, we shall see that a sys- tem of neutral type can be unstable even if it has only stable poles. Recall that a transfer function G dened on the right half-plane is said to be proper, if sup Re s¿0; |s|¿R |G(s)| ¡ ∞, for some R¿ 0, and to be strictly proper, if lim R→∞ sup Re s¿0; |s|¿R |G(s)| = 0. These extend the 0167-6911/$ - see front matter c  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2003.09.014