Improvement on delay dependent absolute stability of Lurie control systems with multiple time delays Phan T. Nam a, * , Pubudu N. Pathirana b a Department of Mathematics, Quynhon University, Binhdinh, Vietnam b School of Engineering, Deakin University, Geelong, Australia article info Keywords: Lurie control system Absolute stability Delay-dependent Linear matrix inequality abstract Absolute stability of Lurie control systems with multiple time-delays is studied in this paper. By using extended Lyapunov functionals, we avoid the use of the stability assump- tion on the main operator and derive improved stability criteria, which are strictly less con- servative than the criteria in [2,3]. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction Consider the following Lurie control system with multiple time-delays: _ xðtÞ¼ AxðtÞþ P m i¼1 B i xðt  s i Þþ bf ðrðtÞÞ; rðtÞ¼ c T xðtÞ; xðhÞ¼ /ðhÞ; h 2 ½ maxfs i g; 0; 8 > > > < > > > : ð1:1Þ where xðtÞ2 R n is the state, A; B i ði ¼ 1; 2; ... ; mÞ2 R nn ; b; c 2 R n ; s i P 0 ði ¼ 1; 2; ... ; mÞ and initial condition is x 0 ðhÞ¼ /ðhÞ2 Cð½maxfs i g; 0; R n Þ. The nonlinearity f() satisfies f ðÞ 2 K½0; 1 ¼ ff ðÞjf ð0Þ¼ 0; 0 < rf ðrÞ < 1; r – 0g: In the paper [2], the authors considered absolute stability of Lurie control system with multiple time-delays (1.1). The proposed result is less conservative due to decomposing the matrix B i ¼ B i1 þ B i2 ði ¼ 1; 2; ... ; mÞ and using the operator Dðx t Þ¼ xðtÞþ P m i¼1 B i1 R t ts i xðsÞds to represent the system in the form of a descriptor system with discrete and distributed de- lays. However, their condition is still dependent on the stability of Dðx t Þ. Therefore, the main purpose of this paper is to re- duce the stability of Dðx t Þ. Hence, we will get an improvement on the result in [2]. The following lemma is needed for our main result. Lemma 1.1 [1]. Assume that S 2 R nn is a symmetric positive definite matrix. Then for every Q 2 R nn , 2hQy; xihSy; yi 6 hQS 1 Q T x; xi; 8x; y 2 R n : If we take S ¼ I then we have j2hQy; xij 6 kyk 2 þkQxk 2 . We also need the main theorem in [2] and its proof for our main result. 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.090 * Corresponding author. E-mail address: pthnam@yahoo.com (P.T. Nam). Applied Mathematics and Computation 216 (2010) 1024–1027 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc