View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 235 (2011) 3086–3095 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Letter to the editor Stability analysis on delayed neural networks based on an improved delay-partitioning approach Tao Li a,∗, Aiguo Song a, Mingxiang Xue b, Haitao Zhang b a School of Instrument Science and Engineering, Southeast University, Nanjing 210096, PR China b Key Laboratory of Measurement and Control of CSE (School of Automation, Southeast University), Ministry of Education, Nanjing 210096, PR China article info a b s t r a c t Article history: In this paper, the asymptotical stability is investigated for a class of delayed neural Received 29 July 2009 networks (DNNs), in which one improved delay-partitioning idea is employed. By choosing Received in revised form 5 April 2010 an augmented Lyapunov–Krasovskii functional and utilizing general convex combination method, two novel conditions are obtained in terms of linear matrix inequalities (LMIs) Keywords: and the conservatism can be greatly reduced by thinning the partitioning of delay intervals. Delayed neural networks (DNNs) Moreover, the LMI-based criteria heavily depend on both the upper and lower bounds on Asymptotical stability time-delay and its derivative, which is different from the existent ones. Though the results Delay-partitioning idea LMI approach are not presented via standard LMIs, they still can be easily checked by resorting to Matlab LMI Toolbox. Finally, three numerical examples are given to demonstrate that our results can be less conservative than the present ones. ' 2010 Elsevier B.V. All rights reserved. 1. Introduction Neural networks have been found many successful applications in various fields due to their strong capability of handling formidable problems and improving systems' performance. Presently, owing to the fact that in biological and artificial neural systems, there inevitably exist integration and communication delays which may induce oscillation, instability, or other poor performances, great efforts have been imposed on stability analysis on neural networks with time-delay, and many elegant results have been proposed in the relevant literature. In recent years, because the Lyapunov–Krasovskii functional method can fully apply the information on time-delay of systems, the delay-dependent stability analysis has become an important topic of primary significance, in which the main purpose is to derive an allowable delay upper bound such that DNNs are asymptotically or exponentially stable [1–19]. Among those present approaches, the delay-partitioning idea has been proven to be more effective and constructive than the earlier ones, which was illustrated in [10–19]. Since Gu [20] initially proposed an effective delay-partitioning idea in 2001, many researchers have employed and further improved Gu's idea to analyze the stability of delayed systems including delayed neural networks [21–23,10]. Later, another novel delay-partitioning idea was put forward in [11] and achieved some developments [12,24–26,13–17], which was proven to be more concise and effective than the ones in [20–23,10]. Moreover, since the idea [11,12,24] cannot effectively deal with the variable delay, some researchers have improved the idea to analyze the stability of time-delayed systems including DNNs in [25,26,13–17], in which time-varying delays were addressed. However, there still exist some following points for the ideas in [25,26,13–17] waiting for further improvements. Firstly, these ideas cannot efficiently analyze the interval variable delay, especially as the lower bound of the delay is available precisely. Secondly, as for the variable delay, the delay-partitioning ideas have not fully employed the information on every ∗ Corresponding author. Tel.: +86 25 83795609; fax: +86 25 83794974. E-mail addresses: [email protected] (T. Li), [email protected] (A. Song). 0377-0427/$ – see front matter ' 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2010.10.002 T. Li et al. / Journal of Computational and Applied Mathematics 235 (2011) 3086–3095 3087 subintervals of delay intervals, which could be illustrated by the constructions of the Lyapunov–Krasovskii functionals [25,26,13–17]. Thirdly, most works in [25,26,13–19] have not taken the lower bound of delay derivative into consideration. In fact, the available lower bound of delay derivative can play an important role in reducing conservatism of stability criteria derived in [27]. Inspired by the above discussion, in this paper, by making great efforts to improve the delay-partitioning idea described in [13–17], we investigate the asymptotical stability for neural networks with interval variable delay, in which the lower bound of delay derivative is involved. Together with one augmented Lyapunov–Krasovskii functional and generalized convex combination technique, two novel conditions are formulated in terms of LMIs and their feasibility can be easily checked with the help of Matlab LMI Toolbox. Finally, three numerical examples are given to illustrate that the proposed results in the paper are less conservative than the existing ones. h XY i hXY i Notations: The symmetric term in a symmetric matrix is denoted by ∗, i.e., D . Y T Z ∗ Z 2. Problem formulations Consider the delayed neural networks described by the following form: zP.t/ D −b.z.t// C Ag.z.t// C Bg.z.t − τ.t/// C L; (1) T n where z DTz1;:::; znU 2 R is a real n-vector denoting the state variables associated with the neurons, b.z/ D T T Tb1.z1/; : : : ; bn.zn/U is the behaved function, g.z/ DTg1.z1/; : : : ; gn.zn/U represents the neuron activation function, T n L DTl1;:::; lnU 2 R is a constant input vector, and A; B are the appropriately dimensional constant matrices. The following assumptions on system (1) are made throughout this paper: H1. Here, τ.t/ denotes the interval time-varying delay satisfying 0 ≤ τ0 ≤ τ.t/ ≤ τm; µ0 ≤τ. P t/ ≤ µm; (2) and introduce τNm D τm − τ0, µN m D µm − µ0. P H2. Each function βi.·/ V R ! R is locally Lipschitz and there exist γi such that βi.z/ ≥ γi > 0 for all z 2 R. Here, we denote Γ D diagfγ1; : : : ; γng. C − · H3. For the constants σj ; σj , the bounded function gj. / in (1) satisfies the following condition − gj(α/ − gj(β/ C σ ≤ ≤ σ ; 8α; β 2 R; α 6D β; j D 1; 2;:::; n; j α − β j N D f C Cg D f − −g and we introduce the denotations Σ diag σ1 ; : : : ; σn , Σ diag σ1 ; : : : ; σn , C C − C C − C − C − nσ1 σ1 σn σn o Σ1 D diag σ σ ; : : : ; σ σ ; Σ2 D diag ;:::; : (3) 1 1 n n 2 2 ∗ DT ∗ ∗UT It is clear that under H1–H3, system (1) has one equilibrium point z z1 ;:::; zn . The equilibrium point of system (1) can be shifted to the origin by the transformation x D z − z∗, which converts system (1) to xP.t/ D −β.x.t// C Af .x.t// C Bf .x.t − τ.t///; (4) T T where x DTx1;:::; xnU is the state vector of transformed system (4), β.x/ DTβ1.x1/; : : : ; βn.xn/U , f .x/ D T UT D C ∗ − ∗ D C ∗ − ∗ D f1.x1/; : : : ; fn.xn/ ; and βi.xi/ b.xi zi / b.zi /, fi.xi/ gi.xi zi / gi.zi /; i 1; 2;:::; n. Note that the function fi.·/ satisfies fi.0/ D 0, and − fi(α/ C σ ≤ ≤ σ ; 8α 2 R; α 6D 0; i D 1; 2;:::; n: (5) i α i Then the problem to be addressed in the paper can be equivalently formulated as developing a condition ensuring that system (4) is asymptotically stable. In order to obtain the stability criterion for system (4), the following lemma will be introduced. Lemma 1 ([28]). Suppose that Ω; Ξ ; Ξ .i D 1; 2/ are the constant matrices of appropriate dimensions, α 2 T0; 1U, and 1i 2i β 2 T0; 1U, then Ω C αΞ11 C .1 − α/Ξ12 C βΞ21 C .1 − β/Ξ22 < 0 holds, if the following inequalities Ω C Ξ11 C Ξ21 < 0; Ω C Ξ11 C Ξ22 < 0; Ω C Ξ12 C Ξ21 < 0, and Ω C Ξ12 C Ξ22 < 0 hold simultaneously. 3. Delay-dependent stability criteria Firstly, we can represent system (4) as xP.t/ D y.t/; y.t/ D −β.x.t// C Af .x.t// C Bf .x.t − τ.t///: (6) 3088 T. Li et al. / Journal of Computational and Applied Mathematics 235 (2011) 3086–3095 − D τ0 D τNm D τ.t/ τ0 For given positive integers m; l, then denoting % m , δ l , ρ.t/ l , and using assumptions H1–H3, we can construct the Lyapunov–Krasovskii functional: V .x.t// D V1.x.t// C V2.x.t// C V3.x.t//; (7) where n Z xi n Z xi n Z xi D T C X − C X T − − U C X T C − U V1.x.t// x .t/Px.t/ 2 qi βi.s/ γis ds 2 ki fi.s/ σi s ds 2 fi σi s fi.s/ ds; iD1 0 iD1 0 iD1 0 Z t [ ]T [ ][ ] Z t−τ0 [ ]T [ ][ ] γ .s/ P1 H1 γ .s/ σ .s/ P2 H2 σ .s/ V2.x.t// D ds C ds g(γ .s// ∗ Q1 g(γ .s// h(σ .s// ∗ Q2 h(σ .s// t−% t−τ0−δ l l t−τ −.i−1)δ T X X Z 0 [ x.s/ ] [X Y ][ x.s/ ] C 1ij 1ij ds f .x.s// ∗ Z1ij f .x.s// iD1 jDi t−τ0−.j−1)δ−ρ.t/ l l t−τ −.i−1)δ−ρ.t/ T X X Z 0 [ x.s/ ] [X Y ][ x.s/ ] C 2ij 2ij ds; f .x.s// ∗ Z2ij f .x.s// iD1 jDi t−τ0−jδ m Z −.i−1/% Z t l l Z −τ0−.i−1)δ Z t X T X X T V3.x.t// D y .s/Viy.s/dsdθ C y .s/Wijy.s/dsdθ iD1 −i% tCθ iD1 jDi −τ0−jδ tCθ with Q D diagfq1;:::; qng, K D diagfk1;:::; kng; F D diagff1;:::; fng, ln × ln constant matrices P1; Q1; H1, mn × mn constant matrices P2; Q2; H2, n × n constant matrices P, Vi, X1ij; Y1ij; Z1ij; X2ij; Y2ij; Z2ij; Wij, and γ T .s/ D xT .s/ ··· xT .s − .m − 1/%/ ; gT (γ .s// D f T .x.s// ··· f T .x.s − .m − 1/%// ; σ T .s/ D xT .s/ ··· xT .s − .l − 1)δ/ ; hT (σ .s// D f T .x.s// ··· f T .x.s − .l − 1)δ// : n hPg Hg i hXgij Ygiji o Denoting a parameter set Φ D P; Q ; K; F; Vd; ; ; Wij; d D 1;:::; mI g D 1; 2I i; j D 1;:::; l , then we ∗ Qg ∗ Zgij give one proposition which is essential to the following proof.