This study is concerned with the problem of new delay-dependent exponential stability criteria for neural networks (NNs) with mixed time-varying delays via introducing a novel integral inequality approach. Specifically, first, by taking fully the relationship between the terms in the Leibniz-Newton formula into account, several improved delay-dependent exponential stability criteria are obtained in terms of linear matrix inequalities (LMIs). Second, together with some effective mathematical techniques and a convex optimization approach, less conservative conditions are derived by constructing an appropriate Lyapunov-Krasovskii functional (LKF). Third, the proposed methods include the least numbers of decision variables while keeping the validity of the obtained results. Finally, three numerical examples with simulations are presented to illustrate the validity and advantages of the theoretical results.
1. Introduction
Over the course of the past decade, neural networks have become an important area of research and attracted increasing attention due to their extensive applications in many practical systems, such as power systems [1], pattern recognition [2], signal detection [3], landmark recognition [4], and other scientific areas [5–7].
Moreover, it is inevitable to introduce time delay into the signals transmitted among neurons because the processes of transcription and translation are not instantaneous. However, it is a well-known fact that time delay as a source of instability and poor performance usually appears in many dynamical systems, for instance, Cohen-Grossberg neural networks, cellular neural networks, BAM neural network, chaotic neural networks, H∞ filtering, and nonlinear systems [8–53]. Therefore, stability analysis for neural networks with delays has been an attractive subject of research in recent years [8–42, 50–53].
Furthermore, neural networks (NNs) often have a spatial nature due to the presence of many parallel pathways of a variety of axon sizes and lengths. Thus, in order to have a more accurate model, a distributed delay over a certain time of duration needs to be included in NNs such that the distant past has less influence compared to the recent behavior of the state. Therefore, there has been a growing interest in the study of neural networks with discrete and distributed delays during the past two decades. To date, some results on delay-dependent exponential stability for neural networks with mixed time-varying delays have been reported in [18–23, 34–36]. In [18], the authors considered the global asymptotic stability for a class of delayed cellular neural networks with mixed time-varying delays by using LMIs approach, Lyapunov theory, and Leibniz-Newton formula. However, the activation functions in [18] were assumed to be monotonically nondecreasing. In [19], several delay-dependent sufficient conditions are obtained to guarantee the global asymptotic and exponential stability of the addressed neural networks by employing appropriate LKF and linear matrix inequality (LMI) technique. In [20], an exponential stability criterion is proposed by constructing an augmented LKF, where the discrete delay d(t) must be differentiable. In [22, 23], some improved delay-dependent stability criteria are derived in terms of linear matrix inequalities by dividing the discrete delay interval into multiple segments. Different from [35], the appropriate LKF not only divides the discrete delay interval [0,d] into two ones [0,d/2] and [d/2,d], but also divides the discrete delay interval [0,d] into three ones [0,d(t)/2], [d(t)/2,d(t)], and [d(t),d]. Although this approach seems to be effective for achieving less conservative conditions, it can increase the larger numbers of computed variables. Hence, there exists great room for further improvement. To the best of our knowledge, it is of a great significance for the current research to find a more effective approach to get rid of the strict constraint and obtain less conservative conditions.
Motivated by the above discussion, combining effective mathematical techniques and a convex optimization approach, we choose a more general type of LKF to study the delay-dependent exponential stability criteria for neural networks (NNs) with mixed time-varying delays in the paper. Some improved delay-dependent stability conditions derived benefit mainly from using firstly a new integral inequality approach, which is proved to be less conservative than the celebrated Jensen’s inequality and showed having a great potential efficient in practice. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the proposed method. Besides, the main merit of this method lies in containing the least numbers of decision variables while keeping the validity of the obtained results. Finally, the stability criteria obtained turn out to be less conservative than some recently reported ones via three numerical examples.
Notation. Notations used in this paper are fairly standard: Rn denotes the n-dimensional Euclidean space, and Rn×m is the set of all n×m dimensional matrices; I is the identity matrix of appropriate dimensions, and AT is the matrix transposition of the matrix A. By X>0 (resp., X≥0), for X∈Rn×n, we mean that the matrix X is real symmetric positive definite (resp., positive semidefinite); diag{r1,r2,…,rn} denotes block diagonal matrix with diagonal elements ri, i=1,2,…,n, and the symbol ∗ represents the elements below the main diagonal of a symmetric matrix; A→ is defined as A→=A+AT.
2. Preliminaries
Consider the following neural networks with mixed time-varying delays:(1)u˙t=-A0ut+A1gut+A2gut-ht+A3∫t-rtgusds+I,where ut=u1t,u2t,…,untT∈Rn is the neural state vector and gut=g1u1t,g2u2t,…,gnuntT∈Rn is the neuron activation function; I=I1,I2,…,InT∈Rn is an external constant input vector, A0=diaga01,a02,…,a0n>0, and A1, A2, and A3 are the constant matrices of appropriate dimensions.
Assumption A.
The time-varying delay h(t) is continuous and differential function satisfying (2)0≤ht≤h,h˙t≤hD<1.
Assumption B.
For the constants li- and li+ the bounded activation function gi· in (1) satisfies the following condition:(3)li-≤giα-giβα-β≤li+,∀α,β∈R,α≠β,i=1,2,…,n.We denote L+=diag{l1+,l2+,…,ln+}, L-=diag{l1-,l2-,…,ln-}, L1=diag{l1+l1-,l1+l2-,…,ln+ln-}, L2=diagl1++l1-/2,l2++l2-/2,…,ln++ln-/2, and L=diagmaxl1+,l1-,…,maxln+,ln-=diagl1,…,ln.
Under Assumption B, by using Brouwer’s fixed-point theorem [25], it can be easily proven that there exists one equilibrium point for system (1). Assuming that u∗=u1∗,…,un∗T is an equilibrium point of system (1). For convenience, we firstly shift the equilibrium point u∗ to the origin by letting zt=ut-u∗ and fzt=gut-gu∗, and then system (1) can be converted to(4)z˙t=-A0zt+A1fzt+A2fzt-ht+A3∫t-rtfzsds,where fxt=f1z1,f2z2,…,fnznT. It is easy to check that the function fi· satisfies fi0=0, and(5)li-≤fiαα≤li+,∀α∈R,α≠0,i=1,2,…,n.Due to the influence of external factors, ∫t-rtfzsds cannot express the actual state of the accurate information. Therefore, by translating r to function rt0≤rt≤r, we have(6)z˙t=-A0zt+A1fzt+A2fzt-ht+A3∫t-rttfzsds.In the paper, we will attempt to formulate some practically computable criteria to check the global exponential stability of system (6). The following lemmas are useful in deriving the criteria.
Definition 1 (see [<xref ref-type="bibr" rid="B19">22</xref>]).
The equilibrium point 0 of system (6) is said to be globally exponentially stable, if there exist scalars k>0 and β>0 such that(7)zt≤βe-ktsup-h≤s≤0zs,∀t>0.
Lemma 2 (see [<xref ref-type="bibr" rid="B19">22</xref>]).
The following inequalities are true:(8)0≤∫0zitfis-li-sds≤fizit-li-zitzit,0≤∫0zitli+s-fisds≤li+zit-fizitzit.
Lemma 3 (see [<xref ref-type="bibr" rid="B21">24</xref>]).
For any positive definite matrix SGGTS>0 and r≥0, 0≤r(t)≤r, the following inequalities hold:(9)-r∫t-rtαTsSαsds≤-∫t-rttαsds∫t-rt-rtαsdsTSGGTS∫t-rttαsds∫t-rt-rtαsds.
Lemma 4 (see [<xref ref-type="bibr" rid="B40">54</xref>]).
For any constant matrix R∈Rn×n, R=RT>0, a scalar function h≔ht>0, and a vector-valued function z˙:-h,0→Rn such that the following integrations are well:(10)-h∫t-htz˙TsRz˙sds≤ztzt-hT-RRR-Rztzt-h.
Lemma 5 (see [<xref ref-type="bibr" rid="B14">17</xref>]).
For any positive semidefinite matrices,(11)R=R11R12R13R12TR22R23R13TR23TR33≥0,the integral inequality holds as follows:(12)∫t-htz˙TsR33z˙Tsds≤∫t-htztzt-hz˙sTR11R12R13R12TR22R23R13TR23T0ztzt-hz˙sds.
3. Main Results
In this section we will give sufficient conditions under which system (6) is globally exponentially stable.
Theorem 6.
For given scalars r>0, h>0, and 1>hD, the origin system (6) with the neuron activation function fzt satisfying condition (5) and the time-varying delay ht satisfying (2) is globally exponentially stable with the exponential convergence rate index k if there exist P>0, Zi>0i=1,2, Ri>0i=1,2,3,4, diagonal matrices H1=diagh11,h12,…,h1n>0, H2=diagh21,h22,…,h2n>0, H3=diagh31,h32,…,h3n>0, H4=diagh41,h42,…,h4n>0, R=diagr1,r2,…,rn>0, G=diagg1,g2,…,gn>0, any matrices Tii=1,2,3,4, and(13)SMMTS>0,X=X11X12X13X12TX22X23X13TX23TX33>0,Y=Y11Y12Y13Y12TY22Y23Y13TY23TY33>0,R3+1-hDR4-X33>0,R3-Y33>0such that the following symmetric linear matrix inequality holds:(14)Ψ=Ψ11Ψ12Ψ130Ψ15T2A2-A0T4T0T2A30∗Ψ2200Ψ25T1A2-T4T0T1A30∗∗Ψ33Ψ340H3L2000∗∗∗Ψ4400H4L20∗∗∗∗Ψ55T3A2+A1TT4T0T3A30∗∗∗∗∗Ψ660T4A30∗∗∗∗∗∗Ψ7700∗∗∗∗∗∗∗-e-2krS-e-2krM∗∗∗∗∗∗∗∗-e-2krS≤0,where(15)Ψ11=2kP+4kL+G-L-R+R1+R2+e-2khhX11+X13→-e-2khhR3+1-hDR4-X33-H2L1-T2A0→,Ψ12=P+L+G-L-R-A0T1T-T2,Ψ13=e-2khhX12-X13+X23T-e-2khhR3+1-hDR4-X33,Ψ15=2kR-G+H2L2+T2A1-A0T3T,Ψ22=hR3+hR4-T1→,Ψ25=R-G+T1A1-T3,Ψ33=-1-hDR2-e-2khhR3+1-hDR4-X33+R3-Y33+e-2khhX22-X23→+hY11+Y13→-H3L1,Ψ34=e-2khhR3-Y33+e-2khhY12-Y13+Y23T,Ψ44=-R1-e-2khhR3-Y33+e-2khhY22-Y23→-H4L1,Ψ55=Z1+Z3+r2S-H1-H2+T3A1→,Ψ66=-1-hDe-2khZ2-H3+T4A2→,Ψ77=-e-2khZ2-H4.
Proof.
Consider an augmentation of LKF for system (6) as follows:(16)Vzt=V1zt+V2zt+V3zt+V4zt+V5zt+V6zt,where(17)V1zt=e2ktzTtPzt,V2zt=2e2kt∑i=1nri∫0zitfis-li-sds+∑i=1ngi∫0zitli+s-fisds,V3zt=∫t-hte2kszTsR1zsds+∫t-htte2kszTsR2zsds,V4zt=∫-h0∫t+θte2ksz˙TsR3z˙sdsdθ+∫-ht0∫t+θte2ksz˙TsR4z˙sdsdθ,V5zt=∫t-hte2ksfTzsZ1fzsds+∫t-htte2ksfTzsZ2fzsds,V6zt=r∫-r0∫t+θte2ksfTzsSfzsdsdθ.The time derivative of V(zt) along with the trajectory of system (6) is given as (18)V˙zt=V˙1zt+V˙2zt+V˙3zt+V˙4zt+V˙5zt+V˙6zt,where(19)V˙1zt=2ke2ktzTtPzt+2e2ktzTtPz˙t,V˙2zt=4ke2ktfTzt-zTtL-Rzt+zTtL+-fTztGzt+2e2ktfTztR-G+zTtL+G-L-R·z˙t,V˙3zt=e2ktzTtR1+R2zt-e2ktzTt-h·R1zt-h-1-h˙te2ktzTt-htR2zt-ht≤e2ktzTtR1+R2zt-e2ktzTt-h·R1zt-h-1-hDe2ktzTt-htR2zt-ht,V˙4zt=e2ktz˙TthR3+htR4z˙t-∫t-hte2ksz˙TsR3z˙sds-1-h˙t·∫t-htte2ksz˙TsR5z˙sds≤e2kthz˙TtR3+R4·z˙t-∫t-htte2ksz˙TsR3+1-hDR4-X33·z˙sds-∫t-ht-hte2ksz˙TsR3-Y33z˙sds-∫t-htte2ksz˙TsX33z˙sds-∫t-ht-hte2ksz˙Ts·Y33z˙sds≤e2kthz˙TtR3+R4z˙t-e2kt-h∫t-httz˙TsR3+1-hDR4-X33·z˙sds-e2kt-h∫t-ht-htz˙TsR3-Y33z˙sds-e2kt-h∫t-httz˙TsX33z˙sds-e2kt-h∫t-ht-htz˙TsY33z˙sds.
Using Lemma 4, we can have(20)-e2kt-h∫t-httz˙TsR3+1-hDR4-X33z˙sds≤-e2kt-hhztzt-htTR3+1-hDR4-X33-R3+1-hDR4-X33-R3+1-hDR4-X33R3+1-hDR4-X33ztzt-ht=-e2kt-hhzTtR3+1-hDR4-X33zt+2e2kt-hhzTtR3+1-hDR4-X33zt-ht-e2kt-hhzTt-htR3+1-hDR4-X33zt-ht,-e2kt-h∫t-ht-htz˙TsR3-Y33z˙sds≤-e2kt-hhzt-htzt-hTR3-Y33-R3-Y33-R3-Y33R3-Y33zt-htzt-h=-e2kt-hhzTt-htR3-Y33zt-ht+2e2kt-hhzTt-htR3-Y33zt-h-e2kt-hhzTt-hR3-Y33zt-h.Using Lemma 5, we may get(21)-e2kt-h∫t-httz˙TsX33z˙sds≤e2kt-h∫t-httztzt-htz˙sT·X11X12X13X12TX22X23X13TX23T0ztzt-htz˙sds=e2kt-hzTthtX11+X13T+X13zt+2e2kt-hzTthtX12-X13+X23Tzt-ht+e2kt-hzTt-hthtX22-X23T-X23zt-ht≤e2kt-hzTthX11+X13T+X13zt+2e2kt-hzTthX12-X13+X23T·zt-ht+e2kt-hzTt-hthX22-X23T-X23zt-ht,-e2kt-h∫t-ht-htz˙TsY33z˙sds≤e2kt-h∫t-ht-htzt-htzt-hz˙sT·Y11Y12Y13Y12TY22Y23Y13TY23T0zt-htzt-hz˙sds=e2kt-hzTt-hth-htY11+Y13T+Y13zt-ht+2e2kt-hzTt-hth-htY12-Y13+Y23T·zt-h+e2kt-hzTt-hh-htY22-Y23T-Y23zt-h≤e2kt-hzTt-hthY11+Y13T+Y13zt-ht+2e2kt-hzTt-hthY12-Y13+Y23Tzt-h+e2kt-hzTt-hhY22-Y23T-Y23zt-h,V˙5zt≤e2ktfTztZ1+Z2fzt-e2kt-hfTzt-hZ1fzt-h-1-hD·e2kt-hfTzt-htZ2fzt-ht.By Lemma 3, we can obtain(22)V˙6zt≤r2e2ktfTztSfzt-r∫t-rte2ksfTzsSfzsds≤r2e2ktfTztSfzt-e2kt-r∫t-rttfzsds∫t-rt-rtfzsdsT·SMMTS∫t-rttfzsds∫t-rt-rtfzsds.From (5), for any n×n diagonal matrices Hi>0(i=1,2,…,4), the following inequality holds:(23)0≤e2ktzTtLH1Lzt-fTztH1fzt+e2kt-zTtH2L1zt+2zTtH2L2fzt-fTztH2fzt+e2kt-zTt-htH3L1zt-ht+2zTt-htH3L2fzt-ht-fTzt-htH3fzt-ht+e2kt-zTt-hH4L1zt-h+2zTt-hH4L2fzt-h-fTzt-hH4fzt-h.Furthermore, for arbitrary matrices T1, T2, T3, and T4 with appropriate dimensions, we have(24)0=2z˙TtT1+zTtT2+fTztT3+fTzt-htT4·-z˙t-A0zt+A1fzt+A2fzt-ht+A3∫t-rttfzsds.The combination of (19)–(24) gives(25)Vxt≤ξTtΨξt,where ξt=[ztz˙tzt-htzt-hfztfzt-htfzt-h∫t-rttfzsds∫t-rt-rtfzsds].
From (14), we know that V˙zt<0, which means the asymptotically stability of system (5). This completes the proof.
Furthermore, setting d=maxh,r, we can have(26)V1z0≤λmaxPzs2≤λmaxP·sup-d≤s≤0zs2,V2z0≤2fz0-L-z0TRz0+2L+z0-fz0TGz0≤2λmaxL+-L-λmaxR+λmaxG·sup-d≤s≤0zs2,V3z0≤hλmaxR1+λmaxR2sup-d≤s≤0zs2,V4z0≤λmaxR3∫-h0∫t+θtz˙Tsz˙sdsdθ+λmaxR4∫-ht0∫t+θtz˙Tsz˙sdsdθ.It is easy to have(27)z˙Tsz˙s=-A0zs+A1fzs+A2fzs-hs+A3∫s-rssfzθdθT·-A0zs+A1fzs+A2fzs-hs+A3∫s-rssfzθdθ=zTsA0TA0zs+fTzsA1TA1fzs+fTzs-hs·A2TA2fzs-hs+∫s-rssfTzθdθA3TA3∫s-rssfzθdθ-2zTsA0TA1fzs-2zTs·A0TA2fzs-hs-2zTs·A0TA3∫s-rssfzθdθ+2fTzs·A1TA2fzs-hs+2fTzs·A1TA3∫s-rssfzθdθ+fTzs-hs·A2TA3∫s-rssfzθdθ.According to 2aTb≤aTXa+bTX-b with X>0, (28)z˙Tsz˙s≤4λmaxA0TA0+λmax2LλmaxA1TA1+λmax2LλmaxA2TA2+r2λmax2LλmaxA3TA3sup-d≤s≤0zs2,V5z0≤hλmax2LλmaxZ1+λmaxZ2·sup-d≤s≤0zs2,V6z0≤rλmaxS·∫-r0∫t+θtfTzsfzsdsdθ≤r32λmax2L·λmaxSsup-d≤s≤0zs2.Thus according to (26)–(28), there exists a positive constant γ such that(29)Vz0<γsup-d≤s≤0zs2,where(30)γ=λmaxP+2λmaxL+-L-λmaxR+λmaxG+hλmaxR1+λmaxR2+2h2λmaxR3+λmaxR4λmaxA0TA0+λmax2LλmaxA1TA1+λmax2LλmaxA2TA2+r2λmax2LλmaxA3TA3+hλmax2L·λmaxZ1+λmaxZ2+r32·λmax2LλmaxS.On the other hand, we have(31)Vzt≥e2ktλminPzt2.Therefore(32)zt≥γλminPe-ktsup-d≤s≤0zs.Then, from Definition 1, system (6) is exponentially stable with convergence rate k, and the proof is completed.
Remark 7.
In the paper, the reduced conservatism of Theorem 6 benefits primarily from a new integral inequality, which is proved to be less conservative than the celebrated Jensen’s inequality, and takes fully the relationship between the terms in the Leibniz-Newton formula within the framework of LMIs into account. In order to lower the conservatism of stability criteria, we further deal with the integral terms of e2kt-h∫t-h(t)tz˙TsR3+1-hDR4-X33z˙sds and e2kt-h∫t-ht-htz˙TsR3-Y33z˙sds via Lemma 4. Different from that of [17], this kind of processing method can reduce ulteriorly the conservatism of stability criteria.
Remark 8.
As a matter of fact, Theorem 6 gives a stability criterion for system (6) with h(t) satisfying 0≤ht≤h, 0≤h˙t≤hD, where hD is given constant. In many cases, hD is unknown. Considering this situation, a rate-independent corollary for the delay ht satisfying 0≤ht≤h is derived by setting R2=0, R4=0, and Z2=0 in the proof of Theorem 6.
Theorem 9.
For given scalars 0<r and h>0, the origin of system (6) with the neuron activation function fzt satisfying condition (5) is globally exponentially stable with the exponential convergence rate index k if there exist P>0, Z1>0, RiT=Ri>0(i=1,3), diagonal matrices H1=diagh11,h12,…,h1n>0, H2=diagh21,h22,…,h2n>0, H3=diagh31,h32,…,r3n>0, H4=diagh41,h42,…,h4n>0, R=diagr1,r2,…,rn>0, G=diagg1,g2,…,gn>0, any matrices Ti(i=1,2,…,4), and(33)SMMTS>0,X=X11X12X13X12TX22X23X13TX23TX33>0,Y=Y11Y12Y13Y12TY22Y23Y13TY23TY33>0,R3-X33>0,R3-Y33>0such that the following symmetric linear matrix inequality holds:(34)Φ=Φ11Φ12Φ130Φ15T2A2-A0T4T0T2A30∗Φ2200Φ25T1A2-T4T0T1A30∗∗Φ33Φ340H3L2000∗∗∗Φ4400H4L20∗∗∗∗Φ55T3A2+A1TT4T0T3A30∗∗∗∗∗Φ660T4A30∗∗∗∗∗∗Φ7700∗∗∗∗∗∗∗-e-2krS-e-2krM∗∗∗∗∗∗∗∗-e-2krS≤0,where(35)Φ11=2kP+4kL+G-L-R+R1+e-2khhX11+X13→-e-2khhR3-X33-H2L1-T2A0→,Φ12=P+L+G-L-R-A0T1T-T2,Φ13=e-2khhX12-X13+X23T-e-2khhR3-X33,Φ15=2kR-G+H2L2+T2A1-A0T3T,Φ22=hR3-T1→,Φ25=R-G+T1A1-T3,Φ33=-1-hDR2-e-2khh2R3-X33-Y33+e-2khhX22-X23→+hY11+Y13→-H3L1,Φ34=e-2khhR3-Y33+e-2khhY12-Y13+Y23T,Φ44=-R1-e-2khhR3-Y33+e-2khhY22-Y23→-H4L1,Φ55=Z1+r2S-H1-H2+T3A1→,Φ66=-H3+T4A2→,Φ77=-e-2khZ2-H4.The other procedure is straight forward from the proof of Theorem 6, so we omit it.
Remark 10.
In the paper, we make full use of the relationship between ∫t-rttfzsds and ∫t-rt-rtfzsds, which can reduce the conservatism of stability criteria once again. However, these useful terms of ∫t-rttfTzsdsM∫t-rt-rtfzsds and ∫t-rt-rtfTzsdsS∫t-rt-rtfzsds were always ignored in [18–20], which may lead to considerable conservatism to certain extent.
Remark 11.
Due to constructing a simple type of Lyapunov-Krasovskii functional and taking full advantage of effective mathematical techniques, the conservatism of improved delay-dependent stability criteria obtained is reduced to a great degree in this study. Compared with those in previous articles [22, 23], we employ a few free variables and do not use a delay decomposition method and add some zero terms, that is, not only dividing the discrete delay interval [0,d] into two ones [0,d/2] and [d/2,d], but also dividing the discrete delay interval [0,d] into three ones [0,d(t)/2], [d(t)/2,d(t)], and [d(t),d] and adding the following equalities:(36)2e2kt-d/2ζ1TtUzt-dt2-zt-d2-∫t-d/2t-dt/2z˙sds=0,2e2kt-d/2ζ1TtVzt-d4-zt-dt2-∫t-d/2t-d/4z˙sds=0,2e2kt-d/2ζ1TtHzt-zt-d4-∫t-d/4tz˙sds=0,2e2kt-dζ1TtWzt-dt-zt-d-∫t-dt-dtz˙sds=0,2e2kt-dζ1TtMzt-d2-zt-dt-∫t-dtt-d/2z˙sds=0,2e2kt-dd2ζ1TtZζ1t-∫t-dt-dtζ1TtZζ1tds-∫t-dtt-d/2ζ1TtZζ1tds=0,2e2kt-d/2d2ζ1TtYζ1t-∫t-d/2t-dt/2ζ1TtYζ1tds-∫t-dt/2t-d/4ζ1TtYζ1tds-∫t-d/4tζ1TtYζ1tds=0.By using this method, the conservatism of the obtained stability condition in [22, 23] is reduced to some degree. However, the computing complexity is also improved since more variables are involved. Besides, we provided a comparison of the numbers of the variables involved in [22, 23] and our paper in Table 1. From Table 1, it is clear to see that the number of decision variables in our paper is much less than those in [22, 23]. Thus, it also expounds validity and applicability of the proposed method.
Comparison of the numbers of the involved variables.
Method/number of decision variables
[22]
[23]
This paper
Theorem 6
41n2+14n
52n2+18n
14n2+13n
Theorem 9
792n2+252n
1012n2+332n
252n2+232n
Remark 12.
In many actual applications, maximum allowable time-delay upper bounds h are of interest. In Theorems 6 and 9, with a fixed hD and r, k can be obtained through following optimization procedure:(37)Maximizeh,Subject to14 or 34.
Besides, maximum allowable time-delay upper bounds k obtained are very valuable. In Theorems 6 and 9, with a fixed hD and r, h can be also acquired through following optimization procedure:(38)Maximizek,Subject to14 or 34.Inequalities (37) and (38) are a convex optimization problem and can be obtained efficiently by using the MATLABLMI Toolbox.
4. Numerical Examples
In this section, three examples are given to demonstrate the feasibility and effectiveness of the main results derived above.
Example 1.
Consider a delayed neural network in (6) with parameters as follows:(39)A0=600050007,A1=1.2-0.80.60.5-1.50.7-0.8-1.2-1.4,A2=-1.40.90.5-0.61.20.80.5-0.71.1,A3=1.80.7-0.81.80.7-0.8-0.4-0.61.2.Case A (set L-=diag-1.2,0,-2.4, L+=diag0,1.4,0). For different h and hD, the allowable upper bounds of the exponential convergence rate index k calculated by Theorem 6 in this paper and Theorem 1 in [20–24] are listed in Table 2. According to Table 2, this example is given to indicate significant improvements over some existing results.
Allowable upper bounds of k for k for Case A in Example 1.
Method
[20]
[21]
[24]
[23]
[22]
Theorem 6
h=0.5, r=0.2, hD=0
0.46
0.58
0.56
0.67
0.80
0.88
h=0.5, r=0.2, hD=0.5
0.21
0.35
0.35
0.45
0.50
0.65
h=0.6, r=0.2, hD=0.5
0.06
0.20
0.33
0.29
0.34
0.45
h=0.8, r=0.2, hD=0.5
0.00
0.05
0.10
0.11
0.14
0.21
Besides, for the parameters listed above, let r=0.2, h=0.5, hD=0.5, and k=0.65. Then we can obtain the following feasible parameters by Theorem 6 in our paper. Due to the limitation of the length of this paper, we only provide a part of the feasible solutions here as follows: (40)P=18.3158-9.17568.2601-9.175615.4632-3.51618.2601-3.516112.6661,Z1=11.60820.1957-0.28760.195711.0661-0.9006-0.2876-0.900611.3939,Z2=19.9317-6.4931-3.0095-6.493121.7679-1.9799-3.0095-1.979930.9316,R1=14.8956-0.47541.0203-0.475414.94580.36531.02030.365317.8456,R2=20.54132.79250.10172.792521.49621.67930.10171.679325.1932,R3=5.6240-3.08142.1045-3.08145.4295-1.28192.1045-1.28195.1175,R4=1.9845-1.32070.9292-1.32071.8417-0.61560.9292-0.61561.6339,T1=4.0278-1.91740.9670-1.61454.2443-0.26591.1456-1.13963.9531,T2=9.0432-0.70860.7264-0.828011.01770.56880.5356-0.977510.0672,T3=-2.05410.5035-1.2211-1.27542.2408-1.71751.01631.5246-3.3814,X11=13.8117-0.59930.7694-0.599313.56600.15500.76940.155014.4814,X12=-1.26960.23760.03130.2855-1.26320.2238-0.13260.1963-0.6930,X13=0.1994-0.46290.4020-0.3868-0.0134-0.12620.4274-0.22640.0949,X22=11.6577-0.96130.8217-0.961311.2995-0.04870.8217-0.048711.7214,X23=0.61720.1352-0.1692-0.00920.88820.01320.01190.00080.9916,X33=3.3075-1.91741.3511-1.91743.0607-0.84791.3511-0.84792.8448,Y11=11.6312-0.98200.8535-0.982011.2608-0.07360.8535-0.073611.6236,Y12=-1.38250.2516-0.06640.2417-1.51930.1776-0.04360.1741-1.3352,Y13=-0.5087-0.13030.1473-0.0614-0.6698-0.03600.1100-0.0658-0.6497,Y22=13.8730-0.49840.5534-0.498413.69080.08970.55340.089714.0093,Y23=-0.19070.3597-0.29020.3244-0.11400.1213-0.28740.1574-0.1037,Y33=2.8308-1.58331.1085-1.58332.6778-0.69061.1085-0.69062.5054.Case B (set L-=diag0,0,0, L+=diag{0.5,0.5,0.5}). When k=0.5 and r=1, the maximum allowable upper bounds h obtained from Theorem 6 for different values hD are listed in Table 3. From Table 3, it can be seen clearly that our results have less conservatism over the results acquired by [20, 22, 34–36] and illustrate the effectiveness and the advantage of the proposed methods. The active functions are chosen as f1zs=0.25zs+1-zs-1, f2zs=0.25zs+1-zs-1, and f3zs=0.25zs+1-zs-1. Moreover, the globally exponentially stable with the initial value [-2.5,-0.5,2.5] is shown in Figures 1–3.
Maximum allowable time delay upper bounds h for Case B in Example 1.
Method
[20]
[34]
[35]
[36]
[22]
Theorem 6
hD=0
2.2931
3.3294
5.8415
7.2820
9.6503
9.8356
hD=0.5
2.1010
2.6363
4.0130
5.8958
8.0394
8.2175
hD=0.8
1.7939
1.7200
3.6574
4.0847
5.8974
6.0561
State trajectories of z(t) in the plane for h=9.8356 in Example 1.
Phase trajectory of model of z(t) in the plane for h=9.8356 in Example 1.
Phase trajectory of model of z(t) in the space for h=9.8356 in Example 1.
Example 2.
Consider a delayed neural network in (6) with parameters as follows:(41)A0=2.30003.40002.5,A1=0.9-1.50.1-1.210.20.20.30.8,A2=0.80.60.20.50.70.10.20.10.5,A3=0.30.20.10.10.20.10.10.10.2.Let L-=diag0,0,0 and let L+=diag0.2,0.2,0.2.
Case A. For h=r the corresponding upper bounds of h for unknown hD obtained by Theorem 9 and the results in [18–20] are listed in Table 4. According to Table 4, this example shows that the stability criterion in this paper gives much less conservative results than those in the previous literatures.
The active functions are set as f1zs=0.1zs+1-zs-1, f2zs=0.1zs+1-zs-1, and f3zs=0.1zs+1-zs-1; the globally exponentially stable with the initial value [-0.4,-0.2,0.5] is shown in Figures 4–6.
Case B. For h=1, the corresponding upper bounds of r for unknown hD obtained by Theorem 9 are listed in Table 5. From Table 5, it is shown clearly that our results have significant improvement over the existing results.
Maximum allowable time delay upper bounds h=r in Example 2.
Method
[19]
[18]
[20]
Theorem 9
h=r
1.833
3.597
5.068
5.220
Maximum allowable time delay upper bounds r for different values k in Example 2.
Method
0.1
0.3
0.5
0.8
[22]
9.082
4.774
3.313
2.217
Theorem 9
9.235
4.963
3.518
2.363
State trajectories of z(t) in the plane for h=5.220 in Example 2.
Phase trajectory of model of z(t) in the plane for h=5.220 in Example 2.
State trajectories of z(t) on the space for h=5.220 in Example 2.
Example 3.
Consider a delayed neural network in (6) with parameters as follows:(42)A0=3.99002.99,A1=1.1880.090.091.188,A2=0.0090.140.050.09,A3=0.45-0.20.30.42.Let L-=diag{0,0} and let L+=diag1,1.
When k=0.5 and r=0.1, the corresponding upper bounds of h for different hD obtained by Theorem 6 are listed in Table 6. The active functions are chosen as f1(z(s))=0.5(|z(s)+1|-|z(s)-1|) and f2(z(s))=0.5(|z(s)+1|-|z(s)-1|); the globally exponentially stable with the initial value [-0.6,0.8] is given in Figures 7–9. Therefore, we can say that the results in this paper are less conservative than those in [24].
Comparison of the maximum delay h between different methods for various hD in Example 3.
hD
0
0.2
0.5
0.7
[24]
3.377
3.336
3.268
3.264
Theorem 6
3.518
3.427
3.358
3.386
State trajectories of z(t) in the plane for h=3.518 in Example 3.
Phase trajectory of model of z(t) in the plane for h=3.518 in Example 3.
State trajectories of z(t) on the space for h=3.518 in Example 3.
5. Conclusions
In this paper, the delay-dependent exponential stability problem for NNs with mixed time-varying delays has been investigated. By using a new integral inequality approach to express the relationship between the terms in the Leibniz-Newton formula within the framework of LMIs for the first time, several less conservative delay-dependent exponential stability criteria are obtained. Moreover, combining effective mathematical techniques and a convex optimization approach, new delay-dependent exponential stability conditions are derived by constructing a proper LKF. Finally, three numerical examples are given to illustrate the feasibility and effectiveness of the proposed methods. The foregoing methods have the potential to be useful for the further study of neural network. Meanwhile, it is expected that these approaches can be further applied to other delayed systems.
Disclaimer
The authors promise that this paper has not been published and will not be simultaneously submitted or published elsewhere.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by National Basic Research Program of China (2010CB732501), National Natural Science Foundation of China (61273015), the National Defense Pre-Research Foundation of China (Grant no. 9140A27040213DZ02001), the Fundamental Research Funds for the Central Universities (ZYGX2014J070), and the Program for New Century Excellent Talents in University (NCET-10-0097).
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