Research Article Global Exponential Stability of Discrete-Time Multidirectional Associative Memory Neural Network with Variable Delays

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1 Iteratioal Scholarly Research Network ISRN Discrete Mathematics Volume 202, Article ID 8375, 0 pages doi:0.5402/202/8375 Research Article Global Expoetial Stability of Discrete-Time Multidirectioal Associative Memory Neural Network with Variable Delays Mi Wag, Tieju Zhou, ad Xiaola Zhag College of Sciece, Hua Agricultural Uiversity, Hua, Chagsha 4028, Chia Correspodece should be addressed to Tieju Zhou, htjzhou@26.com Received 3 July 202; Accepted 20 September 202 Academic Editors: C.-K. Li ad W. F. Smyth Copyright q 202 Mi Wag et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. A discrete-time multidirectioal associative memory eural etworks model with varyig time delays is formulated by employig the semidiscretizatio method. A sufficiet coditio for the existece of a equilibrium poit is give. By calculatig differece ad usig iequality techique, asufficiet coditio for the global expoetial stability of the equilibrium poit is obtaied. The results are helpful to desig global expoetially stable multidirectioal associative memory eural etworks. A example is give to illustrate the effectiveess of the results.. Itroductio The multidirectioal associative memory MAM eural etworks were first proposed by the Japaese scholar M. Hagiwara i 990. The MAM eural etworks have foud wide applicatios i areas of speech recogitio, image deoisig, patter recogitio, ad other more complex itelliget iformatio processig. So they have attracted the attetio of may researchers 2 7. I 5, we proposed a mathematical model of multidirectioal associative memory eural etwork with varyig time delays as follows, which cosists of the m fields, ad there are k euros i the field k k, 2,...,m : dx ki dt I ki a ki x ki t f x t τ ki t )),. where k, 2,...,m, i, 2,..., k, x ki t deote the membrae voltage of the ith euro i the field k at time t, a ki > 0 deote the decay rate of the ith euro i the field k, f is

2 2 ISRN Discrete Mathematics a euroal activatio fuctio of the ith euro i the field k, is the coectio weight from the jth euro i the field p to the ith euro i the field k, I ki is the exteral iput of t is the time delay of the syapse from the j euro i the field p to the ith euro i the field k at time t. We studied the existece of a equilibrium poit by usig Brouwer fixed-poit theorem ad obtaied a sufficiet coditio for the global expoetial stability of a equilibrium poit by costructig a suitable Lyapuov fuctio. However, discrete-time eural etworks are more importat tha their cotiuoustime couterparts i applicatios of eural etworks. Oe ca refer to 8 0 i order to fid out the research sigificace of discrete-time eural etworks. To the best of our kowledge, few studies have cosidered the stability of discrete-time MAM eural etworks. I this paper, we first formulate a discrete-time aalogues of the cotiuous-time etwork., ad i a ext study the existece ad the global expoetial stability of a equilibrium poit for the discrete-time MAM eural etwork. the ith euro i the field k, adτ ki 2. Discrete-Time MAM Neural Network Model ad Some Notatios I this sectio we formulate a discrete-time MAM eural etwork model with time-varyig delays by employig the semidiscretizatio techique 8. Let N m k k, Z deote the itegers set, Z {, 2, 3,...}, Z 0 {0,, 2,...}, ad Z a, b {a, a,...,b} where a, b Z. Let h be a fixed positive real umber deotig a uiform discretioary step size ad u deote the iteger part of the real umber u. Ift h, h Z 0, the t/h. By replacig the time t of the etwork. with h, we ca formulate the followig approximatio of the etwork. : dx ki dt I ki a ki x ki t p f x h τ ki h h h 2. for t h, h Z ki 0. Deote τ h /h κki, we rewrite 2. as follows: dx ki dt a ki x ki t I ki f x h κ ki h )). 2.2 Multiplyig both sides of 2.2 by e a kit,weobtai d x ki e ) a kit e a kit I ki dt f x h κ ki )) h. 2.3

3 ISRN Discrete Mathematics 3 Itegratig the 2.3 over h, t t h, we have x ki t e a kit x ki h e a kih ea kit e a kih a ki I ki f x h κ ki )) h. 2.4 Lettig t h i 2.4, weobtai x ki h x ki h e a kih e a kih I ki a ki f x h κ ki )) h. 2.5 If we adopt the otatios x ki x ki h, e a kih, e a kih e akih wki, I ki I ki, a ki a ki 2.6 the we obtai the discrete-time aalogue of the cotiuous-time etwork. as follows: x ki I ki x ki f x κ ki )) 2.7 for Z 0, k Z,m, i Z, k. Obviously, 0 < <. Throughout this paper, for ay k Z,m, p Z,m p / k, i Z, k, j Z, p, we assume that the euroal activatio fuctios f ki ad the time delays sequeces κ ki satisfy the followig coditios, respectively: H There exist L ki > 0 such that f ki x f ki y L ki x y for each x, y R, H2 0 < κ ki sup Z κ ki <. 0 The iitial coditios associated with 2.7 are of the form x ki ϕ ki, 2.8 where k Z,m, i Z, k, Z κ ki, 0, κ ki max p m,p / k max j p κ ki. For coveiece sake, set col b ki b,...,b,b 2,...,b 22,...,b m,...,b mm T, x col x ki, f x col f ki x ki. For ay matrixes U u ij ad V v ij,weusetheotatio

4 4 ISRN Discrete Mathematics U V to mea that u ij v ij for all i, j, ad U u ij. Let matrix A diag α,..., α, α 2,..., α m,..., α mm, L diag L,...,L,L 2,...,L m,...,l mm, O W 2 W m W 2 O 22 W 2m W, 2.9 W m W m2 O mm where W kp w k p w k p2 wp k p w k2 p2 wp k2 p, 2.0 w k k p p w k2 p w k k p w k k p2 O kk are zero matrixes k, p Z,m. 3. The Existece ad Global Expoetial Stability of a Equilibrium Poit I this sectio, we will give two theorems about the existece ad the global expoetial stability of a equilibrium poit of the discrete-time MAM eural etwork 2.7. Lemma 3.. If 0 <b<, 0 x b,thebe x x b. Proof. Defie a fuctio g t t l t 0 <t. Fromg t t /t > 0for0<t<, we kow that the fuctio g t is icreasig o the iterval 0,. Therefore, g b <g 0 for 0 <b<. So we have b< l b. Defie a fuctio h x be x x b for x 0 agai. Obviously h x be x, h x be x.fromh x > 0, the h x is icreasig. Because there exists a uique critical umber x 0 l b, we kow that h x <h l b 0for0 x l b. It shows that h x is deceasig o 0, l b. So we have h x be x x b h 0 0. I view of b< l b,we obtai be x x b for 0 x b. With a similar method of 5, we ca prove the followig Theorem 3.2. Theorem 3.2. Suppose that all the euroal activatio fuctios f ki k Z,m, i Z, k are cotiuous ad the coditio H) holds. If B A W L is a osigular M matrix, the there exists a equilibrium poit of the discrete-time MAM eural etwork 2.7. Proof. Let β ki m p p, / k j wki f 0 / I ki /. Obviously, β ki 0. Deote β col β ki, r col r ki A W L Aβ. Because Aβ 0adB A W L is a osigular

5 ISRN Discrete Mathematics 5 M matrix, by Lemma A3 i, we have r A W L Aβ 0. That is r ki 0 for ay k Z,m, i Z, k. From the defiitio of r, we have E A W L r β. Therefore, m L r β ki r ki. 3. Let Ω {x col x ki x ki r ki,r ki } with a orm x max k m max i k { x ki }. Obviously, Ω is a bouded closed compact subset. Defie a fuctio F : Ω R N as F x col F ki x, where F ki x m f ) x Iki. 3.2 From the coditio H ad 3., we have F ki x m m m L x f 0 ) I ki L r f 0 ) I ki L r β ki r ki. 3.3 Thus F x is a self-map from Ω to Ω. By Brouwer fixed-poit theorem, there exists at least a x Ω, such that F x x.thatis x ki I ki x ki f ) x. 3.4 Therefore x is a equilibrium poit of the MAM eural etwork 2.7. Next we prove the global expoetial stability of the equilibrium poit of the discretetime MAM eural etwork 2.7. Theorem 3.3. Suppose that all the euroal activatio fuctios f ki k Z,m, i Z, k are cotiuous ad the coditios H) ad H2) hold. If B A W L is a osigular M matrix, the the equilibrium poit of the MAM eural etwork 2.7 is global expoetial stable. Proof. Let x col x be a equilibrium poit for the MAM eural etwork 2.7, x ki col x ki, be a arbitrary solutio of 2.7.Setu col u ki, where u ki x ki x ki 3.5

6 6 ISRN Discrete Mathematics for k Z,m, i Z, k. Defie fuctios F z f z x ) ) f x, 3.6 where p Z,m, j Z, p. Obviously, F 0 0 ad from the coditio H, we have F z L z 3.7 for ay z R. By 3.5, the MAM eural etwork 2.7 is reduced to the form u ki u ki F u κ ki )), 3.8 where k Z,m, i Z, k. Obviously, there exists a equilibrium poit u 0ofthe system 3.8. From 2.8 ad 3.5, the iitial coditios associated with 3.8 are of the form u ki ψ ki ϕ ki x ki, 3.9 where k Z,m, i Z, k, Z κ ki, 0. Let ψ col ψ ki, ψ max k Z,m max i Z,k sup κki 0 ψ ki. Because B A W L is a osigular M matrix, the there exist costats ξ ki > 0 i Z, k,k Z,m such that ξ ki p L ξ > Defie the fuctios H ki λ λ ξ ki e λ p L ξ e λκki, 3. where λ R, k Z,m, i Z, k. Apparetly H ki λ is strictly mootoe decreasig ad cotiuous fuctio. I view of 3.0, it is clear that H ki 0 > 0,H ki < 0. Therefore there exist λ ki 0, such that H ki λ ki 0 k Z,m, i Z, k. Takig α mi k Z,m mi i Z,k {λ ki } < mi k Z,m mi i Z,k { }, we have H ki α α ξ ki e α p L ξ e ακki for i Z, k, k Z,m.

7 ISRN Discrete Mathematics 7 Set y ki e α u ki. By calculatig Δ y ki y ki y ki alog the solutios of system 3.8, we have Δ yki e α u ki e α u ki e α e α u ki eα )) F u κ ki. 3.3 By usig the iequality 3.7, we have Δ yki e α e α u ki eα ) u L κ ki e α yki L e ακki y κ ki ) e α y ki ) L e ακki y κ ki e α e α yki L e ακki sup κ ki s y s. p p p 3.4 By Lemma 3., we have Δ y ki α e α y ki L e ακki sup κ ki s y s. p 3.5

8 8 ISRN Discrete Mathematics Let ξ mi k Z,m mi i Z,k {ξ ki }, ξ max k Z,m max i Z,k {ξ ki } ad l 0 where δ is a positive costat. Therefore, whe s Z κ ki, 0, we have yki s e αs ψki s ψ <ξki l 0 δ ψ /ξ, 3.6 for i Z, k, k Z,m. We assert that yki <ξki l for Z, i Z, k ad k Z,m. If the assertio is false, the there exist k, i, ada miimum time t 0 Z such that y ki t 0 ξ ki l 0, Δ y ki t 0 0, ad y ξ l 0 whe Z κ ki,t 0.From 3.2 ad 3.5, ad oticed that α < 0, we obtai Δ yki t 0 α ξ ki e α p L ξ e ακki l 0 < It coflicts with Δ y ki t 0 0. Therefore, yki <ξki l for Z. The we have u ki <ξ ki l 0 e α δ ξ ψ e α M ψ e α, ξ 3.20 where M δ ξ/ξ >. So the zero solutio of the system 3.8 is global expoetial stable; thus the equilibrium poit of the discrete-time MAM eural etwork 2.7 is global expoetial stable. 4. A Example Cosider the followig discrete-time MAM eural etwork with three fields: )) x I α x w 2 f 2 x 2 κ 2 )) w 3 f 3 x 3 t κ 3 )) x 2 I 2 α 2 x 2 w 2 2 f 2 x 2 κ 2 2 )) w 2 3 f 3 x 3 t κ 2 3,,

9 ISRN Discrete Mathematics The state trajectories of the th euro o the first field with three iitial values The state trajectories of the 2th euro o the first field with three iitial values The state trajectories of the th euro o the first field 2 with three iitial values The state trajectories of the th euro o the first field 3 with three iitial values Figure : The globally expoetial stability of the equilibrium of the MAM etwork 4. with 0 cases radom iitial values. )) x 2 I 2 α 2 x 2 w 2 f x κ 2 )) w 2 2 f 2 x 2 κ 2 2 w 2 3 f 3 x 3 κ 2 3 )), )) x 3 I 3 α 3 x 3 w 3 f x κ 3 )) w 3 2 f 2 x 2 κ 3 2 w 3 2 f 2 x 2 κ 3 2 )), 4. where the euroal sigal decay rates α 0., α 2 0.2, α 2 0.2, α 3 0., the exteral iputs I, I 2, I 2, I 3, ad coectio weights w 2 0.2, w 3 0.3, w , w , w2 0.25, w , w , w3 0.2, w , w The euroal activatio fuctios f ki x x, the time delays κ ki 5 si π/2. Obviously, sigal trasfer fuctios f ki x are cotiuous, ad they satisfy the coditio H, ad the costat L ki. The time delays κ ki satisfy the coditio H2, ad κ ki 6. By calculatig, we have B A W L

10 0 ISRN Discrete Mathematics It is easy to verify that the matrix B is a osigular M matrix. The by Theorems 3.2 ad 3.3, there exists a equilibrium poit which is globally expoetially stable for the MAM eural etwork 4.. The umerical simulatio is give i Figure i which te cases of iitial values are take at radom. From Figure, we ca kow that the MAM eural etwork 4. coverges globally expoetially to the equilibrium poit x,x 2,x 2,x 3 T 2.735, 2.68,.9635,.8782 T o matter what it starts from the iitial states. 5. Coclusios I this paper, we have formulated a discrete-time aalogue of the cotiuous-time multidirectioal associative memory eural etwork with time-varyig delays by usig semidiscretizatio method. Some sufficiet coditios for the existece ad the global expoetial stability of a equilibrium poit have bee obtaied. Our results have show that the discrete-time aalogue iherits the existece ad global expoetial stability of equilibrium poit for the cotiuous-time MAM eural etwork. Refereces M. Hagiwara, Multidirectioal associative memory, i Proceedigs of the Iteratioal Joit Coferece o Neural Networks, vol., pp. 3 6, Washigto, DC, USA, M. Hattori ad M. Hagiwara, Associative memory for itelliget cotrol, Mathematics ad Computers i Simulatio, vol. 5, o. 3-4, pp , M. Hattori, M. Hagiwara, ad M. Nakagawa, Improved multidirectioal associative memories for traiig sets icludig commo terms, i Proceedigs of the Iteratioal Joit Coferece o Neural Networks, vol. 2, pp , Baltimore, Md, USA, J. Huag ad M. Hagiwara, A combied multi-wier multidirectioal associative memory, Neurocomputig, vol. 48, pp , M. Wag, T. Zhou, ad H. Fag, Global expoetial stability of mam eural etwork with varyigtime delays, i Proceedigs of the Iteratioal Coferece o Computatioal Itelligece ad Software Egieerig CiSE, 200), vol., pp. 4, Wuha, Chia, T. Zhou, M. Wag, H. Fag, ad X. Li, Global expoetial stability of mam eural etwork with time delays, i Proceedigs of the 5th Iteratioal Coferece o Bio-Ispired Computig: Theories ad Applicatios BIC-TA, 202), vol., pp. 6 0, Chagsha, Chia, T. Zhou, M. Wag, ad M. Log, Existece ad expoetial stability of multiple periodic solutios for a multidirectioal associative memory eural etwork, Neural Processig Letters, vol. 35, pp , S. Mohamad, Global expoetial stability i cotiuous-time ad discrete-time delayed bidirectioal eural etworks, Physica D, vol. 59, o. 3-4, pp , S. Mohamad ad K. Gopalsamy, Expoetial stability of cotiuous-time ad discrete-time cellular eural etworks with delays, Applied Mathematics ad Computatio, vol. 35, o., pp. 7 38, S. Mohamad ad A. G. Naim, Discrete-time aalogues of itegrodifferetial equatios modellig bidirectioal eural etworks, Joural of Computatioal ad Applied Mathematics, vol. 38, o., pp. 20, Z.-H. Gua, C. W. Cha, A. Y. T. Leug, ad G. Che, Robust stabilizatio of sigular-impulsivedelayed systems with oliear perturbatios, IEEE Trasactios o Circuits ad Systems I, vol. 48, o. 8, pp. 0 09, 200.

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