pth moment exponential stability of stochastic fuzzy Cohen Grossberg neural networks with discrete and distributed delays

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1 Nonlinear Analysis: Modelling and Control Vol. 22 No ISSN pth moment exponential stability o stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays Changjin Xu Peiluan Li Guizhou Key Laboratory o Economics System Simulation Guizhou University o Finance and Economics Guiyang China xcj403@126.com Received: July / Revised: January / Published online: June Abstract. In this paper stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays are investigated. By using Lyapunov unction and the Itô dierential ormula some suicient conditions or the pth moment exponential stability o such stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays are established. An example is given to illustrate the easibility o our main theoretical indings. Finally the paper ends with a brie conclusion. Methodology and achieved results is to be presented. Keywords: stochastic uzzy Cohen Grossberg neural networks global pth moment exponential stability discrete delays distributed delay Itô dierential ormula. 1 Introduction It is well known that Cohen Grossberg neural networks have been widely applied in various ields such as signal processing associative memory and optimization problems [6]. Many scholars argue that in these applications or neural networks it is o prime importance to ensure that the designed neural networks are stable [26]. In hardware implementation time delays inevitably occur due to the inite switching speed o the ampliiers and communication time. The qualitative research and analysis o Cohen Grossberg neural networks with delays has been investigated by numerous authors. Much richer dynamics has been reported [ ]. Considering that the synaptic transmission is a noisy process brought about by random luctuations rom the release o neurotransmitters and other probabilistic causes we think that it is o great signiicance to consider stochastic eects on the stability o neural networks described by stochastic unctional dierential equations [6]. In recent years numerous authors deal with the dynamical behavior o stochastic neural networks see e.g. [ ]. Since Yang and Yang [50] irst This work is supported by National Natural Science Foundation o China Nos and ) and Project o High-Level Innovative Talents o Guizhou Province [2016]5651). c Vilnius University 2017

2 532 C. Xu P. Li introduced uzzy cellular neural networks a lot o scholars have ound that uzzy neural networks have important applications in image processing and many results have been reported on stability and periodicity o uzzy neural networks [ ]. In addition we shall point out that neural networks usually have a spatial nature due to the presence o an amount o parallel pathways o variety o axon sizes and length. A distribution o conduction velocities along these pathways will lead to a distribution o propagation delays. Thus the time-varying delays and continuous distributed delays are more appropriate to uzzy cellular networks [ ]. To the best o our knowledge there are very ew papers that deal with the stability o stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays [ ]. Inspired by the analysis above in this paper we consider the ollowing stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays: dx i t) a i xi t) )[ b i xi t) ) n α ij t) n β ij t) c ij t) j xj t τij t) )) K ij t s)g j xj s) ) ds K ij t s)g j xj s) ) ds I i t) σ ij xj t) ) dω j t) 1) where n corresponds to the number o units in the neural networks respectively x i t) corresponds to the state o the ith neuron j and g j are signal transmission unctions τ ij t) denotes the transmission delay along the axon o the jth unit rom the ith unit and satisies 0 τ ij t) τ ij τ ij is a positive constant). a i x i t)) denotes an ampliication unction at time t b i x i t)) is an appropriately behaved unction at time t such that the solutions o model 1) remain bounded I i t) Ĩt) n T ijt)u j t) n H ijt)u j t). α ij t) β ij t) T ij and H ij t) are elements o uzzy eedback MIN template and uzzy eedback MAX template uzzy eed-orward MIN template and uzzy eed-orward MAX template respectively and stands or the uzzy AND and uzzy OR operation respectively u j t) denotes the external input o the ith neurons. Ĩt) is the external bias o ith unit. K ij ) is the delay kernel unction σ ij ) is the diusion coeicient σ i σ i1 σ i2... σ in ) ωt) ω 1 t) ω 2 t)... ω n t)) T is an n-dimensional Brownian motion deined on a complete probability space Ω F {F t } t 0 P) with a iltration {F t } t 0 satisying the usual conditions i.e. it is right continuous and F 0 contains all P-null sets). Here we would like to emphasize that pth moment exponential stability o stochastic delayed uzzy neural networks plays an important role in biological and artiicial neural networks. It can eectively portray the dynamics o neural networks [ ]. ] dt

3 pth moment exponential stability o stochastic uzzy Cohen Grossberg neural networks 533 Thus the research on pth moment exponential stability o stochastic delayed uzzy neural networks has important practical meanings. In addition we point out that the exponential stability in general sense and the pth moment exponential stability are dierent. The ormer is aimed at all dierential equations and the latter is aimed at stochastic dierential equations. General speaking a stochastic dierential equation is exponentially stable traditionally implies a stochastic dierential equation is pth moment exponentially stable. In particular i p 2 then we say that a stochastic dierential equation is exponentially stable in mean square. The key task o this article is to discuss the pth moment exponential stability o system 1). In recent years there are many papers that deal with pth moment exponential stability o stochastic neural networks [324346]. It is worth pointing out that most neural networks involve negative eedback terms or uzzy terms and do not possess ampliication unctions behaved unctions and uzzy terms. Model 1) o this paper has ampliications unction and behaved unctions which dier rom most neural networks with negative eedback term. Up to now there are rare papers that consider pth moment exponential stability this kind o stochastic uzzy neural networks. The main advantages o this article consist o our aspects: i) the study o pth moment exponential stability or stochastic delayed uzzy Cohen Grossberg neural networks with ampliication unctions and behaved unctions is proposed; ii) a set o new suicient criteria that ensure the pth moment exponential stability o system 1) by using Lyapunov unction and the Itô dierential ormula are established; iii) the key ideas o this article are also suitable or handling some other similar stochastic uzzy Cohen Grossberg neural networks; iv) to the best o our knowledge it is the irst time to deal with the pth moment exponential stability or stochastic delayed uzzy Cohen Grossberg neural networks with ampliication unctions behaved unctions and uzzy terms. The remainder o the paper is organized as ollows: in Section 2 the basic deinitions and lemmas are introduced. In Section 3 the suicient condition or the pth moment p 2) exponential stability or system 1) is established by using the Lyapunov unction method and Itô dierential inequality. In Section 4 an illustrative example is given. A brie conclusion is drawn in Section 5. 2 Preliminaries For convenience we introduce some notations. Let C C[ 0] R n ) be the Banach space o continuous unction which map into R n with the topology o uniorm convergence. For any xt) x 1 t) x 2 t)... x n t)) T R n we deine x x p n x it) p ) 1/p 1 < p < ). The initial conditions o system 1) are xs) ϕs) τ s 0 ϕ L p F τ 0] R n ) where L p F τ 0] Rn ) is R n -value stochastic process ϕs) τ s 0 ϕs) is F 0 measurable 0 τ E[ ϕs) p ] ds <. Throughout this paper we always make the ollowing assumptions: H1) There exist positive constants a i and ā i such that 0 < a i a i x) ā i or x R i n. Nonlinear Anal. Model. Control 224):

4 534 C. Xu P. Li H2) j ) and g j ) are Lipschitz continuous on R with Lipschitz constants L j Lg j j n i.e. or all x y R one has j x) j y) L j x y g j x) g j y) L g j x y. H3) b i ) CR R) and there exist positive constants µ i such that b i u) b i v) u v µ i or u v i n. H4) σxt)) σ ij x j t))) n n i j n) there exist nonnegative numbers ϱ i i n) such that tr[σ T x)σx)] n ϱ ix 2 i. H5) The delay kernel K ij : [0 ) [0 ) is a real-valued nonnegative continuous unction and satisies K ijt s) ds ρ ij where ρ ij is a positive constant and i j n. Let C 12 [τ ) R n ; R ) denote the amily o all nonnegative unctions V t x) on [τ ) R n which are continuous once and dierentiable in t and twice dierentiable in x. I V t x) C 12 [τ ) R n ; R ) in view o the Itô ormula we deine an operator LV associated with 1) as { [ LV t x) V t t x) V x t x) a i t) b i t) c ij t) j xj t τij t) )) where V t t x) n α ij t) n β ij t) 1 2 tr[ σ T V xx t x)σ ] K ij t s)g j xj s) ) ds K ij t s)g j xj s) ) ds I i t) V t x) V t x) V x t x) V xx t x) t x i ] dt } ) V t x) x i x j. n n Deinition 1. The equilibrium x o system 1) is said to be global pth moment exponentially stable i there exist positive constants M 1 λ > 0 such that E xt) x p ) M ϕ x p L eλtt0) t > t 0 x 0 R n where xt) x 1 t) x 2 t)... x n t)) T is any solution o system 1) p 2 is a constant when p 2 it is said to be exponential stability in mean square.

5 pth moment exponential stability o stochastic uzzy Cohen Grossberg neural networks 535 Lemma 1. See [50].) Let x and y be two states o system 1). Then n n α ij t)g j x) α ij t)g j y) n αij t) gj x) g j y) n n n β ij t)g j x) β ij t)g j y) βij t) gj x) g j y). Lemma 2. See [7].) I a i > 0 i n) denote p nonnegative real numbers then a 1 a 2 a m ap 1 ap 2 ap m p where p 1 denotes an integer. A particular orm o the above inequality is a p 1 1 a 2 p 1)ap1 p ap 2 p. Lemma 3 [Hölder inequality]. See [38].) Let x) and gx) be two continuous unctions and Ω a set a and b satisy 1/b 1/a 1 or any a 0 b 0 i a > 1 then the ollowing inequality holds: x)gx) ds ) 1/a x) a ds gx) b ds ) 1/b. Ω Ω Ω 3 pth moment exponential stability In this section we shall present suicient conditions or the global pth moment exponential stability o system 1). Theorem 1. Suppose that H1) H5) and the ollowing assumption hold true: H6) there exist a positive diagonal matrix M diagθ 1 θ 2... θ n ) and two constants 0 < Π 2 0 < u < 1 such that 0 < Π 2 Π 2 t) uπ 1 t) t t 0 where { Π 1 t) min 1 i n pa i µ i ā cij i t) L ) j p 1) p 1)ā i cij t) L j p 1)ā αij i t) βij t) ) ρ ij L g j p 1)p 2) ϱ j 2 θ j p 1)ϱ i } Nonlinear Anal. Model. Control 224):

6 536 C. Xu P. Li Π 2 t) max 1 i n { n θ j ā i cij t) L j p 1) ) } then x x 1 x 2... x n) T is a unique equilibrium which is globally pth moment exponentially stable where p 2 denotes a positive constant. When p 2 the equilibrium x o system 1) has exponential stability in mean square. Proo. Similar to [47 57] we can easily prove the existence and uniqueness o the equilibrium or system 1). Here we omit it. Let x x 1 x 2... x n) T be the unique equilibrium o system 1). Set y i t) x i t) x i σ ij σ ij y i t) x j ) σ ijx j ). Then it ollows rom 1) that dy i t) a i yi t) x i ) [ b i yi t) x i ) bi x i ) c ij t) j x j t τij t) )) j x j ) ) n α ij t) n β ij t) K ij t s) g j x j s) ) g j x j ) ) ds K ij t s) g j xj s) ) g j x j ) ) ds σ ij yj t) ) dω j t) t t 0 i n. 2) Deine a Lyapunov unction V by V t yt) ) θ yi i t) p ] dt θ xi i t) x p i p 2. 3) Calculating the operator LV t yt)) and using Lemma 2 associated with system 2) we have LV t yt) ) p [ θ yi i t) p1 sgn { y i t) }{ a i yi t) x ) i b i yi t) x i ) bi x i ) n α ij t) c ij t) j xj t τij t) )) j x j ) ) K ij t s) g j x j s) ) g j x j ) ) ds

7 pth moment exponential stability o stochastic uzzy Cohen Grossberg neural networks 537 p p n β ij t) pp 1) 2 K ij t s) g j xj s) ) g j x j ) ) ds yi t) p2 n σ ij yi t) ) θ yi i t) p1 a i yi t) x ) i µi y i t) sgn { y i t) } ]} θ yi i t) p1 a i yi t) x ) [ n i c ij t) j yj t τij t) )) sgn { y i t) } p p p p p αij t) ρijgj xj s) ) g j x j ) { sgn yi t) } βij t) ρijgj xj s) ) g j x j ) { sgn yi t) }] pp 1) 2 θ yi i t) p2 n θ yi i t) p1 a i µ yi i t) θ yi i t) p1 ā i θ yi i t) p1 ā i pp 1) 2 n n y i t) p2 σ 2 ij sgn { y i t) } cij t) L yi t τij t) ) j αij t) βij t) ) ρ ij L g j yi t) n y i t) p1 a i µ i y i t) ϱ j y 2 i t) ā i c ij t) L j p 1) y i t) p y i t τij t) ) p y i t) p1 ā i pp 1) 2 n θ yi i t) p2 α ij t) β ij t) ) ρ ij L g j y i t) n ϱ j y 2 i t) Nonlinear Anal. Model. Control 224):

8 538 C. Xu P. Li where Π 1 t) min 1 i n {pa i µ i p 1)ā i cij t) L j ā cij i t) n L j p1) p1)ā αij i t) βij t) ) ρ ij L g j p 1)p 2) ϱ j 2 n Π 1 t)v t yt) ) Π 2 t) { pa i µ i } θ j p 1)ϱ i y i t) p θ j ā i cij t) L j p 1) yi t τij t) ) p sup tτ s t p 1)ā i cij t) L j V s ys) ) 4) ā cij i t) L ) n j p 1) p 1)ā αij i t) βij t) ) ρ ij L g j p 1)p 2) ϱ j 2 { n Π 2 t) max 1 i n θ j p 1)ϱ i θ j ā i cij t) L j p 1) ) }. Applying the Itô ormula or t t 0 we have V t ξ yt ξ) ) V t yt) ) tξ 0 LV s ys) ) ds tξ 0 } V y s ys) ) σ s ys) ) dωs). 5) Since E[V x s ys))σs ys)) dωs)] 0 taking expectations on both sides o 5) and applying 4) we get V t ξ yt ξ) ) V t yt) ) tξ The Dini derivative D is t [ Π 1 t)e V s ys) )) Π 2 t)e D E V t yt) )) lim ξ 0 sup sup sτ ς s V ς yς) ))] ds. 6) EV t ξ yt ξ)) V t yt))). ξ

9 pth moment exponential stability o stochastic uzzy Cohen Grossberg neural networks 539 Denote zt) EV t yt))). It ollows rom 6) that In view o Lemma o [17] we obtain That is where D zt) Π 1 t)zt) Π 2 t) z t p. zt) zt0 ) p e λtt0). E [ xt) x p ] M ϕ x p e λtt0) t t 0 M max 1 i n min 1 i n > 1 and λ is the unique positive solution o the ollowing equation: λ Π 1 t) Π 2 t)e λτ. Thus the equilibrium x o system 1) is pth moment exponentially stable. The proo o Theorem 1 is completed. 4 An illustrate example In this section we present numerical examples to illustrate the eectiveness o the obtained results. Consider the ollowing stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays: dx 1 t) a 1 x1 t) )[ b 1 x1 t) ) 2 c 1j t) j xj t τ1j t) )) 2 α 1j t) 2 β 1j t) 2 σ 1j xj t) ) dω j t) K ij t s)g j xj s) ) ds K 1j t s)g j xj s) ) ds I 1 t) dt dx 2 t) a 2 x2 t) )[ b 2 x2 t) ) 2 c 2j t) j xj t τ2j t) )) 2 α 2j t) K 2j t s)g j xj s) ) ds ] 7a) Nonlinear Anal. Model. Control 224):

10 540 C. Xu P. Li 2 β 2j t) 2 σ 2j xj t) ) dω j t) K 2j t s)g j xj s) ) ] ds I 2 t) dt 7b) where j x) g j x) x 1 x 1 )/2 K ij t) te t and [ ] [ ] a1 x 1 t)) a 2 x 2 t)) 4 2 cos x1 t) 3 2 sin x 2 t) b 1 x 1 t)) b 2 x 2 t)) 12x 1 t) 14x 2 t) [ ] [ ] [ ] [ ] c11 t) c 12 t) α11 t) α 12 t) c 21 t) c 22 t) α 21 t) α 22 t) [ ] β11 t) β 12 t) β 21 t) β 22 t) [ ] τ11 t) τ 12 t) τ 21 t) τ 22 t) [ ] [ ] [ ] σ11 t) σ 12 t) σ 21 t) σ 22 t) [ ] I1 t) I 1 t) [ ] 3 4t. 1 2t [ ] 0.3x 0.2x 0.1x 0.4x Let ϱ ϱ then it is easy to see that that H1) H5) are satisied. Let p 2 then we can obtain Π Π There exists a positive constant 0 < u 0.8 < 1 such that 0 < Π < uπ Thus all the assumptions in Theorem 1 are ulilled. Thus we can conclude that system 7) has a unique equilibrium point x which is pth moment exponentially stable. The results are illustrated in Fig. 1 a) Figure 1. Transient response o state variables: a) x 1 t) b) x 2 t). b)

11 pth moment exponential stability o stochastic uzzy Cohen Grossberg neural networks Conclusions In this paper applying Lyapunov unction and the Itô dierential ormula we investigate the pth moment exponential stability or a class o stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays. Some simple suicient conditions checking the pth moment exponential stability o the stochastic uzzy Cohen Grossberg neural networks with discrete and distributed delays have been obtained. The obtained criteria play an important role in designing pth moment exponential stability o stochastic uzzy Cohen Grossberg neural networks. Reerences 1. P. Balasubramaniam M. Kalpana R. Rakkiyappan State estimation or uzzy cellular neural networks with time delay in the leakage term discrete and bounded distributed delays Comput. Math. Appl. 6210): P. Balasubramaniam M. Kalpana R. Rakkiyappan Stationary oscillation o interval uzzy cellular neural networks with mixed delays under impulsive perturbations Neural Comput. Appl ): P. Balasubramaniam R. Rakkiyappan R. Sathy Delay dependent stability results or uzzy BAM neural networks with Markovian jumping parameters Expert Syst. Appl. 381): P. Balasubramaniam M. Syed Ali Stability analysis o Takagi Sugeno uzzy Cohen Grossberg BAM neural networks with discrete and distributed time-varying delays Math. Comput. Modelling 531 2): P. Balasubramaniam M. Syed Ali S. Arik Global asymptotic stability o stochastic uzzy cellular neural networks with multiple time-varying delays Expert Syst. Appl. 3712): H. Bao Dynamic analysis o stochastic uzzy Cohen Grossberg neural networks with timevarying delays Adv. Dier. Equ ): J.D. Cao J. Liang Boundedness and stability or Cohen Grossberg neural network with timevarying delays J. Math. Anal. Appl. 2962): J.D. Cao R. Rakkiyappan K. Maheswari A. Chandrasekar Exponential H iltering analysis or discrete-time switched neural networks with random delays using sojourn probabilities Sci. China Technol. Sci. 593): L. Chen R. Wu D. Pan Mean square exponential stability o impulsive stochastic uzzy cellular neural networks with distributed delays Expert Syst. Appl. 385): Y. Du S. Zhong N. Zhou Global asymptotic stability o Markovian jumping stochastic Cohen Grossberg BAM neural networks with discrete and distributed time-varying delays Appl. Math. Comput. 243: Y. Du S. Zhong N. Zhou K. Shi J. Cheng Exponential stability or stochastic Cohen Grossberg BAM neural networks with discrete and distributed time-varying delays Neurocomputing 127: Nonlinear Anal. Model. Control 224):

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