S Fuzzy Memristive Neural Networks with Time-Varying ... - IEEE Xplore

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 26, NO. 3, JUNE 2018

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Lagrange Stability for T–S Fuzzy Memristive Neural Networks with Time-Varying Delays on Time Scales Qiang Xiao

and Zhigang Zeng

Abstract—The existed results of Lagrange stability for neural networks (NNs) are scale-free, and hence, conservativeness appears naturally. A class of Takagi–Sugeno (T–S) fuzzy memristive NNs (FMNNs) with time-varying delays is considered on time scales. First, a class of FMNNs is formulated using characteristics of memristors and T–S fuzzy rules. Then some new scale-limited criteria of global exponential stability in Lagrange sense are obtained for FMNNs with bounded feedback functions on the basis of inequalities on time scales and inequality scaling techniques. Also, novel criteria for Lurie-type feedback functions are given, which mainly employ the constructed scale-limited generalized Halanay inequality. Moreover, by matrix-norm strategies, some matrixnorm-based scale-limited criteria are derived for bounded and Lurie-type feedback functions, respectively. It also can be seen that the matrix-norm-based criteria are in accordance with the matrixmeasure-based conditions provided the time scale is specified as real set. All scale-limited criteria for Lagrange stability not only include continuous-time criteria and its discrete-time analogues, but also contain more complex cases such as the arbitrary combination of them. In the end, some numerical simulations exhibit the validity of the obtained results. Index Terms—Lagrange stability, memristive neural networks (MNNs), Takagi–Sugeno (T–S) fuzzy logics, time scale.

I. INTRODUCTION EMRISTIVE neural networks (MNNs) are suitable to characterize the nonvolatile feature of the memory cell due to hysteresis effects. MNNs have been intensively studied in bioinspired engineering [1]–[3], stability and chaos [4]–[6] since memristor was first postulated by Chua [7] and presented by Williams’s team [8]. Among which, Guo et al. [6] pointed out that the number of equilibria in n-neuron MNNs can be vastly 2 increased by 22n times, which exhibited the huge attractive potential and applications of MNNs.

M

Manuscript received December 21, 2016; revised March 28, 2017; accepted May 8, 2017. Date of publication May 12, 2017; date of current version May 31, 2018. This work was supported in part by the Guangdong Innovative and Entrepreneurial Research Team Program under Grant 2014ZT05G304, in part by the Natural Science Foundation of China under Grant 61673188 and Grant 61761130081, in part by the National Key Research and Development Program of China under Grant 2016YFB0800402, and in part by the Science and Technology Support Program of Hubei Province under Grant 2015BHE013. (Corresponding author: Zhigang Zeng.) The authors are with the Guangdong HUST Industrial Technology Research Institute, Guangdong Province Key Lab of Digital Manufacturing Equipment, Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science & Technology, Wuhan 430074, China (e-mail: [email protected]; zgzeng@ hust.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2017.2704059

, Senior Member, IEEE

Stability of NNs is prerequisite for most applications and implementations; it has been studied for several decades due to its wide theoretical and practical applications in image processing, object detection, pattern recognition, associative memory, and combinatorial optimization [9]–[12], and so forth. It is known to all that Lyapunov stability, which is also said to be monostability, is an important issue in applications of NNs and it has been received tremendous investigations during the past years [13], [14]. The nature of monostability cannot process multiobjective decision-making issues, emulate and explain biological behaviors [15], and it also could be computationally restrictive [16]. Unlike Lyapunov stability, Lagrange stability means the stability of the entire system. That is to say, all the solutions of the system will be infinitely approaching to an global attractive set (GAS) if their initial values are outside the GAS, and they will be stayed in the GAS all the time if their initial values are in the GAS. Thus, all the solutions of the system are bounded by the GAS. As Lagrange stability is considered based on the boundedness of solutions and the existence of GAS, there are not any equilibria, periodic solutions, or chaotic attractors outside the GAS. Some Lagrange stability criteria depending on the network parameters were obtained, which also indicated that Lagrange stability did not exclude multistability in view of simulation results [15]. Actually, Lyapunov stability could be viewed as a special case of Lagrange stability, provided the GAS reduces to a singleton, and then, the attractive set is the corresponding singleton. Noting that time delays are frequently encountered in engineering. Due to the finite speed of information processing and the inherent communication time of neurons, they often lead to complex dynamic behaviors or may even cause system unstable. Therefore, it matters to study dynamics of NNs with time delays in theory and practice. In addition, the feedback function often plays an important role in dynamics of NNs. Many a criterion for important dynamic properties not only depends on the neurosystem parameters, but also relates to the types of feedback functions. In this paper, time-varying delays and two general types of feedback functions (the bounded [17]–[19] and the Lurie-type feedback functions [20]) are considered. We also note that, in recent decades, investigations of NNs with fuzzy logics have been emerged since they are closely related to complexity of NNs. For instance, in [21], fuzzy complex networks with both partial and discrete-time couplings were considered in which much less information was needed compared with the perfect communication.

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The unknown dynamics were approximated by fuzzy logics that contributed to solving the consensus of multiagent system in [22]. Among numerous kinds of fuzzy logics, the Takagi– Sugeno (T–S) fuzzy model [23] that could approximate smooth nonlinear functions to any arbitrary accuracy is frequently used to analyze and synthesize nonlinear systems [24]. Recent years have witnessed prosperous developments of T–S fuzzy systems, such as pinning synchronization, consensus control, fault detection, H∞ filtering [21]–[27], and the references therein. The robust H∞ filtering for two-dimensional uncertain improved T–S fuzzy systems with mixed delays were presented, and sufficient conditions were acquired for existence and explicit expressions of desired filters [26]. Generally, two major directions are mainly adopted to analyze dynamics of NNs, namely, continuous-time analysis and its discrete-time analogues, they are often investigated separately. For example, a class of discrete-time recurrent NNs with unsaturating piecewise linear activation functions were investigated, in which the nondivergence, global attractivity, and complete stability of the system were addressed [28], and its generalized continuous-time analogue was studied in [29]. Sometimes, the results concerning discrete NNs can be carried easily from the corresponding results from their continuous analogues [30]. In addition, from a modeling point of view, it may be more realistic to model a phenomenon by a dynamic system that incorporates both continuous and discrete time, and there exist a large amount of neurosystems that include both continuous-time states and its discrete-time analogues. Hence, it is of necessity and importance to study artificial NNs in continuous-time and discrete-time simultaneously. This would be naturally integrated by notion of time scales, which was introduced by S. Hilger in 1988 and primitively aimed at unifying the continuous and discrete analysis [31]. The needs for obtaining separate results for discrete and continuous cases can be avoided under the umbrella of time scales. In recent years, many excellent works have been done on time scales [30], [32], [33]. In [32], nondivergence, attractivity, and multiperiodic dynamics were concerned for threshold linear networks on time scales, and they contained the results both in [28] and [29]. Accordingly, investigations of dynamics for neurosystems on time scales contain abundant different results and some repetitive works can be eliminated. Nevertheless, to the best of authors’ knowledge, very few results of Lagrange stability for fuzzy MNNs (FMNNs) have been opened, not to mention the research on time scales. Inspired by the above arguments, a class of T–S fuzzy MNNs with time-varying delays on time scales is proposed and investigated. Mainly stated, the novelties of the paper are as follows. 1) Scale-limited generalized Halanay inequality is first proposed, and the classical Halanay inequality is just a special case of it. The Lagrange stability is discussed on time scales with respect to different types of feedback functions. 2) The scale-limited criteria for delayed FMNNs not only contain the continuous-time case and its discrete-time analogues, but also can tackle the case of any combination of them. Accordingly, they can provide a wider scope of applications.


3) The scale-limited criteria in the paper contain pure algebraic criteria (for bounded feedback functions) and matrix-based algebraic criteria (for Lurie-type feedback functions). The approaches in the paper could be used to studied related dynamics of NNs, such as stability and synchronization. The rest structure of the paper is as follows. In Section II, some preliminaries are introduced, and the model is formulated. Then, the main results are given in Section III. Simulation examples demonstrate the effectiveness of the obtained results in Section IV. Conclusions are made for the paper in Section V. In closing, the proves of main results are given in six appendixes. II. PRELIMINARIES AND MODEL FORMULATION A. Preliminaries In this section, we recall some preliminaries and calculus on time scales [34], [35], which are needed for later sections. A time scale T is an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. The jump operator σ, ρ : T → T are defined by σ(t) = inf{w ∈ T : w > t}, ρ(t) = sup{w ∈ T : w < t}, respectively. The graininess function μ : T → [0, +∞) is defined by μ(t) := σ(t) − t. A point t ∈ T is said to be right (left)-dense if σ(t) = t (ρ(t) = t); A point t ∈ T is said to be right (left)-scattered if σ(t) > t (ρ(t) < t). If T has a left-scattered maximum m then T k := T /{m}, otherwise T k := T . A function f : T → R is called rd-continuous if it is continuous at rightdense points and the left-hand sided limits exist at left-dense points. The set of rd-continuous functions f : T → R is denoted by Crd = Crd (T , R). Generally, for a function f : T → R, f σ (t) = f (σ(t)), f ρ (t) = f (ρ(t)). If p(t) is rd-continuous and 1 + μ(t)p(t) > 0 for all t ∈ T , then p ∈ R+ . Let f : T → R and t ∈ T k . Then, the number f Δ (t) (when it exists), with the property that, for any ε > 0, there exists a neighborhood U of t such that |f (σ(t)) − f (s) − f Δ (t)(σ(t) − s)| ≤ ε|σ(t) − s|

∀s ∈ U

is called the Δ-derivative of f at t. f is said to be Δ-differentiable + f (t) is said to be the provided f Δ (t) exists for all t ∈ T k . DΔ upper-right (Dini) Δ-derivative of f (t) if, given ε > 0, there exists a right neighborhood Uε ∈ U of t such that f (σ(t)) − f (s) + < DΔ f (t) + ε σ(t) − s

∀s ∈ Uε , s > t.

Definition 1: Let p ∈ R. The exponential function ep (t, s) is defined on T by t ξμ(τ ) (p(τ ))Δτ for s, t ∈ T k ep (t, s) = exp s

with the cylinder transformation ξh (z) defined by ξh (z) = {

lo g (1 + z h ) h

z

, h= 0 h=0

p(t) For any p(t) ∈ R+ , (p)(t) is defined by − 1+μ(t)p(t) . Moreover, it follows that ep (t, t0 )ep (t, t0 ) = 1. In addition, if p > 0, then ep (t, s) < 1 for any s < t and s, t ∈ T .

XIAO AND ZENG: LAGRANGE STABILITY FOR T–S FMNNS WITH TIME-VARYING DELAYS ON TIME SCALES

Lemma 1 (see [34]): Let G(t) ∈ Crd , constants p > 0, q > + G(t)≤ −pG(t) + q, implies 0 with −p ∈ R+ , then DΔ G(t) −

q ≤ p

G(t0 ) −

q p

Plant Rule r: IF ζ1 (t) is Σ1r and · · · and ζq (t) is Σqr THEN xΔ i (t) = − cir xi (t) +

e−p (t, t0 )

B. Model A general class of MNNs with time-varying delays on time scales is described by the following Δ-differential equation: = − ci (t)xi (t) +

n

aij (xi (t))gj (xj (t))

j =1

+

n

dij (xi (t))gj (xj (t − τj (t))) + Ii (t)

(1)

j =1

where i ∈ In , t ∈ [0, +∞)T , n is the number of neurons, xi (t) ∈ R is the voltage of the capacitor Ri at instant t, ci (t) ∈(0, +∞) is the rate with which the ith neuron resets its potential to the resting state when isolated from all the other neurons in the network, gj (·) ∈ R denotes the feedback function, τi (t) ∈ R is the delay between neurons which are unknown but satisfies inf t∈T {τi (t)} ≥ 0, Ii (t) ∈ R represents the bounded external input of ith neuron at instant t, aij (xi (t)) and dij (xi (t)) stand for the memristive synaptic weights without and with time delays, and aij (xi (t)) =

Pij × ij , Ri

dij (xi (t)) =

Qij × ij Ri

in which ij = −1 if i = j, otherwise ij = 1, Pij and Qij denote the menductances of memristor Jij and Kij . Jij represents the memristor between the feedback function gj (xj (t)) and xi (t), Kij denotes the memristor between the feedback function gj (xj (t − τj (t))) and xi (t). According to the feature of the memristor and the current– voltage characteristic, aij (xi (t)) =

dij (xi (t)) =

⎧ ⎨ aij , ⎩

a ij ,

⎧ ⎨ dij , ⎩

d ij ,

ij gjΔ (xj (t)) − xΔ i (t) ≤ 0 ij gjΔ (xj (t)) − xΔ i (t) > 0

(2)

ij gjΔ (xj (t − τj (t))) − xΔ i (t) ≤ 0 ij gjΔ (xj (t − τj (t))) − xΔ i (t) > 0

n


j =1

for any t0 , t ∈ T and t ≥ t0 . Particularly, if G(t) ≥ q/p for t ∈ [t0 , +∞)T , then G(t) approaches q/p exponentially as t increases.

xΔ i (t)

1093

(3)

for i, j ∈ In , where aij , a ij , dij , and d ij are constants. In fact, (2)–(3) on ckl (xk (t)) and dkl (xk (t)) can be derived via voltage difference [15]. In this paper, we involve MNNs with T–S fuzzy sets on time scales, in which the rth rule is as follows:

+

n

dij (xi (t))gj (xj (t − τj (t))) + Ii (t),

j =1

where ζl (t)(l ∈ Iq ) are the premise variables, Σlr (r ∈ Im , l ∈ Iq ) are fuzzy sets and m is the number of IF-THEN rules. Let fr (ζ(t)) be the normalized membership function, i.e., πr (ζ(t)) , r ∈ Im fr (ζ(t)) = m r =1 πr (ζ(t)) where πr (ζ(t)) = Πqj =1 Σjr (ζj (t)), Σjr (ζj (t)) is the grade of membership of ζj (t) in Σjr . Based m on the fundamental prop≥ 0 and erties that πr (ζ(t)) r =1 πr (ζ(t)) > 0 for any t ∈ f (ζ(t)) = 1. [0, +∞)T , then m =1 After introducing fuzzy module, T–S FMNNs (1) can be represented by xΔ i (t) = −

m =1

+

n

f (ζ(t))ci xi (t) +

n


j =1

dij (xi (t))gj (xj (t − τj (t))) + Ii (t).

(4)

j =1

Remark 1: From (2) and (3), one can see that FMNNs (4) are differential equations with discontinuous parameters, the traditional control and analytical techniques cannot be used directly. To tackle this problem, solutions of all the systems in this paper are considered in Filippov’s sense [36]. Remark 2: FMNNs (4) are quite general, which include several well-known NNs, see [15], [37]. C. Notations ˇ Π} ˆ denotes closure of the Throughout this paper, co{Π, ˇ and Π. ˆ T is a time convex hull generated by real numbers Π scale with sup{T } = +∞, R is the real set, Z is the integer set. In is the index set {1, 2, . . . , n}. | · | denotes absolute value. The superscript “T” represents the transpose. 1 = (1, 1, . . . , 1)T ∈ Rn . For any real interval [a, b] on R, [a, b]T = [a, b] ∩ T . diag(m1 , m2 , . . . , mn ) denotes the diagonal matrix generated by numbers m1 , m2 , . . . , mn . a∗ij = max{|aij |, |a ij |}, d∗ij = max{|dij |, |d ij |}, a∗ = max{a∗ij },

= (a∗ )n ×n ,

= (d∗ )n ×n , d∗ = max{d∗ij }, A D Ga = ij ij ∗ ∗ a max{ki }, Gd = d max{ki }, K = diag(k1 , k2 , . . . , kn ), c¯ = maxi,r {cir }, c = mini,r {cir }, c˜i = maxr {cir }, and ci = minr {cir } for any i, j ∈ In and r ∈ Im . For a function y(t), without other statements, y¯(t) = sups∈[t−τ ,t] T {y(s)}, τ = maxi {¯ τi } for i ∈ In . and y¯ = supt∈T {y(t)}. S([−τ − μ ¯, 0]T , Rn ) denotes a Banach space of all continuous ¯, 0]T → functions F = (F1 (s), F2 (s), . . . , Fn (s))T : [−τ − μ Rn with default norm defined by F (s) = maxi

1094


sups∈[−τ −¯μ ,0] T {|Fi (s)|}, and denoted by S for simplicity. For a given positive constant W, SW is defined as the subset {ν ∈ S : ν ≤ W}. C is the set of all nonnegative functionals K : S → [0, +∞), mapping bounded sets in S into bounded sets in [0, +∞). The bounded feedback functions are given by B {g(·)|gi ∈ S(R, R),

∃ki > 0

∀xi ∈ R,

i ∈ In

s.t. |gi (xi )| ≤ ki } where the constants ki , (i ∈ In ) are generally said to be saturation constants. Then, the Lurie-type feedback functions are defined as L {g(·)|gi ∈ U,

∃ki > 0

∀xi ∈ R,

i ∈ In

s.t. xi gi (xi ) ≤ ki x2i } where the constants ki , (i ∈ In ) are generally said to be Lurie constants, and + U {G ∈ S(R, R)|sG(s) ≥ 0 and DΔ G(s) ≥ 0, s ∈ R}.

D. Properties By theories of set-valued maps and differential inclusions [36], [38], from (4) xΔ i (t) ∈ −

m


+

co{aij , a ij }gj (xj (t))

j =1

=1 n

n

co{dij , d ij }gj (xj (t − τj (t))) + Ii (t)

or equivalently, there exist a†ij ∈ co{aij , a ij }, d†ij ∈ co{dij , d ij } (i, j ∈ In ) such that = −

m


+

d†ij gj (xj (t − τj (t))) + Ii (t).

(5)

j =1

Vector function X (t) = (X1 (t), X2 (t), . . . , Xn (t))T ∈ Rn is said to be a solution of FMNNs (4) with initial value φ(t) ∈ S in Filippov’s sense provided it is absolutely continuous on any compact interval of [0, +∞)T , and XiΔ (t) ∈ −

m =1

+

n

f (ζ(t))ci Xi (t) +

n

(6)

Now, some concepts are given which will be used later. Definition 2: The trajectory of FMNNs (4) is said to be uniformly stable in Lagrange sense (or uniformly bounded), if for any H > 0, there exists a constant K > 0 such that |x(t; φ)| < K for all φ ∈ SH and t ∈ [0, ∞)T . Definition 3: If there exist a radially unbounded and positive definite function V (x), a functional K ∈ C, positive constants ς and , such that for any solution x(t; φ) of FMNNs (4), V (x) > ς, t ∈ [0, +∞)T , implies V (x) − ς ≤ K(ϕ)e (t, 0) (or V (x) − ς ≤ K(ϕ)e− (t, 0) if − ∈ R+ ), then the trajectory of (4) is said to be globally exponentially attractive with respect to V , and the compact set Ω := {x ∈ Rn |V (x) ≤ ς} is a global exponential attractive set (GEAS) of (4). Definition 4: FMNNs (4) is called globally exponentially stable in Lagrange sense (or Lagrange stability), if it is both uniformly bounded and globally exponentially attractive. Definition 5: For vector x ∈ Rn , the vector norm is defined as follows n 12 n 2 |xi |, x 2 = |xi | x 1 = i=1

i=1

x ∞ = max |xi |. 1≤i≤n

A 1 = max j

co{aij , a ij }gj (Xj (t))

j =1

co{dij , d ij }gj (Xj (t − τj (t))) + Ii (t)

j =1

for i ∈ In . Let x(t) = (x1 (t), x2 (t), . . . , xn (t))T , I(t) = (I1 (t), I2 (t), . . . , In (t))T , g(x(t)) = (g1 (x1 (t)), g2 (x2 (t)), . . . , gn (xn (t)))T , g τ(x(t)) = (g1 (x1 (t − τ1 (t))), g1(x2 (t − τ2 (t))), . . . , gn (xn (t − τn (t))))T , A† = (a†ij )n ×n , D† = (d†ij )n ×n , C m m = diag( m =1 f (ζ(t))c1 , =1 f (ζ(t))c2 , . . ., =1

n

1 |aij |, A 2 = λm ax (AT A) 2

i=1

A ∞ = max i

a†ij gj (xj (t))

j =1

=1 n

n

xΔ (t) = −Cx(t) + A† g(x(t)) + D† g τ (x(t)) + I(t).

For constant matrix A = (aij )n ×n ∈ Rn ×n , the matrix norm is defined as follows

j =1

xΔ i (t)

f (ζ(t))cn ), then the compact form of FMNNs (4) is

n

|aij |

j =1

where λm ax (M ) is the maximum eigenvalue of matrix M . Before ending this section, two lemmas are constructed, which will play critical roles in acquiring the main results. Lemma 2 (Scale-limited Halanay inequality): Assume that constants k1 , k2 satisfy k1 > k2 > 0, −k1 ∈ R+ , y(t) is a non¯, t0 ]T , and the negative rd-continuous function on [t0 − τ − μ following inequality holds: + y(t) ≤ −k1 y(t) + k2 y¯(t) DΔ

(7)

for t ∈ [t0 , +∞)T , then y(t) ≤ y¯(t0 )eλ (t, t0 ) with λ > 0 and λ + k2 exp (λτ ) < k1 .

(8)

Proof: See Appendix A. Remark 3: From Appendix A, one can see that (8) would be equation rather than inequality if we specify time scale as R, thus the conventional Halanay inequality is a special case of this novel scale-limited Halanay inequality. Lemma 3: Assume that constant matrix = diag(1 , 2 , . . . , n ) with i > 0, −i ∈ R+ for i ∈ In , (t) = diag(1 (t), 2 (t), . . . , n (t)) with i (t) ∈ Crd for i ∈ In , Θ(t) = diag(Θ1 (t), Θ2 (t), . . ., Θn (t)) with Θi (t) ∈ Crd for


i ∈ In . Then for any p in {1, 2, ∞} and time scale T , + Θ(t) ≤ −Θ(t) + (t), DΔ

t ∈ Tk

implies + DΔ Θ(t) p ≤ − min{i } Θ(t) p + (t) p ,

t ∈ Tk.

i∈In

(9) Moreover, for some r, + Θr (t) ≤ −r Θr (t) + r (t), DΔ

≤ −r |Θr (t)| + |r (t)|,

α2 = c,

t∈T . k

(10)

Proof: See Appendix B. III. MAIN RESULTS

In this section, we always assume that φ(t) is the initial value of FMNNs (4), where φ ∈ S([−τ − μ ¯, 0]T , Rn ). A. Bounded Feedback Functions for FMNNs on Time Scales In this section, bounded feedback functions are considered, i.e., g(·) ∈ B, the saturation constants ki > 0 for i ∈ In . Theorem 1: Suppose that g(·) ∈ B is satisfied. If mini {−α12 i } ∈ R+ and α11 > 0, then FMNNs (4) is globally exponentially stable in Lagrange sense. Moreover, the compact sets Ω11 and Ω12 are GEASs of (4). Besides, if −c¯ ∈ R+ , then FMNNs (4) is globally exponentially stable in Lagrange sense, Ω13 and Ω14 are GEASs of (4), where n γ11 γ12 i 2 , Ω12 = xx2i ≤ xi ≤ Ω11 = x α11 α12 i i=1 n γ13 γ14 i Ω13 = x , Ω14 = x|xi | ≤ |xi | ≤ α13 α14 i i=1 where x = (x1 , x2 , . . . , xn )T ∈ Rn and α11 = 2c − 2 − 3¯ μc¯2 , γ11

μc˜2i α12 i = 2ci − 2 − 3¯ ⎛ ⎞2 n n I¯i2 = n(1 + 3¯ μ)(a∗ + d∗ )2 ⎝ kj ⎠ + (1 + 3¯ μ) j =1

⎛ γ12 i = (1 + 3¯ μ)(a + d ) ⎝ ∗

∗ 2

n

i=1

⎞2

α13 = c,

γ13 = (a∗ + d∗ )

i=1

α14 i = ci ,

γ14 i

ki +

n

Proof: See Appendix D. Remark 5: In previous works about stability of NNs, the main approach is constructing various Lyapunov function (or functional), then the corresponding criteria are usually in the form of linear matrix inequalities or algebraic expressions. However, decent Lyapunov function (or functional) may not be established easily. For this purpose, Theorem 2 presents a new criterion of global exponential stability in Lagrange sense for FMNNs on time scales, in which the vector norms and the corresponding induced matrix norms are used instead of constructing Lyapunov function. B. Lurie-Type Activation Functions for FMNNs on Time Scales Lurie-type feedback functions are involved here, i.e., g(·) ∈ L, the saturation constants ki > 0 for i ∈ In . Theorem 3: Suppose that g(·) ∈ L is satisfied. If α31 > β31 and −α31 ∈ R+ , then FMNNs (4) is globally exponentially stable in Lagrange sense. Moreover, the compact set Ω31 is a GEAS of (4). In addition, if α32 > β32 and −α32 ∈ R+ , then Ω32 is a GEAS, where n γ31 2 Ω31 = x xi ≤ α31 − β31 i=1 n γ32 Ω32 = x |xi | ≤ α32 − β32 i=1 where x = (x1 , x2 , . . . , xn )T ∈ Rn and α31 = 2c − 1 − 2nGa − nGd − 4¯ μc¯2 − 4n2 μ ¯G2a γ31 =

n

[1 + 4¯ μ]I¯i2 (t)

i=1

α32 = c − nGa , β32 = nGd , γ32 =

|I¯i |

n

|I¯i |.

i=1

i=1

= (a + d )ki + |I¯i | ∗

p + D

p ) K p + I ¯ p. γ2 = ( A

β31 = nGd + 4n2 μ ¯G2d ,

kj ⎠ + (1 + 3¯ μ)I¯i2

j =1 n

then Ω12 ⊂ Ω∗12 ⊂ Ω11 . Namely, Ω12 is more precise than Ω11 . Likewise, we can derived that Ω14 is more precise than Ω13 . Theorem 2: Suppose that g(·) ∈ B is satisfied. If −c ∈ R+ , then FMNNs (4) is globally exponentially stable in Lagrange sense. Moreover, the compact set Ω2 is a GEAS of (4), where γ2 Ω2 = x x p ≤ α2 where x ∈ Rn , p = 1, 2, ∞, and

t ∈ Tk

implies + DΔ |Θr (t)|

1095

∗

for i, j ∈ In . Proof: See Appendix C. Remark 4: The relationship between Ω11 and Ω12 is inα11 ≤ α12 i , and γ11 = teresting. Evidently, for any i ∈ In , n γ1 2 i n n ∗ n 2 i=1 γ12 i . Let Ω12 := {x ∈ R | i=1 xi ≤ i=1 α 1 2 }, i

Proof: See Appendix E. Theorem 4: Suppose that g(·) ∈ L is satisfied. If α4 > β4 and −c ∈ R+ , then FMNNs (4) is globally exponentially stable in Lagrange sense. Moreover, the compact set Ω4 is a GEAS of (4), where γ4 Ω4 = x x p ≤ α4 − β4

1096


TABLE I ALLOWABLE UPPER BOUND OF TIME DELAY FOR DIFFERENT k 1 , k 2 , AND T UNDER THE SAME UPPER BOUND OF CONVERGENT RATE

TABLE II UPPER BOUND OF CONVERGENT RATE FOR DIFFERENT k 1 , k 2 , AND T UNDER THE SAME UPPER BOUND OF TIME DELAY

T

k1

k2 = 1

k 2 = 0.5

k 2 = 0.1

k 2 = 0.05

k 2 = 0.01

T

k1

k2 = 1

k 2 = 0.5

k 2 = 0.1

k 2 = 0.05

k 2 = 0.01

R

2 1.5 2 1.5 2 1.5

2.9389 1.3118 2.9000 1.2000 2.5000 1.0000

6.4047 4.7776 6.3000 4.7000 6.0000 4.5000

14.452 12.825 14.300 12.600 13.500 12.000

17.918 16.291 17.700 16.100 17.000 15.000

25.965 24.338 25.700 24.000 24.500 23.000

R

2 1.5 2 1.5 2 1.5

−0.8214 −0.4137 −0.8147 −0.4122 −0.7810 −0.4025

−1.3456 −0.9012 −1.3342 −0.8967 −1.2570 −0.8714

−1.8551 −1.3685 −1.8495 −1.3659 −1.7646 −1.3438

−1.9265 −1.4334 −1.9234 −1.4319 −1.8565 −1.4181

−1.9851 −1.4865 −1.9844 −1.4862 −1.9546 −1.4828

0.1Z 0.5Z

where x ∈ Rn and

p K p , α4 = c − A

p K p , β4 = D

¯p γ4 = I

for p = 1, 2, ∞. Proof: See Appendix F. Remark 6: The criteria of Lagrange stability in Theorems 3 and 4 are obtained by scale-limited generalized Halanay inequality, which does not appear in literatures the time before; thus, the results obtained in the paper are novel. Remark 7: Theorems 1–4 can be used to tackle the Lagrange stability problem on time scales, the obtained criteria not only cover the continuous-time analysis and its discrete-time analogue, but also can tackle many complex cases such as the timevarying μ(t). The method in [39] and [40] even cannot deal with discrete-time case, not to mention the ones on any time scales. Therefore, the results can improve or extend the previous works [15], [37], [39]–[41], they are probably mighty and superior in applications and implementations of discontinuous signal processing, irregular signal transmissions, etc. Remark 8: With regard to all the results aforementioned, one can see that different time scales could affect the Lagrange stability and the bounds of GEAS by μ ¯, hence all of them can be called scale-limited Lagrange stability criteria for FMNNs (4). Remark 9: From Appendixes E and F, one can see that the time delay considered in the paper will not affect the ultimate property of solutions of (4). However, the upper bound of convergent rate λ of solutions in the paper will be affected by the upper bound of time delay τ , which can be obtained by (8). Therefore, to obtain better performance of the solutions, τ should not be too large. Moreover, λ can be smaller if the time-delay term vanishes. In addition, λ will be influenced by different time scales as well. Given different k1 and k2 and three kinds of common time scales (R, 0.1Z, and 0.5Z), to obtain the same upper bound of convergent rate (λ ≡ −0.2), the correspondingly allowable maximum time delay (τ ) is listed in Table I . Conversely, given the same upper bound of time delay (τ = 0.2) for different k1 , k2 and three kinds of time scales listed above, the upper bound of convergent rate (λ) is listed in Table II . IV. NUMERICAL EXAMPLES Two examples are given to illustrate the effectiveness and validity of the obtained Lagrange stability criteria.

0.1Z 0.5Z

Example 1: Consider two-neuron delayed FMNNs on time scale T1 with two fuzzy rules [42]: Plant Rule 1: IF xi (t) is Σ11 , THEN xΔ i (t) = − c1i xi (t) +

n


j =1

+

n

dij (xi (t))gj (xj (t − τ1 (t))) + Ii (t).

(11)

j =1

Plant Rule 2: IF xi (t) is Σ12 , THEN xΔ i (t) = − c2i xi (t) +

n


j =1

+

n

dij (xi (t))gj (xj (t − τ2 (t))) + Ii (t)

(12)

j =1

where Σ11 is xi (t) ≤ 0, and Σ12 is xi (t) > 0 for i = 1, 2. Time scale T1 = 0.1Z. c11 = 4.1, c12 = 4.3, c21 = 4.2, c22 = 4.5. τ1 (t) = τ2 (t) = ττ = 0.1, I1 (t) = cos(t), I2 (t) = sin(t) for any t ∈ [0, ∞)T1 . The memristive synaptic weights are a 11 = −0.5, a11 = −0.1, a 12 = 0.1, a12 = 0.3, a 21 = 0.6, a21 = 0.4, a 22 = 0.9, a22 = 0.7, d 11 = 0.3, d11 = 0.25, d 12 = −0.7, d12 = −0.1, d 21 = −0.6, d21 = −0.5, d 22 = 0.5, d22 = 0.3. Two cases are considered for feedback functions accordingly. Case 1: g(·) ∈ B. We specify the bounded feedback functions as ⎧ −0.5, ⎪ ⎪ ⎪ ⎪ ⎨ 0.5x gi (x) = , ⎪ 3 ⎪ ⎪ ⎪ ⎩ 0.5,

x ∈ (−∞, −3) x ∈ [−3, 3]

(13)

x ∈ (3, +∞)

for i = 1, 2, then saturation constants k1 = k2 = 16 . By some calculations, α11 = 0.13 > 0, maxi=1,2 {α12 i } = max{0.65, 0.33} = 0.65 < 10 and c¯ = 4.5 < 10, which satisfy all the conditions of Theorems 1 and 2. Then, FMNNs (11)–(12) with bounded feedback functions (13) is globally exponentially


Fig. 1. State phases of FMNNs (11)–(12) with bounded feedback functions on time scale T1 .

stable in Lagrange sense, and the compact sets Ω11 = x ∈ R2 x21 + x22 ≤ 26.720 Ω12 = x ∈ R2 x21 ≤ 2.557, x22 ≤ 5.138 Ω13 = x ∈ R2 |x1 | + |x2 | ≤ 0.373 Ω14 = x ∈ R2 |x1 | ≤ 0.182, |x2 | ≤ 0.182 Ω12 = x ∈ R2 x 1 ≤ 11.200 Ω22 = x ∈ R2 x 2 ≤ 11.024 2 Ω∞ = x ∈ R ≤ 11.464 x ∞ 2

Fig. 2. State phase of FMNNs (11)–(12) with Lurie-type feedback functions on time scale T1 .

for i = 1, 2, then saturation constants k1 = k2 = 18 . By calculating, α31 = 0.219 > 0, β31 = 0.188, γ31 = 2.8, α32 = 2.388, β32 = 0.175, γ32 = 2. So, we have α31 > β31 , α31 < 10, α32 > β32 , α32 < 10, which satisfy all the conditions of Theorem 3. Then, FMNNs (11)–(12) with Lurie-type feedback functions (14) is globally exponentially stable in Lagrange sense, and the compact sets Ω31 = x ∈ R2 x21 + x22 ≤ 87.500 Ω32 = x ∈ R2 |x1 | + |x2 | ≤ 0.904 are GEASs. Moreover, by Theorem 4, the following three compact sets

are GEASs. We can see that there exist serious difference among those attractive sets, Ω14 is the most accurate attractive set and Ω∞ 2 is the roughest attractive set. What should be mentioned here is that although Ω12 , Ω22 , Ω∞ 2 are not as accurate as Ω13 and Ω14 , the method used in Theorem 2 is indispensable for some cases. Fig. 1 shows the state trajectories and the attractive sets of FMNNs (11)–(12) with six initial randomly occurred values. Now, we consider [15, model (10)]. Taking the first set of parameters as shown in this example and the feedback function is (13); then, we get M1 = 1.3, M2 = 1.433, which implies that the GAS Ω1[11] = {x ∈ R2 ||x1 | + |x2 | ≤ 2.733}. Obviously, the range of Ω1[11] is larger than Ω13 , implying that the attractive set in this paper are more precise than that in [11] in some cases. Case 2: g(·) ∈ L. Specifying the Lurie-type feedback functions as gi (x) = (x + tanh(x))/16

1097

(14)

Ω14 = x ∈ R2 x 1 ≤ 0.460 Ω24 = x ∈ R2 x 2 ≤ 0.451 2 x = x ∈ R ≤ 0.851 Ω∞ ∞ 4 are GEASs of FMNNs (11)–(12). Fig. 2 shows the state trajectories and the attractive sets of FMNNs (11)–(12) with six initial randomly generated values. Now we consider [15, model (10)] again. Taking the first set of parameters as shown in this example the feedback function is (14). Letting ε1 = ε2 = 1, p = 2; then, we get M1 = 1.225, M2 = 1.325, which implies that the GAS Ω2[11] = {x ∈ R2 ||x1 | + |x2 | ≤ 2.55}. Obviously, the range of Ω2[11] is larger than Ω32 , implying that the attractive set in this paper are more precise than that in [11] in some cases.

1098


Example 2: In this example, a class of FMNNs with three neurons and two fuzzy rules is constructed as follows: xΔ i (t) = −

2 =1

+

3


3


j =1

dij (xi (t))gj (xj (t − τj (t))) + Ii (t)

(15)

j =1

for i ∈ I3 , where the parameters are given as follows: The membership functions for Rules 1 and 2 are f1 (t) = 1 1 2 (1 − cos(t)) and f2 (t) = 2 (1 + cos(t)), I1 (t) = sin(2t), I2 (t) = cos(t), I3 (t) = cos(2t), τk (t) = ττ (t) =mod(t, 0.1) for any t ∈ [0, ∞)T2 , k ∈ I3 . Time scale T2 = ∪∞ k =1 [0.1k, 0.1k + 0.05]. c11 = 6.5, c12 = 7.4, c21 = 7.5, c22 = 6.8, c31 = 7.1, c32 = 6.9. aij (xi (t)) and dij (xi (t)) (i, j = 1, 2) are the same as in Example 1, and a 13 = −0.3, a13 = −0.1, a 31 = −0.1, a13 = −0.2, a 32 = a 23 = 0.3, a23 = 0.5, 0.4, a32 = 0.1, a33 = 0.1, a33 = 0.9, d13 = −0.1, d13 = −0.2, d 23 = 0.4, d23 = 0.9, d 31 = 0.8, d31 = 0.2, d 32 = 0.5, d32 = 0.1, d 33 = 0.3, d33 = 0.7. The feedback functions are defined by (13) for i ∈ I3 . By some calculations, α11 = 2.368 > 0, maxi∈I3 {α12 i } = max{2.786, 3.163, 4.024} = 4.024 < 20 and c¯ = 7.5 < 20 which satisfy all the conditions of Theorems 1 and 2. Then, FMNNs (15) with bounded feedback functions (13) (i ∈ I3 ) is globally exponentially stable in Lagrange sense, and the compact sets Ω11 = x ∈ R3 x21 + x22 + x23 ≤ 2.638 Ω12 = x ∈ R3 x21 ≤ 0.809, x22 ≤ 0.713, x23 ≤ 0.560 Ω13 = x ∈ R3 |x1 | + |x2 | + |x3 | ≤ 0.600 Ω14 = x ∈ R3 |x1 | ≤ 0.200, |x2 | ≤ 0.191, |x3 | ≤ 0.188 Ω12 = x ∈ R3 x 1 ≤ 0.239 Ω22 = x ∈ R3 x 2 ≤ 0.237 3 = x ∈ R ≤ 0.251 Ω∞ x ∞ 2 are GEASs. Fig. 3 shows the state trajectories and the attractive sets of FMNNs (15) with 20 initial randomly occurred values. The sphere in Fig. 3 is denoted by {x ∈ R3 |x21 + x22 + x23 ≤ 0.01}, we can see that all the trajectories enter into the sphere ultimately. If we specify feedback functions as rectified linear unit-type ones ⎧ ⎨ x, x≥0 (16) gi (x) = 16 ⎩ 0, x 0 such that (8) holds due to k2 < k1 . λ + k2 exp (λτ ) < k1 . Then, Then for any t ∈ T , 1+μ(t) λ −k1 + k2 eλ (t − τ, t) − λ < 0.

(17)

Let constant c > 1, and take x(t) = c¯ y (t0 )eλ (t, t0 ),

lim

h→0 +

Θ(t + h) p − Θ(t) p h

= lim

Θ(t) + hΘΔ (t) + o(h)1 p − Θ(t) p h

= lim

Θ(t) + hΘΔ (t) p − Θ(t) p h

h→0 +

t ∈ [t0 − τ − μ ¯, +∞)T

(18)

h→0 +

one can see that y(t) < x(t) for all t ∈ [t0 − τ − μ ¯, t0 ]T , since ¯, t0 ]T . c > 1 and eλ (t, t0 ) ≥ 1 for all t ∈ [t0 − τ − μ Now, we assert that h(t) := y(t) − x(t) < 0

We prove (9) at first. For any instant t ∈ T , it is either rightdense or right-scattered. Hence, we could consider these two cases separately for p = 1, 2, ∞. Case 1: t is right-dense, calculating the following expression, then

∀t ∈ [t0 − τ − μ ¯, +∞)T

≤ lim

(1 − h min{i }) Θ(t) p − Θ(t) p i∈In

h→0 +

h

+ (t) p

= − min{i } Θ(t) p + (t) p . i∈In

(19)

otherwise, there exists a left-dense point tˇ such that for any + sufficiently small j > 0, it gives h(tˇ) = 0, DΔ h(tˇ − j)> 0 and ˇ ¯, t)T or a left-scattered point tˇ h(s) < 0 for all s ∈ [t0 − τ − μ ¯, tˇ)T and h(tˇ) ≥ 0. If such that h(s) < 0 for all s ∈ [t0 − τ − μ it is the first case, from (7), (17), and (18), + DΔ h(tˇ − j)

≤ −k1 y(tˇ − j) + k2 y¯(tˇ − j) − (λ)c¯ y (t0 )eλ (tˇ − j, t0 ) < −k1 x(tˇ − j) + k2 x(tˇ − j − τ ) − (λ)c¯ y (t0 )eλ (tˇ − j, t0 ) + O(j) = c¯ y (t0 )eλ (tˇ − j, t0 )[−k1 + k2 eλ (tˇ − j − τ, tˇ − j) − (λ)] + O(j) where O(j) denotes the infinitesimal quantity when j → 0. By + virtue of (17), for sufficiently small j, it follows DΔ h(tˇ − j)≤ 0,

According to the definition of Dini Δ-derivative of x(t) p for right-dense point, inequality (9) holds. Case 2: t is right-scattered, calculating the following expression, together with −i ∈ R+ for i ∈ In , then Θσ (t) p − Θ(t) p μ(t) =

Θ(t) + μ(t)ΘΔ (t) p − Θ(t) p μ(t)

≤

(1 − μ(t) mini∈In {i }) Θ(t) p − Θ(t) p + (t) p μ(t)

= − min{i } Θ(t) p + (t) p . i∈In

Therefore, according to the definition of Dini derivative of x(t) p for right-scattered point, inequality (9) holds. Together with cases 1 and 2, inequality (9) holds for any t ∈ T . For inequality (10), it also can be derived similarly.

1100


APPENDIX C PROOF OF THEOREM 1

gives

First, we prove the global exponential stability in Lagrange + sense of FMNNs n (4) 2when mini {−α12 i } ∈ R and α11 > 0. Let V11 (t) = i=1 xi (t), computing the Δ-derivative of V11 (t) along the positive half trajectory of (4) gives n

Δ (t) = V11

Δ Δ (t) = [2xi (t) + μ(t)xΔ V12 i (t)]xi (t) i ⎛ ⎞2 n ≤ − 2ci x2i + x2i (t) + (a∗ + d∗ )2 ⎝ kj ⎠ + x2i (t) + I¯i2 j =1

⎛

⎡

+ 3μ(t) ⎣c˜2i x2i (t) + (a∗ + d∗ )2 ⎝

Δ [2xi (t) + μ(t)xΔ i (t)]xi (t)

= −2

fr (θ(t))cir x2i (t)

n

+2

i=1 r =1

+2

n

xi (t)

i=1

+ μ(t)

n

⎡

+

d†ij gj (xj (t − τj (t))) + 2

⎣−

m

n

a†ij gj (xj (t))

j =1

j =1

n

xi (t)Ii (t)

i=1

fr (θ(t))cir xi (t) +

r =1

i=1 n

xi (t)

i=1 n

n

a†ij gj (xj (t))

j =1

⎤2 d†ij gj (xj (t − τj (t))) + Ii (t)⎦

n m

n

fr (θ(t))cir x2i (t) + 2

i=1 r =1

+2

n

|xi (t) I¯i | + 3¯ μ

i=1

+ 3¯ μ

n

|xi (t)|

i=1

⎛

I¯i2 + 3¯ μn(a∗ + d∗ )2 ⎝

n

(a∗ + d∗ )kj

fr (θ(t))cir

n

x2i (t)

(21)

r =1

n

⎤

(|a†ij gj (xj (t))| + |d†ij gj (xj (t − τj (t)))|) + |I¯i |⎦

≤ −α13 V13 (t) + γ13 .

kj ⎠

γ11 γ11 ≤ V11 (0) − e−α 1 1 (t, 0) α11 α11

e−α 1 2 i (t, 0).

j =1

⎞2

According to Lemma 1, γ13 γ13 V13 (t) − ≤ V13 (0) − e−α 1 3 (t, 0), α13 α13

By Lemma 1, for all t ∈ [0, ∞)T , (20)

2 so we have V11 (t) ≤ n max For any H > 0, φ ∈ SH , ni {φi (0)}. 2 let K1 = nH , we have i=1 x2i (t) ≤ K1 , which implies the uniform stability in Lagrange sense of FMNNs (4). In addition,

γ11 ≤ V11 (0) = φ2i (0) := K11 (φ) α11 i=1 n

V11 (0) −

γ12 i γ12 i ≤ V12 i (0) − α12 i α12 i

According to Definition 3, Ω12 is a GEAS of FMNNs (4). It remains to prove the global exponential stability in Lagrange sense of FMNNs (4) when −¯ c ∈ R+ . To this end, let n V13 = i=1 |xi (t)|, by Lemma 3, it follows ! m n + fr (θ(t))cir |xi (t)| DΔ V13 (t) ≤ −

+

≤ − α11 V11 (t) + γ11 .

V11 (t) −

V12 i (t) −

i=1

j =1

i=1

By Lemma 1 again

2

r =1

kj ⎠ + I¯i2 ⎦

≤ − α12 i V12 i + γ12 i .

j =1

i=1

m n

⎤

Let K12 i (φ) = φ2i (0), i ∈ In , then K12 i ∈ C. From (21) γ12 i V12 i (t) − ≤ K12 i (φ)e−α 1 2 i (t, 0). (22) α12 i

j =1

≤ −2

⎞2

j =1

i=1 n m

n

then K11 ∈ C, from (20), V11 (t) − αγ 11 11 ≤ K11 (φ)e−α 1 1 (t, 0). By Definition 3, FMNNs (4) is globally exponentially attractive and Ω11 is a GEAS. Therefore, FMNNs (4) is globally exponentially stable in Lagrange sense. Next, we prove that Ω12 is also a GEAS of FMNNs (4). To this end, let V12 i (t) = x2i (t), i ∈ In , computing the Δderivative of V12 i (t) along the positive half trajectory of (4)

(23)

so V13 (t) ≤ n maxi {|φi (0)|}. any H > 0, φ ∈ SH , let K13 = nH, we have For n i=1 |xi (t)| ≤ K13 , which implies the uniform stability in Lagrange sense of FMNNs (4). In addition, γ13 ≤ V13 (0) = |φi (0)| := K13 (φ) α13 i=1 n

V13 (0) −

(24)

then K13 ∈ C, from (23), V13 (t) − αγ 11 33 ≤ K13 (φ)e−α 1 3 (t, 0). By Definition 3, FMNNs (4) is globally exponentially attractive and Ω13 is a GEAS. Therefore, FMNNs (4) is globally exponentially stable in Lagrange sense. In the sequel, to show that Ω14 is a GEAS, we employ n radially unbounded and positive definite function V14 i = |xi (t)|, i ∈ In , by Lemma 3, it gives + V1 4i (t) ≤ − DΔ

m

fr (θ(t))cir |xi (t)| + [|a†ij gj (xj (t))|

r =1

+ |d†ij gj (xj (t − τj (t)))|] + |I¯i | ≤ − α14 V14 (t) + γ14 .


Through Lemma 1, V14 i (t) −

γ14 i α14 i

γ14 i ≤ V14 i (0) − α14 i

+ Gd e−α 1 4 i (t, 0).

n n

+ 4μ(t)

n

+ 4μ(t)

n i=1

according to Definition 3, Ω14 is a GEAS of FMNNs (4).

≤ +

It’s clear that for any g(·) ∈ B,

(26)

Then by (26) and Lemma 3, for any t ∈ [0, ∞)T , + x(t) p ≤ −α2 x(t) p + γ2 . DΔ Applying Lemma 1, γ2 γ2 V2 (t) − ≤ V2 (0) − e−α 2 (t, 0) α2 α2

(27)

¯ so we have V2 (t) ≤ φ(0) p. For any H > 0, φ ∈ SH , let K2 = nH, we have x(t) p ≤ K2 for p = 1, 2, ∞, which implies the uniform stability in Lagrange sense of FMNNs (4). In addition,

then K2 ∈ C, and from (27), V2 (t) − αγ 22 ≤ K2 (φ)e−α 2 (t, 0). By Definition 3, FMNNs (4) is globally exponentially attractive and Ω2 is a GEAS. Thus, FMNNs (4) is globally exponentially stable in Lagrange sense.

First, we prove the uniformly stable in Lagrange sense of FMNNs (4). To this end, let V31 (t) = ni=1 x2i (t) (hence V31 (t) = V11 (t)), calculating the Δ-derivation of V31 (t) along the positive half trajectory of (4), then by Cauchy–Schwarz inequality, fr (θ(t))cir x2i (t) + 2Ga

i=1 r =1

+ Gd

i=1 j =1

+

n i=1

i=1

+ 4μ(t)

c¯2 x2i (t)

i=1

a∗ij kj |xj (t)|⎠

j =1

⎛ ⎝

n

⎞2 d∗ij kj |xj (t − τj (t))|⎠

j =1

[−2c + 1 + 2nGa + nGd + 4¯ μc¯2 + 4n2 μ ¯G2a ]x2i (t)

n

[1 + 4¯ μ]Ii2 (t) +

n i=1

n

[nGd + 4n2 μ ¯G2d ]x2i (t − τi (t))

i=1

≤ −α31 V31 (t) + β31 V¯31 (t) + γ31 . that is Then, according to Lemma 2, γ31 γ31 V31 (t) − ≤ V¯31 (0) − eα 3 1 (t, 0) α31 − β31 α31 − β31 (28) where ε31 > 0 and it satisfies ε31 + β31 eε 3 1 τ < α31 . any H > 0, φ ∈ SH , let K3 = nH2 , we have For n 2 i=1 xi (t) ≤ K3 , which implies the uniform stability in Lagrange sense of FMNNs (4). In addition, V¯31 (0) −

γ31 ≤ V¯13 (0) = φ¯2i (0) := K31 (φ) α31 − β31 i=1 n

then K31 ∈ C, and from (28) n

x2i (t) −

Ii2 (t)

2 |xi (t)|

γ31 ≤ K31 (φ)eα 3 1 (t, 0). α31 − β31

By Definition 3, FMNNs (4) is globally exponentially attractive and Ω31 is a GEAS. Therefore, FMNNs (4) is globally exponentially stable in Lagrange sense. Next, we prove that Ω32 is also a GEAS of FMNNs (4). Let V32 (t) = ni=1 |xi (t)|, it follows from Lemma 3 that ! m n + V32 (t) ≤ fr (θ(t))cir |xi (t)| − DΔ i=1

APPENDIX E PROOF OF THEOREM 3

Ii2 (t)

n

i=1

γ2 ≤ V2 (0) = φ(0) p := K2 (φ), α2

n

i=1

n

Δ (t) V31

p K p + D

p K p + I ¯ p. ≤ A

x2i (t)

n

x2i (t) + 4μ(t)

⎞2

i=1

A† g(x(t)) + D† g τ (x(t)) + I(t) p

n n

⎝

n

i=1

APPENDIX D PROOF OF THEOREM 2

Δ V31 (t) ≤ −2

⎛

i=1

γ14 i ≤ K14 i (φ)e−α 1 4 i (t, 0), V14 i (t) − α14 i

n m

x2j (t − τj (t)) +

i=1 j =1

(25)

Let K14 i (φ) = |φi (0)|, i ∈ In , then K14 i ∈ C. From (25)

V2 (0) −

1101

+

n

r =1

(|a†ij gj (xj (t))|

⎤ +

|d†ij gj (xj (t

− τj (t)))|) + I¯i ⎦

j =1

≤ −α32 V32 (t) + β32 V¯32 (t) + γ32 . Then, according to Lemma 2, γ32 γ32 ≤ V¯32 (0) − V32 (t) − eε 3 2 (t, 0) α32 − β32 α32 − β32 (29) ε3 2 τ where ε32 > 0 and it satisfies ε + β e < α . 32 32 32 Let K32 (φ) = ni=1 |φ¯i (0)|, then K32 ∈ C. From (29) V32 (t) −

γ32 ≤ K32 (φ)eα 3 2 (t, 0) α32 − β32

1102


according to Definition 3, Ω32 is a GEAS of FMNNs (4).

APPENDIX F PROOF OF THEOREM 4 It is clear that for any g(·) ∈ L, A† g(x(t)) + D† g τ (x(t)) + I(t) p

p K p x(t) p + D

p K p ¯ ¯ p . (30) ≤ A x(t) p + I Then, together with (6), (30), and Lemma 3, for any t ∈ + x(t) p ≤ −α4 x(t) p + β4 ¯ x(t) p + γ4 . [0, ∞)T , DΔ Applying Lemma 2, one gets γ4 γ4 V4 (t) − ≤ V¯4 (0) − (31) eε 4 (t, 0) α4 − β4 α4 − β4 where ε4 > 0 and it satisfies ε4 + β4 eε 4 τ < α4 . So, we have ¯ V4 (t) ≤ φ(0) p. For any H > 0, φ ∈ SH , let K4 = nH, we have x(t) p ≤ K4 , which implies the uniform stability in Lagrange sense of FMNNs (4). In addition, γ4 ¯ V¯4 (0) − ≤ V4 (0) = φ(0) p := K4 (φ) α4 − β4 then, K4 ∈ C, and from (31) γ4 ≤ K4 (φ)eε 4 (t, 0). V4 (t) − α4 − β4 By Definition 3, FMNNs (4) is globally exponentially attractive and Ω4 is a GEAS. Thus, FMNNs (4) is globally exponentially stable in Lagrange sense. REFERENCES [1] K. D. Cantley, A. Subramaniam, H. J. Stiegler, R. A. Chapman, and E. M. Vogel, “Hebbian learning in spiking neural networks with nanocrystalline silicon TFTs and memristive synapses,” IEEE Trans. Nanotechnol., vol. 10, no. 5, pp. 1055–1073, Sep. 2011. [2] Y. V. Pershin and M. D. Ventra, “Experimental demonstration of associative memory with memristive neural networks,” Neural Netw., vol. 23, no. 7, pp. 881–886, Sep. 2010. [3] T. J. Walls and K. K. Likharev, “Self-organization in autonomous, recurrent, firing-rate crossNets with quasi-Hebbian plasticity,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 4, pp. 819–824, Apr. 2014. [4] A. L. Wu and Z. G. Zeng, “Exponential stabilization of memristive neural networks with time delays,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 12, pp. 1919–1929, Dec. 2012. [5] R. X. Li and J. D. Cao, “Finite-time stability analysis for Markovian jump memristive neural networks with partly unknown transition probabilities,” IEEE Trans. Neural Netw. Learn. Syst., to be published, doi: 10.1109/TNNLS.2016.2609148. [6] Z. Y. Guo, J. Wang, and Z. Yan, “Attractivity analysis of memristor-based cellular neural networks with time-varying delays,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 4, pp. 704–717, Apr. 2014. [7] L. O. Chua, “Memristor—The missing circuit element,” IEEE Trans. Circuit Theory, vol. 18, no. 5, pp. 507–519, Sep. 1971. [8] D. B. Stukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 80–83, May 2008. [9] X. Y. Liu, Z. G. Zeng, and S. P. Wen, “Implementation of memristive neural networks with full-function Pavlov associative memory,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 63, no. 9, pp. 1454–1463, Sep. 2016. [10] Y. Hayakawa and K. Nakajima, “Design of the inverse function delayed neural networks for solving combinatorial optimization problems,” IEEE Trans. Neural Netw., vol. 21, no. 2, pp. 224–237, Feb. 2010.

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Qiang Xiao received the B.Eng. degree in digital media technology from Hunan University, Changsha, China, in 2013, and the M.S. degree in mathematics from Jimei University, Xiamen, China, in 2016. He is currently working toward the Ph.D. degree in control science and engineering from School of Automation, Huazhong University of Science and Technology, Wuhan, China. His current research interests include memristive neural networks, multiagent systems, and control theory.

1103

Zhigang Zeng (SM’07) received the Ph.D. degree in systems analysis and integration from Huazhong University of Science and Technology, Wuhan, China, in 2003. He is currently a Professor with the School of Automation, Huazhong University of Science and Technology, Wuhan, China, and also with the Key Laboratory of Image Processing and Intelligent Control of the Education Ministry of China, Wuhan, China. He has published more than 100 international journal papers. His current research interests include theory of functional differential equations and differential equations with discontinuous right-hand sides, and their applications to dynamics of neural networks, memristive systems, and control systems. Dr. Zeng has been an Associate Editor of theIEEE TRANSACTIONS ON NEURAL NETWORKS from 2010 to 2011, the IEEE TRANSACTIONS ON CYBERNETICS since 2014, the IEEE TRANSACTIONS ON FUZZY SYSTEMS since 2016. He has been a member of the Editorial Board of Neural Networks since 2012, Cognitive Computation since 2010, Applied Soft Computing since 2013.