# Contraction Mapping Theory and Approach to LMI-Based Stability Criteria of T-S Fuzzy Impulsive Time-Delays Integrodifferential Equations.

1. IntroductionIn this paper, we consider a class of integrodifferential equations, which is originated from Cohen-Grossberg Neural Networks (CGNNs). CGNNs model was proposed originally by Cohen-Grossberg in 1983 [1]. Since then, the stability analysis of CGNNs has attracted extensive attentions owing to its wide applications. Lyapunov function method is a common technique which derived stability criteria of various time-delays dynamic systems [2-13]. However, any method has its limitations. Fixed point technique may be one of common replacements. Brouwer fixed point theorem and Schauder fixed point theorem were always applied to stability analysis of various CGNNs models [14-17]. Of course, Banach fixed point theorem was also employed to derive the stability criteria of various CGNNs models [17-22]. But for most of existing literature, either using the same inverse function translates CGNNs model into another model similar as follows (see, e.g., [17,19-21]),

[x'.sub.i] (t) = -bi ([u.sup.-1] ([x.sub.i] (t))) + [n summation over (I=1)] [[b.sub.ij] (t) [f.sub.j] ([u.sup.-1] ([x.sub.j] (t))) (1)

+ [c.sub.ij] [f.sub.j] ([u.sup.-1] ([x.sup.j] (t-[tau](t))}}],

or the stability criteria are too complex [22, Theorem 3.1], which cannot be programmable for computer software while practical engineering is often involved in large-scale computing. Of course, both methods and results of [17, 19-22] are good and referential. But, in order to innovate, we have to find another way. In [18], a LMI-based stability criterion was given for the following CGNNs:

[mathematical expression not reproducible]. (2)

Theorem A (see [18, Theorem 4]). If (H1)-(H4) are satisfied and there exists a positive constant [alpha] < 1 such that the following LMI condition holds,

[mathematical expression not reproducible], (3)

then System (2) is globally exponentially stable, where the four conditions are proposed as follows.

(H1) For any j [member of] N, there exist constants [[bar.F].sub.j] > 0, [[bar.G].sub.j] > 0, [F.sub.j] > 0, and [G.sub.j] > 0 such that

[mathematical expression not reproducible]. (4)

(H2) For any j [member of] N, [a.sub.j](*) is differentiate, and there exists a constant [mathematical expression not reproducible] such that

[mathematical expression not reproducible], (5)

(H3) There exist nonnegative constants [mathematical expression not reproducible] such that

[mathematical expression not reproducible]. (6)

(H4) For j [member of] N, there exist [[GAMMA].sub.j] (t) and a constant [[gamma].sub.j] > 0 such that

[[GAMMA].sub.j] (t) [greater than or equal to] [[gamma].sub.j], [for all]t [greater than or equal to] 0,

[a.sub.j] (r) [b.sub.j] (r) = [[GAMMA].sub.j] (t) r, r [member of] R.

Remark 1. Denoting [x.sub.i](s) = [a.sub.i](s)[b.sub.i](s), we know from [18, (H4)] that [x.sub.i](s) can only be a linear function on s. So it is the main objective of this paper to make up the deficiency. Below, we try to make the condition of [x.sub.i](s) better so that [x.sub.i](s) may be a nonlinear function on s.

All the good results and methods in existing literature, particularly in [17, 19-22], inspired our current work. Of course, we cannot imitate (1) again, and the condition [18, (H4)] should be replaced by a new suitable condition. Below, we shall propose a new condition which is better than [18, (H4)]. Due to the changes of conditions, we have to formulate new contraction mapping to derive new LMI-based exponential stability criterion for Takagi-Sugeno (T-S) fuzzy impulsive CGNNs with discrete and distributed delays. This is one of main innovations of this paper. In addition, we shall admit weaker condition on the behavior functions than that of existing literature (e.g., [17, 19-22]) published from 2007 to 2016. Moreover, a corollary of our main result is better than [18, Theorem 3.1] due to the above reasons.

Remark 2. Due to the weaker condition on the behavior functions, both our results and methods are novelty (see below "Remark 13" and "Table 1" for details).

For convenience's sake, we introduce the following standard notations.

(i) L = [([l.sub.ij]).sub.nxn] > 0 (<0) is a positive (negative) definite matrix; that is, [y.sup.T] Ly > 0 (<0) for any 0 [not equal to] y [member of] [R.sup.n].

(ii) L = [([l.sub.ij]).sub.nxn] [greater than or equal to] 0 ([less than or equal to] 0) is a semipositive (seminegative) definite matrix; that is, [y.sup.T] Ly [greater than or equal to] 0 ([less than or equal to] 0) for any y [member of] [R.sup.n].

(iii) L [member of] [[L.sub.*], [L.sup.*]] implies that [l.sub.ij*] [less than or equal to] [l.sub.ij] [less than or equal to] [l.sup.*.sub.ij] for all i, j with L = [([l.sub.ij]).sub.nxn], [L.sub.*] = [([l.sub.ij*]).sub.nxn], and [L.sup.*] = [([l.sup.*.sub.ij]).sub.nxn].

(iv) [L.sub.1] [greater than or equal to] [L.sub.2] ([L.sub.1] [less than or equal to] [L.sub.2]) means matrix ([L.sub.1] - [L.sub.2]) is a semipositive (seminegative) definite matrix.

(v) [L.sub.1] > [L.sub.2] ([L.sub.1] < [L.sub.2)] means matrix ([L.sub.1] - [L.sub.2]) is a positive (negative) definite matrix.

(vi) [[lambda].sub.max] ([PHI]), [[lambda].sub.min] ([PHI]) denotes the largest and smallest eigenvalue of matrix [PHI], respectively.

(vii) Denote m = [absolute value of L] = [([absolute value of [l.sub.ij]]).sub.nxn] for any matrix L = [([l.sub.ij]).sub.nxn].

(viii) [absolute value of u] = [([absolute value of [u.sub.1]], [absolute value of [u.sub.2]], ..., [absolute value of [u.sub.n]]).sup.T] for any vector u = [([u.sub.1], [u.sub.2], ..., [u.sub.n]).sup.T] [member of] [R.sup.n].

(ix) u [less than or equal to] ([greater than or equal to]) v implies that [u.sub.i] <(>) [v.sub.i], [for all]i, and u < (>) V implies that [u.sub.i] < (>) [v.sub.i], [for all]i, for any vectors u = [([u.sub.1], [u.sub.2], ..., [u.sub.n]).sup.T] [member of] [R.sup.n] and v = [([v.sub.1], [v.sub.2], ..., [v.sub.n]).sup.T] [member of] [R.sup.n].

(x) I is identity matrix with compatible dimension.

2. Preliminaries

Cohen-Grossberg Neural Networks (CGNNs) with discrete and distributed delays have been investigated in many papers [23-28]. Consider the following impulsive CGNNs with discrete and distributed delays,

[mathematical expression not reproducible]. (8)

with the initial condition

x(s) = [phi](s), s [member of][-[tau],0], (9)

where x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] [R.sup.n] with [x.sub.i] (t) being the state variables of the ith neuron at time t. The neuron active functions f(x) = [([f.sub.1]([x.sub.1] (t)), [f.sub.2]([x.sub.2] (t)), ..., [f.sub.n]([x.sub.n](t))).sup.T] [member of] [R.sup.n] for any given t [member of] R. A(x(t)) = diag([a.sub.1]([x.sub.1](t)), a2(x2(t)),..., an(xn(t))) is n-dimension diagonal matrix with [a.sub.i]([x.sub.i](t)) representing an amplification function, and B(x(t)) = [([b.sub.1]([x.sub.1](t)), ..., [b.sub.n]([x.sub.n](t)).sup.T] [member of] [R.sup.n] with [b.sub.j]([x.sub.j](t)) being an appropriately behavior function. C = [([c.sub.ij]).sub.nxn], D = [([d.sub.ij]).sub.nxn], and E = [([e.sub.ij]).sub.nxn] are the connection weight matrices with [c.sub.ij], [d.sub.ij], [e.sub.ij] representing the strengths of connectivity between cells i and j at time t. Vector functions f(x(t)) = [([f.sub.1]([x.sub.1](t)), ..., [f.sub.n]([x.sub.n](t))).sup.T], g(x(t)) = [([g.sub.1](x1 (t - [tau](t))), ..., [g.sup.n](xn (t - [tau](t)).sup.T], and [eta](x(s)) = [([[eta].sub.1] ([x.sub.1](s)), ..., [[eta].sub.n] ([x.sub.n](s))).sup.T] with activation functions [f.sub.j](*), [g.sub.j](*),[[eta].sub.j](*) telling how the jth neuron reacts to the input. [tau](t) is discrete delay, and [delta](t) is distributed time-delay. [tau] is a positive constant with [delta](t), [tau](t) [member of] [0, [tau]]. [rho](x(t)) = [([[rho].sub.1]([x.sub.1](t)), ..., [[rho].sub.n]([x.sub.n](t))).sup.T] denotes the impulsive function, and the fixed impulsive moments [t.sub.k] (k = 1, 2, ...) satisfy 0< [t.sub.1] < [t.sub.2] < ... with [lim.sub.k[right arrow]+[infinity]] [t.sub.k] = +[infinity]. x([t.sup.+.sub.k]) and x([t.sup.-.sub.k]) stand for the right-hand and left-hand limit of x(t) at time [t.sub.k], respectively. Further, suppose that

[mathematical expression not reproducible]. (10)

In addition, we assume that [phi](s) [member of] C([-[tau], 0]; R), where C([-[tau],0];R) is the space of all the continuous functions defined on [-[tau], 0].

Since practice has shown that fuzzy logic theory is an efficient approach to deal with the analysis and synthesis problems for complex nonlinear system, the fuzzy model is far more important than stochastic model [29-31]. Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear systems using some local linear subsystems.

Below, we describe the T-S fuzzy mathematical model with time-delay as follows.

Fuzzy Rule j. IF [[omega].sub.1] (t) is [[mu].sub.j1], ..., [[omega].sub.s], (t) is [[mu].sub.js] * THEN

[mathematical expression not reproducible], (11)

where [[omega].sub.k] (t) (k = 1,2, ..., [s.sup.*]) is the premise variable, [[mu].sub.jk] (j = 1, 2, ..., r; k = 1, 2, ..., [s.sup.*]) is the fuzzy set that is characterized by membership function, r is the number of the IF-THEN rules, and [s.sup.*] is the number of the premise variables. By way of a standard fuzzy inference method, System (11) is inferred as follows:

[mathematical expression not reproducible], (12)

where [omega](t) = [[[omega].sub.1](t), [[omega].sub.2] (t), ..., [[omega].sub.s] (t)], [h.sub.j] ([omega](t)) = [w.sub.j] ([omega](t))/ [[summation].sup.r.sub.k=1] [w.sub.k] ([omega](t)), and [w.sub.j]([omega](t)) : [R.sup.s] [right arrow] [0,1] (j = 1, 2, ..., [r.sup.*]) is the membership function of the system with respect to the fuzzy rule j. [h.sub.j] can be regarded as the normalized weight of each IF-THEN rule, satisfying [[h.sub.j]([omega](t)) [greater than or equal to] 0 and [[summation].sup.r*.sub.j=1] [h.sub.j]([omega](t)) = l.

Throughout this paper, we assume that B(0) = f(0) = g(0) = [eta](0) = 0 [member of] [R.sup.n] and the following.

(A1) [f.sub.i], [g.sub.i], [[eta].sub.i], [p.sub.i] all are Lipschitz continuous functions with Lipschitz constants [F.sub.i], [G.sub.i], [Q.sub.i], [H.sub.i], respectively. In addition, there are positive constants [F.sup.*], [G.sup.*], [Q.sup.*] such that

[mathematical expression not reproducible]. (13)

(A2) There is positive definite diagonal matrix [bar.A] such that A(x(t)) [member of] [0, [bar.A]], and [a.sub.i](s) is Lipschitz continuous with Lipschitz constant [mathematical expression not reproducible], for all s [member of] R, i = 1, 2, ..., n, where A(x(t)) = diag([a.sub.1]([x.sub.1](t)), [a.sub.2]([x.sub.2](t)), ..., [a.sub.n]([x.sub.n](t))) is n-dimension diagonal matrix.

(A3) For each i, there exists a constant [m.sub.i] >0 such that the function ([a.sub.i](s)[b.sub.i](s)-[m.sub.i]s) is Lipschitz continuous with the Lipschitz constant [l.sub.i].

Remark 3. There are a large number of functions [a.sub.i](*), [b.sub.i](*) satisfying (A2) and (A3).

For example, we denote

[mathematical expression not reproducible]. (14)

Obviously, [phi](s) is continuous for all s [member of] R so that [max.sub.s[member of]R] [phi](s) exists, and [max.sub.s[member of]R][phi] (s) [greater than or equal to] [phi](0) > 0. Let [m.sub.i] = 1 + [max.sub.s[member of]R] [phi](s), and then the constant [m.sub.i] >1. Next, let

[a.sub.i] (s) = [m.sub.i] + [phi](s),

(15) [b.sub.i] (s) = s,

and then

[a.sub.i] (s) [b.sub.i] (s) = [m.sub.i] s + s[phi] (s),

(16) 0 < 1 [less than or equal to] [a.sub.i](s) [less than or equal to] 2[m.sub.i], [b.sub.i] (0) = 0.

Hence,

[mathematical expression not reproducible], (17)

and it is easy to prove that the above function ([a.sub.i](s)[b.sub.i](s)- [m.sub.i]s) is differentiable, and ([a.sub.i](s)[b.sub.i](s) - [m.sub.i]s)' [less than or equal to] 1 for all s [member of] R. So we can set l [less than or equal to] i = 1 such that

[mathematical expression not reproducible]. (18)

In addition, we can prove that [a.sub.i](s) = [m.sub.i] + [phi](s) is also Lipschitz continuous. Indeed,

[absolute value of [a.sub.i] (s) - [a.sub.i] (t)] = [absolute value of [phi](s) - [phi](t)]. (19)

Below, we only need to prove [phi](s) is Lipschitz continuous. Firstly, we claim that [phi](s) is differentiable for all s [member of] R. In fact, we can get by L'Hospital's rule and s = tan r

[mathematical expression not reproducible]. (20)

Hence, [phi](s) is differentiable for all s [member of] R, and [sup.sub.s[member of]R] [absolute value of [phi]'(s)] = [sup.sub.s[not equal to]0] [absolute value of (arctans/s)']. Further, Lagrangian differential mean value theorem derives

[mathematical expression not reproducible]. (21)

Owing to

[mathematical expression not reproducible], (22)

we can conclude that the continuous function (arctan s/s)' is bounded in all s [not equal to] 0, which implies that [sup.sub.s[not equal to]0] [absolute value of (arctans/s)'] is a positive constant, and hence both [phi](s) and [a.sub.i](s) are Lipschitz continuous for all s [member of] R. This has proved that [a.sub.i](s) and [b.sub.i](s) defined as (15) satisfy (A2) and (A3).

Remark4. Let [x.sub.i](s) = [a.sub.i](s)[b.sub.i](s); we know that condition (A3) admits that [x.sub.i](s) is a nonlinear function on s while [18, (H4)] derives that [x.sub.i](s) can only be a linear function on s.

Remark 5. Since (H4) is replaced with (A3), the methods of [18] cannot be applied to this paper, and we have to formulate new contraction mapping, different from [18].

Definition 6. Impulsive fuzzy CGNNs (12) with initial condition (9) are said to be globally exponentially stable if, for any initial condition [phi](s) [member of] C([-[tau], 0],[R.sup.n]), there exists a pair of positive constants a and b such that

[parallel]x(t; s,[phi])[parallel] [less than or equal to] b[e.sup.-at], [for all]t>0, (23)

where the norm [parallel]x(t)[parallel] = [([[summation].sup.n.sub.i=1] [[absolute value of [x.sub.i](t)].sup.2]).sup.1/2].

Lemma 7 (contraction mapping theorem, see [32]). Letting P be a contraction operator on a complete metric space [THETA], then there exists a unique point [theta] [member of] [THETA] for which P([theta]) = [theta].

3. Main Results

If (A1)-(A3) hold, we can derive the following main result.

Theorem 8. Impulsive fuzzy CGNNs (12) with initial condition (9) are global exponential stability if there exists a positive number [alpha] < 1, satisfying the following LMI condition:

[mathematical expression not reproducible], (24)

where [mu] = [inf.sub.k=1, 2, ...] ([t.sub.k+1] - [t.sub.k]) > 0, and

[mathematical expression not reproducible]. (25)

Proof. First of all, we need to formulate integral equations equivalent to (12).

Denote, for convenience, [mathematical expression not reproducible], and then [zeta](x(t)) = [([[zeta].sub.1]([x.sub.1](t)), [[zeta].sub.2]([x.sub.2](t)), ..., [[zeta].sub.n] [x.sub.n](t))).sup.T]. Thereby, we have

[zeta] (x (t)) = A(x (t)) B (x (t)) - Mx (t). (26)

And hence [zeta](0) = A(0)E(0) - M * 0 = 0.

Next, we claim that System (12) with initial condition (9) is equivalent to

[mathematical expression not reproducible], (27)

with initial condition (9).

On one hand, we can prove that the solution of System (27) with initial condition (9) is that of System (12) with initial condition (9).

Indeed, we get by (27)

[mathematical expression not reproducible], (28)

Differentiating both sides of (28) on t results in that for t [greater than or equal to] 0, t [not equal to] [t.sub.k],

[mathematical expression not reproducible], (29)

which generates the first equation of System (12).

Further, let t [right arrow] [t.sup.-.sub.i] in (27) produce x([t.sup.-.sub.i]) = [lim.sub.t[right arrow] [t.sup.- .sub.i]] x(t) = x([t.sub.i]), and let t [right arrow] [t.sup.+.sub.i] in (27) come to x([t.sup.+.sub.i]) = [lim.sub.t[right arrow] [t.sup.+.sub.i]] x(t) = x([t.sub.i]) + p(x(t [summation])), which derives the second equation of (12).

Thus, we have proved the above claim.

On the other hand, we claim that the solution of System (12) with initial condition (9) is that of System (27) with initial condition (9).

In fact, multiplying both sides of the first equation of System (12) with [e.sup.Mt] results in

[mathematical expression not reproducible], (30)

for all t [greater than or equal to] 0, t [not equal to] [t.sub.k]. Moreover, integrating from [t.sub.k-1] + [epsilon] to t [member of] ([t.sub.k-1], [t.sub.k]) gives

[mathematical expression not reproducible], (31)

which yields, after letting [epsilon] [right arrow] [0.sup.+],

[mathematical expression not reproducible], (32)

for all t [member of] ([t.sub.k-1], [t.sub.k]).

Throughout this paper, we assume that [epsilon] is a sufficiently small positive number. Now, taking t = [t.sub.k] - [epsilon] in (32) reaches

[mathematical expression not reproducible], (33)

which yields by letting [epsilon] [right arrow] [0.sup.+]

[mathematical expression not reproducible]. (34)

Combining (32) and the above equation comes to

[mathematical expression not reproducible], (35)

for all t [member of] ([t.sub.k-1], [t.sub.k]], k = 1, 2, ... Thereby, we have

[mathematical expression not reproducible]. (36)

Synthesizing the above analysis derives System (27). Hence, we have proved that each solution of (12) with initial condition (9) is that of (27) with initial condition (9). Now, we have proved that System (12) with initial condition (9) is really equivalent to integral equations (27) with initial condition (9).

To apply the contraction mapping theorem, we firstly define the complete metric space X as follows.

Let X be the space consisting of functions q(t) : [-[tau], [infinity]) [right arrow] [R.sup.n], satisfying the following:

(a) q(t) is continuous on t [member of] [0, +[infinity]) [{[t.sub.k]}.sup.[infinity].sub.k=1]

(b) q(t) = [phi](t), for t [member of] [-[tau],0].

(c) [mathematical expression not reproducible] and [mathematical expression not reproducible] exists, for all k = 1, 2, ...

(d) [e.sup.[beta]t] q(t) [right arrow] 0 [member of] [R.sup.n] as t [right arrow] + [infinity], where [beta] > 0 is a positive constant, satisfying [beta] < [[lambda].sub.min] M.

It is not difficult to verify that the product space X is a complete metric space if it is equipped with the following metric:

[mathematical expression not reproducible]. (37)

Hence, we define the mapping P as follows:

P(x(t))

[mathematical expression not reproducible], (38)

and P(x(s)) = [phi](s) for all s [member of] [-[tau], 0].

Below, we are to prove that P is contraction mapping from X into X.

We may firstly prove P(x) [subset] X. So we need to verify that P(x) satisfies conditions (a)-(d) for all x [member of] x.

Indeed, for x(t) [member of] x, p(x(t)) is continuous on t [member of] [0, +[infinity]) \ [{[t.sub.k]}.sup.[infinity].sub.k=1] owing to (38), and P(*) satisfies condition (a). Further, the definition of P implies that P(*) satisfies condition (b). Besides, we can derive from (38) that

[mathematical expression not reproducible], (39)

which comes to a conclusion that P(^) satisfies condition (c). Remark, the above convergence is under the metric of the metric space X. Below, all the convergence is in this sense. No longer repeat.

Throughout this section, we assume that e is a sufficiently small positive real number.

Below, we need to prove that P(*) satisfies condition (d). Indeed, for x(t) [member of] X, we have

[absolute value of [e.sup.[beta]t] P(x)] = [e.sup.[beta]t] [absolute value of P(x)]

(40)

[less than or equal to] [U.sub.1] + [U.sub.2] + [U.sub.3] + [U.sub.4] + [U.sub.5] + [U.sub.6],

where

[mathematical expression not reproducible]. (41)

Owing to condition (d), for x(t) [member of] x, [e.sup.[beta]t] x(t) [right arrow] 0 as t [right arrow] +[infinity]. So, for any given [epsilon] > 0, there exists sufficiently large [t.sub.*] > [tau] > 0 such that

[absolute value of [e.sup.[beta]t] x(t)] [less than or equal to] [epsilon][xi], [for all]t [greater than or equal to] [t.sub.*], (42)

where the fixed vector

[mathematical expression not reproducible]. (43)

When t > [t.sup.*], we have

[mathematical expression not reproducible]. (44)

It is obvious that

[mathematical expression not reproducible]. (45)

Besides,

[mathematical expression not reproducible], (46)

which together with arbitrariness implies that

[mathematical expression not reproducible]. (47)

Now we have actually proved that [U.sub.1] [right arrow] 0 as t [right arrow] +[infinity].

Similarly, we have

[mathematical expression not reproducible]. (48)

Similarly, we use the methods employed in (44)-(47), obtaining [U.sub.2] [right arrow] 0 as t [right arrow] +[infinity].

Besides,

[mathematical expression not reproducible]. (49)

Similarly, we use the methods employed in (44)-(47), obtaining [U.sub.3] [right arrow] 0 as t [right arrow] +[infinity].

Next,

[mathematical expression not reproducible]. (50)

It is clear that

[mathematical expression not reproducible]. (51)

Besides,

[mathematical expression not reproducible]. (52)

Similarly, we use the methods employed in (44)-(47), obtaining [U.sub.4] [right arrow] 0 as t [right arrow] +[infinity].

Obviously, [mathematical expression not reproducible].

Below, we assume that [t.sub.m] < [t.sub.*] [less than or equal to] [t.sub.m+1] and [t.sub.j] < t [less than or equal to] [t.sub.j+1] with j = 0, 1, 2, . ... Here, [t.sub.0] = 0.

[mathematical expression not reproducible]. (53)

Obviously,

[mathematical expression not reproducible]. (54)

Next,

[mathematical expression not reproducible], (55)

which together with the arbitrariness of [epsilon] implies

[mathematical expression not reproducible]. (56)

So we have proved that [U.sub.6] [right arrow] 0 as t [right arrow] +[infinity].

Synthesizing the above analysis results in [e.sup.[beta]t] P(x(t)) [right arrow] 0 as t [right arrow] +[infinity], for all x(t) [member of] X. Thus, P(*) satisfies all conditions (a)-(d), which derives P(x) [subset] x.

Finally, we claim that P : x [right arrow] x is contraction mapping. Indeed, for any x,y [member of] X,

[mathematical expression not reproducible], (57)

where

[mathematical expression not reproducible], (58)

where we assume [t.sub.j] < t [less than or equal to] [t.sub.j+1] with j = 0, 1, 2, ... Here,

[t.sub.0] = 0.

Combining the above five inequalities and (24) produces

[mathematical expression not reproducible], (59)

and hence

dist (P (x), P(x)) [less than or equal to] [alpha] dist (x, y). (60)

Therefore, P : x [right arrow] x is contraction mapping such that there exists the fixed point x(t) of P in X, which implies that x(t) is the solution for the impulsive fuzzy CGNNs (12) with initial condition (9), satisfying [e.sup.[beta]t] [parallel]x(t)[parallel] [right arrow] 0 as t [right arrow] +[infinity]. So the proof is completed.

Remark 9. From the proof, we know that the formulated contraction mapping is different from that of existing literature involved in the fixed point technique and CGNNs models. One can also understand from Remark 1 that we proposed in this paper a really feasible new condition on amplification function [a.sub.i] and behavior function [b.sub.i]. This implies that our result and methods are novelty versus [18, Theorem 4]. Besides, our conditions and result are also different from those of existing literature [17,19-22] (see below "Table 1" and "Remark 13").

In case of ignoring fuzzy factors and distributed delay (letting E = 0) in (12), we can get the following impulsive CGNNs with discrete delay

[mathematical expression not reproducible]. (61)

Corollary 10. Impulsive CGNNs (61) with initial condition (9) are global exponential stability if there is a positive number [alpha] < 1, satisfying the following LMI condition:

L + [bar.A] [absolute value of C] F + [F.sup.*] A [absolute value of C] + [bar.A] [absolute value of D] G + [G.sup.*] A [absolute value of D] + 1 / [mu] H

(62)

+ MH - [alpha]M <0,

where [mu] = [inf.sub.k=1, 2, ...] ([t.sub.k+1] - [t.sub.k]) > 0.

Remark 11. As a corollary of our main result, the conditions and conclusion are better than the main result of [18] (see Remark 1).

4. Numerical Example

Example 1. Consider the T-S fuzzy impulsive discrete and distributed delays CGNNs model as follows.

Fuzzy Rule 1. IF [[omega].sub.1] (t) is [mathematical expression not reproducible], THEN

[mathematical expression not reproducible], (63)

Fuzzy Rule 2. IF [[omega].sub.2] (t) is [mathematical expression not reproducible], THEN

[mathematical expression not reproducible], (64)

where [r.sup.*] =2, [F.sup.*] = 0.2897, [G.sup.*] = 0.2679, [Q.sup.*] = 0.2102,

[mathematical expression not reproducible]. (65)

Let [mu] = 0.5, [tau] = 11.7, then we can use MATLAB LMI toolbox to solve the LMI condition (24), extracting the feasible datum

[alpha] = 0.9981, (66)

which means 0 < [alpha] <1. Thereby, we can conclude from Theorem 8 that the impulsive fuzzy dynamic equations are globally exponentially stable.

Remark 12. From Example 1, the allowable upper bound of time-delays [tau] = 11.7 and the pulse interval [mu] = 0.5 indicate that the range of feasibility is extensive and broad.

Table 1 shows the comparison of our Theorem 8 and Corollary 10 with other criteria of CGNNs (derived by fixed point theorems) about the behavior function [b.sub.j] and programmability for computer software.

Remark 13. From Table 1, we know that the conditions of our behavior function [b.sub.j] are weaker than those of existing results published from 2007 to 2016. Moreover, our criterion is LMI-based, which is programmable for computer LMI toolbox. In addition, our Corollary 10 is better than [18, Theorem 4], the main result of [18] (see Remark 4 for details).

5. Conclusions

In many existing literatures related to Cohen-Grossberg Neural Networks, various fixed point theorems were applied to derive the stability criteria by imitating System (1). However, in this paper, the authors proposed a new condition on behavior functions and formulate new contraction mapping to derive a new LMI-based exponential stability. To a certain extent, this paper improves the previous related results (see Table 1, Remarks 4 and 13). Moreover, numerical example is presented to illustrate the feasibility and effectiveness of the proposed methods.

http://dx.doi.org/10.1155/2016/5035618

Competing Interests

The authors declare that they have no competing interests.

Authors' Contributions

All authors typed, read, and approved the final manuscript.

Acknowledgments

This work was supported by National Basic Research Program of China (2010CB732501), Scientific Research Fund of Science Technology Department of Sichuan Province (2012JY010), Sichuan Educational Committee Science Foundation (08ZB002, 12ZB349, and 14ZA0274), and the Initial Founding of Scientific Research for Chengdu Normal University Introduction of Talents.

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Ruofeng Rao (1) and Shouming Zhong (2)

(1) Department of Mathematics, Chengdu Normal University, Chengdu 61130, China

(2) Research Institute of Mathematics, Chengdu Normal University, Chengdu 61130, China

Correspondence should be addressed to Ruofeng Rao; ruofengrao@163.com

Received 1 September 2016; Accepted 17 November 2016

Academic Editor: Enrique Llorens-Fuster

Table 1: Comparing our Theorem 8 and Corollary 10 with other existing criteria on CGNNs derived by fixed point theorems. Fixed point theorems Our Theorem 8 (Corollary 10) Contraction mapping theorem [22, Theorem 3.1] Contraction mapping theorem [19, Theorem 1] Contraction mapping theorem [20, Theorems 1 and 2] Contraction mapping theorem [17, Theorems 1-3] Brouwer fixed point theorem [21, Theorem 3.1] Contraction mapping theorem Continuity of Differentiability [b.sub.i] of [b.sub.i] Our Theorem 8 (Corollary 10) Unnecessary Unnecessary [22, Theorem 3.1] Yes, necessary [22, (A2)] [19, Theorem 1] Yes, necessary [19, (A2)] [20, Theorems 1 and 2] Yes, necessary Yes, necessary [17, Theorems 1-3] Yes, necessary [17, (H3)] [21, Theorem 3.1] Yes, necessary Yes, necessary Programmability of criteria Our Theorem 8 (Corollary 10) Yes, LMI-based [22, Theorem 3.1] No [19, Theorem 1] No [20, Theorems 1 and 2] No [17, Theorems 1-3] No [21, Theorem 3.1] No Conditions [19, (A2)], [22, (A2)], and [17, (H3)] are similar to ([b.sub.i](s) - [b.sub.i] (t))/(s - t) [greater than or equal to] constant.

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Title Annotation: | Research Article |
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Author: | Rao, Ruofeng; Zhong, Shouming |

Publication: | Journal of Function Spaces |

Article Type: | Report |

Date: | Jan 1, 2017 |

Words: | 5929 |

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