Volume 2013, Issue 1 261353

Research Article

Open Access

LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

Ruofeng Rao,

Corresponding Author

Ruofeng Rao

[email protected]

orcid.org/0000-0001-7911-1712

Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China yibinu.cn

Search for more papers by this author

Zhilin Pu,

Zhilin Pu

Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China yibinu.cn

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China sicnu.edu.cn

Search for more papers by this author

Ruofeng Rao,

Corresponding Author

Ruofeng Rao

[email protected]

orcid.org/0000-0001-7911-1712

Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China yibinu.cn

Search for more papers by this author

Zhilin Pu,

Zhilin Pu

Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China yibinu.cn

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China sicnu.edu.cn

Search for more papers by this author

First published: 25 September 2013

https://doi.org/10.1155/2013/261353

Citations: 5

Academic Editor: Chuanzhi Bai

Share a link

Email
Wechat
Bluesky

Abstract

By formulating a contraction mapping and the matrix exponential function, the authors apply linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in largescale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. At the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods.

1. Introduction

In this paper, we consider a class of delayed impulsive differential equations, which admits some biomathematics, physics, and engineering backgrounds, including the famous cellular neural networks proposed by Chua and Yang in 1988 [1, 2]. In practice, both time delays and impulse are unavoidable and may cause undesirable dynamic network behaviors such as oscillation and instability. So the stability analysis for delayed impulsive neural networks has become a topic of great theoretic and practical importance in recent years [3–25]. However, the research skill of the above literature is mainly based on Lyapunov theory. And there are many difficulties in applications of corresponding theory to the specific problems [26–32]. Recently, Burton and other authors have applied fixed point theory to investigate the stability of deterministic systems and obtained some more applicable results [11, 26–42]. For example, in [11], the authors used Leray-Schauders fixed point theorem to obtain the stability criteria of neural networks. Besides, the contraction-mapping theory is also an important fixed point theory in studying the stability of dynamics equations (see, e.g., [11, 33, 39–42]). On the other hand, fuzzy logic theory has shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear system [20–25]. In practice, the fuzzy model is far more important than stochastic model. 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. To the best of our knowledge, few authors have used the fixed point theorem to study the stability of Takagi-Sugeno fuzzy differential equations with impulses. In addition, the LMI-based stability criterion of neural networks has never been investigated or obtained via any of the fixed point theories. Such a situation motivates our present study. Motivated by the above related literature [3–9, 11, 26–42], we will not only apply the fixed point theory to study the impulsive Takagi-Sugeno fuzzy dynamics equations but also try to obtain the LMI-based stability criterion by applying the contraction-mapping theory. To the best of our knowledge, it is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. It is worth mentioning that LMI methods have high efficiency and other advantages in large-scale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. In the end of this paper, a numerical example is presented to illustrate the effectiveness of the proposed methods. Finally, a conclusion is given in the final chapter.

2. Preliminaries

Let us consider the following delayed differential equations:

()

equipped with the impulsive condition

()

and the initial condition

()

where x(t) = (x₁(t), x₂(t), …, x_n(t)) ^T ∈ Rⁿ, ϕ(θ) ∈ C[[−τ, 0], Rⁿ]. Functions f(x) = (f₁(x₁(t)), f₂(x₂(t)), …, f_n(x_n(t))) ^T ∈ Rⁿ, g(x(t − τ(t))) = (g₁(x₁(t − τ(t))), g₂(x₂(t − τ(t))), …, g_n(x_n(t − τ(t)))) ^T ∈ Rⁿ, ρ(x(t)) = (ρ₁(x₁(t)), ρ₂(x₂(t)), …, ρ_n(x_n(t))) ^T ∈ Rⁿ, and time delays 0 ⩽ τ(t) ⩽ τ for all i = 1,2, …, n. The fixed impulsive moments t_k (k = 1,2, …) satisfy 0 < t₁ < t₂ < ⋯ with lim _k→∞ t_k = ∞.

, and

stand for the right-hand and left-hand limits of x(t) at time t_k, respectively. We always assume

, for all k = 1,2, …. Similarly as in [33], we assume in this paper that f(0) = g(0) = ρ(0) = 0 ∈ Rⁿ.

Constant matrix B = diag (b₁, b₂, …, b_n) is a positive definite diagonal matrix, and both C = (c_ij) _n×n and D = (d_ij) _n×n are matrices with n × n dimension. It is well gknown that the above equation admits its practical implications. For example, it can serve as a model of impulsive cellular neural networks with time-varying delays. The parameter c_ij denotes the connection weight of the jth neuron on the ith neuron at time t. And the parameter d_ij represents the connection strength of the jth neuron on the ith neuron at time t − τ(t). The constant b_i represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs. f_j(x_j(t)) is the activation function of the jth neuron at time t, and g_j(x_j(t − τ(t))) represents the activation function of the jth neuron at time t − τ(t).

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

Fuzzy Rule j:

IF ω₁(t) is μ_j1 and …ω_s(t) is μ_js, THEN

()

where ω_k(t)(k = 1,2, …, s) is the premise variable, μ_jk (j = 1,2, …, r; k = 1,2, …, s) is the fuzzy set that is characterized by membership function, r is the number of the IF-THEN rules, and s is the number of the premise variables. By the way of a standard fuzzy inference method, the system (4) is inferred as follows:

()

where ω(t) = [ω₁(t), ω₂(t), …, ω_s(t)],

is the membership function of the system with respect to the fuzzy rule j. h_j can be regarded as the normalized weight of each IF-THEN rule, satisfying h_j(ω(t))⩾0 and

For convenience′s sake, we introduce the following standard notations similarly as [10, (iii)–(X)]:

()

In addition, we denote |C | = (|c_ij | ) _n×n for any matrix C = (c_ij) _n×n and |v | = (|v₁ | , |v₂ | , …, |v_n | ) ^T for any v = (v₁, v₂, …, v_n) ^T ∈ Rⁿ. Denote the finite set 𝒩 = {1,2, …, n}.

Throughout this paper, we assume

(A1)
f_j is locally Lipschitz continuous, and there exists a positive constant F_j > 0 such that for all r ∈ R at which f_j is differentiable;
(A2)
g_j is locally Lipschitz continuous, and there exists a positive constant G_j > 0 such that for all r ∈ R at which g_j is differentiable;
(A3)
ρ_j is locally Lipschitz continuous, and there exists a positive constant H_j > 0 such that for all r ∈ R at which ρ_j is differentiable.

Lemma 1 (see [25].)Let f : Rⁿ → Rⁿ be locally Lipschitz continuous. For any given x, y ∈ Rⁿ, there exists an element 𝔴 in the union ∪_z∈[x,y]∂f(z) such that

()

where [x, y] denotes the segment connecting x and y.

Remark 2. From Lemma 1, (A1)–(A3), and [10, equation (27)], we can similarly derive

()

where matrices F = diag (F₁, F₂, …, F_n), G = diag (G₁, G₂, …, G_n), and H = diag (H₁, H₂, …, H_n).

Remark 3. In many previous literature, f_i, g_i (i ∈ 𝒩) are always assumed to be globally Lipschtiz continuous. However, the most common function f_i(r) = r² is not globally Lipschtiz continuous in R¹. Note that we extend the functions f, g from global Lipschtiz continuous functions to locally Lipschtiz continuous functions. Obviously, f_i(r) = r² is a local Lipschitz continuous function.

Similarly as is [11, Definition 2.2], the exponential stability is defined as follows.

Definition 4. Dynamic equation (5) is said to be exponentially stable if, for any initial condition ϕ(s) ∈ C[[−τ, 0], Rⁿ], there exists a pair of positive constants a and b such that

()

where the norm

3. Main Result

Before giving the main result of this paper, we need to define the matrix exponential function as follows.

Definition 5. For a diagonal constants matrix B = diag (b₁, b₂, …, b_n), we denote the matrix exponential function for all t ∈ R.

From the above definition of the matrix exponential function, we are not difficult to obtain the following lemma.

Lemma 6. Let B be a diagonal constants matrix, and let e^Bt be the matrix exponential function of B. Then, we have

(1)
(d/dt)e^Bt = Be^Bt, t ∈ R,
(2)
(d/dt)(e^Btη) = Be^Btη, t ∈ R,

where η = (η₁, η₂, …, η_n) ^T ∈ Rⁿ, and each η_i ∈ R (i = 1,2, …, n) is a constant.

In addition, we need to define the rule on vectors in Rⁿ as follows.

Definition 7. v ⩽ w if v_i − w_i ⩽ 0 for all i ∈ 𝒩, where v = (v₁, v₂, …, v_n) ^T ∈ Rⁿ, w = (w₁, w₂, …, w_n) ^T ∈ Rⁿ.

Now, we present the main result of this paper as follows.

Theorem 8. Assume that there exists a positive constant δ such that inf _k=1,2,…(t_k+1 − t_k)⩾δ. In addition, there exists a constant 0 < λ < 1 such that

()

where B⁻¹ is the inverse matrix of B. Then, the impulsive fuzzy dynamic equation (5) is exponentially stable in the mean square.

Proof. To apply the fixed point theory, we firstly define a complete metric space Ω as follows.

Let Ω be the space consisting of functions q(t):[−τ, ∞) → Rⁿ, satisfying that

(a)
q(t) is continuous on t ≠ t_k (k = 1,2, …),
(b)
and exist, and for all k = 1,2, …,
(c)
q(t) = ϕ(t) for t ∈ [−τ, 0],
(d)
e^βtq(t) → 0 ∈ Rⁿ as t → ∞, where β > 0 is a positive constant, satisfying β < λ_minB.

It is not difficult to verify that the above-mentioned space Ω is a complete metric space if it is equipped with the following metric:

()

where

, and

Remark 9. Here, we consider the above-defined metric, which is different from those of some previous related literature (see, e.g., [33]) so that the LMI-based stability criterion in this paper may be obtained expediently.

Next, we formulate and define a contraction mapping P : Ω → Ω, which may be divided into three steps.

Step 1. Formulating the mapping.

Let x(t) be a solution of the fuzzy equation (5).

Then, for t⩾0, t ≠ t_k, we have

()

Further, we get by the integral nature

()

where η ∈ Rⁿ is the vector to be determined.

From (2) and (3) or x(0) = ϕ(0), it is not difficult to conclude . And, hence,

()

On the other hand, it follows from (14) that

()

Let ε → 0⁺; then we can derive from the two equations above that

()

So, we may define the mapping P on the space Ω as follows:

()

on t⩾0, and Px(s) = ϕ(s) on s ∈ [−τ, 0].

Step 2. We claim that Px(t) ∈ Ω for any x(t) ∈ Ω. That is, Px(t) satisfies the conditions (a)–(d) of Ω.

Indeed, since Px(s) = ϕ(s) on s ∈ [−τ, 0], the condition (c) is satisfied. It is obvious from (17) that Px(t) is continuous on t ≠ t_k and t⩾0. And then the condition (a) is satisfied.

Next, we verify the condition (b).

Indeed, for any given t_k, we can get from (17)

()

where

()

Obviously, π₁ → 0 and π₂ → 0 as ε → 0. In addition, letting ε → 0⁻, we have π₃ → 0. Letting ε → 0⁺, we get by (18)

()

So the condition (b) is satisfied.

Finally, it is followed from (17) that

()

Obviously, e^βte^−Btϕ(0) → 0 ∈ Rⁿ as t → ∞.

On the other hand, it follows from e^βtx(t) → 0 that, for any given ε > 0, there exists a corresponding constant t_* such that |e^βtx(t)| < εu for all t⩾t_*, where

()

In addition, (A3) derives

()

Obviously,

()

Besides,

()

To prove , we only need to prove

()

for the other cases can be similarly proved.

Indeed, we may as well assume t_k < t ⩽ t_k+1 and t_m−1 < t_* ⩽ t_m. Then, we have

()

which implies that (26) holds. And, hence,

Below, we only need to prove

()

In fact, it follows from e^βtx(t) → 0 that, for any given ε > 0, there exists a corresponding constant t^* such that |e^βtx(t)| < εu for all t⩾t^*. Then, we get by (A1)

()

On the one hand,

()

where

On the other hand,

()

where

Now we can conclude from (29)–(31) that

()

Similarly, we can prove

()

Indeed, we can similarly define the corresponding constant T_* for any given ε > 0, satisfying |e^βtx(t)| < εu for all t⩾T_*. Then, we have

()

Similarly, we can prove

()

where

On the other hand,

()

where

From (34)–(36), we can conclude that (33) holds. Hence, the condition (d) is satisfied.

Step 3. Below, we only need to prove that P is a contraction mapping.

Indeed, for any x = x(t) = (x₁(t), x₂(t), …, x_n(t)) ^T, y = y(t) = (y₁(t), y₂(t), …, y_n(t)) ^T ∈ Ω, we estimate |Px(t) − Py(t)| ⩽ K₁ + K₂ + K₃, where

()

From mathematical analysis and computation, we can derive

()

Similarly, we have

()

Combining the above three inequalities results in

()

and hence

()

Therefore, P : Ω → Ω is a contraction mapping such that there exists the fixed point x(t) of P in Ω, which implies that x(t) is the solution for the the impulsive fuzzy dynamic equation (5), satisfying e^βt∥x(t)∥→0 as t → ∞. So the proof is completed.

4. Numerical Example

Example 1. Consider the T-S fuzzy impulsive dynamic equations as follows.

Fuzzy Rule 1:

IF ω₁(t) is , THEN

()

Fuzzy Rule 2:

IF ω₂(t) is , THEN

()

where

()

Let δ = 1.5. Then we can use Matlab LMI toolbox to solve the LMI condition (10) and obtain tmin = −0.0046 < 0 which implies it is feasible (see [10, Remark 29(3)] for details). Further, extracting the datum shows

()

which means 0 < λ < 1. Thereby, we can conclude from Theorem 8 that this impulsive fuzzy dynamic equation is exponentially stable in the mean square.

5. Conclusion

By formulating a contraction mapping and the matrix exponential function, the author applies linear matrix inequality (LMI) technique to investigate and obtain the LMI-based stability criterion of a class of time-delay Takagi-Sugeno (T-S) fuzzy differential equations. It is the first time to obtain the LMI-based stability criterion derived by a fixed point theory. The LMI methods have high efficiency and other advantages in large-scale engineering calculations. And the feasibility of LMI-based stability criterion can efficiently be computed and confirmed by computer Matlab LMI toolbox. A numerical example is presented to illustrate the effectiveness of the proposed methods. In the end of this paper, we have to point out that there are still many difficulties in obtaining the LMI-based stability criteria for some other dynamics equations, such as Cohen-Grossberg neural networks (see, e.g., [11, 19]) and other neural networks. These problems remain open and challenging.

Acknowledgments

This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2010JY0057, 2012JYZ010) and the Scientific Research Fund of Sichuan Provincial Education Department (12ZB349).

References

1 Chua L. O. and Yang L., Cellular neural networks: theory, IEEE Transactions on Circuits and Systems. (1988) 35, no. 10, 1257–1272, https://doi.org/10.1109/31.7600, MR960777, ZBL0663.94022.
10.1109/31.7600
Web of Science® Google Scholar
2 Chua L. O. and Yang L., Cellular neural networks: applications, IEEE Transactions on Circuits and Systems. (1988) 35, no. 10, 1273–1290, https://doi.org/10.1109/31.7601, MR960778.
10.1109/31.7601
Web of Science® Google Scholar
3 Xu D., Wang X., and Yang Z., Further results on existence-uniqueness for stochastic functional differential equations, Science China Mathematics. (2013) 56, no. 6, 1169–1180, https://doi.org/10.1007/s11425-012-4553-1, MR3063963.
10.1007/s11425-012-4553-1
Web of Science® Google Scholar
4 Xu D. and Long S., Attracting and quasi-invariant sets of non-autonomous neural networks with delays, Neurocomputing. (2012) 77, no. 1, 222–228, 2-s2.0-80955157450, https://doi.org/10.1016/j.neucom.2011.09.004.
10.1016/j.neucom.2011.09.004
Web of Science® Google Scholar
5 Xu D. and Xu L., New results for studying a certain class of nonlinear delay differential systems, IEEE Transactions on Automatic Control. (2010) 55, no. 7, 1641–1645, https://doi.org/10.1109/TAC.2010.2048939, MR2675824.
10.1109/TAC.2010.2048939
Web of Science® Google Scholar
6 Xu D. and Wang X., A new nonlinear integro-differential inequality and its application, Applied Mathematics Letters. (2009) 22, no. 11, 1721–1726, https://doi.org/10.1016/j.aml.2009.06.007, MR2560984, ZBL1177.26030.
10.1016/j.aml.2009.06.007
Web of Science® Google Scholar
7 Xu D. and Yang Z., Attracting and invariant sets for a class of impulsive functional differential equations, Journal of Mathematical Analysis and Applications. (2007) 329, no. 2, 1036–1044, https://doi.org/10.1016/j.jmaa.2006.05.072, MR2296904, ZBL1154.34393.
10.1016/j.jmaa.2006.05.072
Web of Science® Google Scholar
8 Xu D., Yang Z., and Huang Y., Existence-uniqueness and continuation theorems for stochastic functional differential equations, Journal of Differential Equations. (2008) 245, no. 6, 1681–1703, https://doi.org/10.1016/j.jde.2008.03.029, MR2436457, ZBL1161.34055.
10.1016/j.jde.2008.03.029
Web of Science® Google Scholar
9 Xu D. and Zhao H., Invariant set and attractivity of nonlinear differential equations with delays, Applied Mathematics Letters. (2002) 15, no. 3, 321–325, https://doi.org/10.1016/S0893-9659(01)00138-0, MR1891554, ZBL1023.34066.
10.1016/S0893-9659(01)00138-0
Web of Science® Google Scholar
10 Rao R., Wang X., Zhong S., and Pu Z., LMI approach to exponential stability and almost sure exponential stability for stochastic fuzzy Markovian-jumping Cohen-Grossberg neural networks with nonlinear p-Laplace diffusion, Journal of Applied Mathematics. (2013) 2013, 21, 396903, MR3064876.
10.1155/2013/396903
Google Scholar
11 Yang X., Huang C., Zhang D., and Long Y., Dynamics of Cohen-Grossberg neural networks with mixed delays and impulses, Abstract and Applied Analysis. (2008) 2008, 14, 432341, https://doi.org/10.1155/2008/432341, MR2471251, ZBL1160.37431.
10.1155/2008/432341
Google Scholar
12 Wan L. and Sun J., Global asymptotic stability of Cohen-Grossberg neural network with continuously distributed delays, Physics Letters A. (2005) 342, no. 4, 331–340, 2-s2.0-21244436276, https://doi.org/10.1016/j.physleta.2005.05.026, ZBL1222.93200.
10.1016/j.physleta.2005.05.026
CAS Web of Science® Google Scholar
13 Rao R. F., Zhong S. M., and Wang X. R., Stochastic stability criteria with LMI conditions forMarkovian jumping impulsive BAM neural networks with mode-dependent time-varying delaysand nonlinear reaction-diffusion, Communications in Nonlinear Science and Numerical Simulation. (2013) https://doi.org/10.1016/j.cnsns.2013.05.024.
10.1016/j.cnsns.2013.05.024
PubMed Web of Science® Google Scholar
14 Song Q. and Zhang J., Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays, Nonlinear Analysis: Real World Applications. (2008) 9, no. 2, 500–510, https://doi.org/10.1016/j.nonrwa.2006.11.015, MR2382394, ZBL1142.34046.
10.1016/j.nonrwa.2006.11.015
Web of Science® Google Scholar
15 Song Q. and Cao J., Impulsive effects on stability of fuzzy Cohen-Grossberg neural networks with time-varying delays, IEEE Transactions on Systems, Man, and Cybernetics B. (2007) 37, no. 3, 733–741, 2-s2.0-34249053502, https://doi.org/10.1109/TSMCB.2006.887951.
10.1109/TSMCB.2006.887951
PubMed Web of Science® Google Scholar
16 Song Q. and Wang Z., Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays, Physica A. (2008) 387, no. 13, 3314–3326, 2-s2.0-40849111420, https://doi.org/10.1016/j.physa.2008.01.079.
10.1016/j.physa.2008.01.079
Web of Science® Google Scholar
17 Bai C., Stability analysis of Cohen-Grossberg BAM neural networks with delays and impulses, Chaos, Solitons & Fractals. (2008) 35, no. 2, 263–267, https://doi.org/10.1016/j.chaos.2006.05.043, MR2357001, ZBL1166.34328.
10.1016/j.chaos.2006.05.043
Web of Science® Google Scholar
18 Li B. and Xu D. Y., Exponential p-stability of stochastic recurrent neural networks with mixeddelays and Markovian switching, Neurocomputing. (2013) 103, 239–246, https://doi.org/10.1016/j.neucom.2012.09.026.
10.1016/j.neucom.2012.09.026
Web of Science® Google Scholar
19 Li B. and Xu D., Existence and exponential stability of periodic solution for impulsive Cohen-Grossberg neural networks with time-varying delays, Applied Mathematics and Computation. (2012) 219, no. 5, 2506–2520, https://doi.org/10.1016/j.amc.2012.08.086, MR2988131.
10.1016/j.amc.2012.08.086
Web of Science® Google Scholar
20 Song Q., Wang Z., and Liang J., Analysis on passivity and passification of T-S fuzzy systems with time-varying delays, Journal of Intelligent & Fuzzy Systems. (2013) 24, no. 1, 21–30, MR3031202.
10.3233/IFS-2012-0504
Web of Science® Google Scholar
21 Balasubramaniam P., Vembarasan V., and Rakkiyappan R., Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays, Communications in Nonlinear Science and Numerical Simulation. (2011) 16, no. 4, 2109–2129, https://doi.org/10.1016/j.cnsns.2010.08.024, MR2736076, ZBL1221.93254.
10.1016/j.cnsns.2010.08.024
Web of Science® Google Scholar
22 Muralisankar S., Gopalakrishnan N., and Balasubramaniam P., An LMI approach for global robust dissipativity analysis of T-S fuzzy neural networks with interval time-varying delays, Expert Systems with Applications. (2012) 39, no. 3, 3345–3355, 2-s2.0-80255134461, https://doi.org/10.1016/j.eswa.2011.09.021.
10.1016/j.eswa.2011.09.021
Web of Science® Google Scholar
23 Rao R. F. and Pu Z. L., Stability analysis for impulsive stochastic fuzzy p-Laplace dynamic equations under Neumann or Dirichlet boundary condition, Boundary Value Problems. (2013) 2013, article 133, https://doi.org/10.1186/1687-2770-2013-133.
10.1186/1687-2770-2013-133
Google Scholar
24 Pu Z. L. and Rao R. F., Stability analysis of Takagi-Sugeno fuzzy Markovian jumping PDEswith p-Laplace under zero-boundary condition, to appear Journal of Applied Mathmatics.
Google Scholar
25 Balasubramaniam P., Vembarasan V., and Rakkiyappan R., Delay-dependent robust asymptotic state estimation of Takagi-Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays, Expert Systems with Applications. (2012) 39, no. 1, 472–481, 2-s2.0-81855205008, https://doi.org/10.1016/j.eswa.2011.07.038.
10.1016/j.eswa.2011.07.038
Web of Science® Google Scholar
26 Luo J., Fixed points and stability of neutral stochastic delay differential equations, Journal of Mathematical Analysis and Applications. (2007) 334, no. 1, 431–440, https://doi.org/10.1016/j.jmaa.2006.12.058, MR2332567, ZBL1160.60020.
10.1016/j.jmaa.2006.12.058
Web of Science® Google Scholar
27 Luo J., Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, Journal of Mathematical Analysis and Applications. (2008) 342, no. 2, 753–760, https://doi.org/10.1016/j.jmaa.2007.11.019, MR2433617, ZBL1157.60065.
10.1016/j.jmaa.2007.11.019
Web of Science® Google Scholar
28 Luo J., Stability of stochastic partial differential equations with infinite delays, Journal of Computational and Applied Mathematics. (2008) 222, no. 2, 364–371, https://doi.org/10.1016/j.cam.2007.11.002, MR2474632, ZBL1151.60336.
10.1016/j.cam.2007.11.002
Web of Science® Google Scholar
29 Luo J. and Taniguchi T., Fixed points and stability of stochastic neutral partial differential equations with infinite delays, Stochastic Analysis and Applications. (2009) 27, no. 6, 1163–1173, https://doi.org/10.1080/07362990903259371, MR2573454, ZBL1177.93094.
10.1080/07362990903259371
Web of Science® Google Scholar
30 Sakthivel R. and Luo J., Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, Journal of Mathematical Analysis and Applications. (2009) 356, no. 1, 1–6, https://doi.org/10.1016/j.jmaa.2009.02.002, MR2524209, ZBL1166.60037.
10.1016/j.jmaa.2009.02.002
Web of Science® Google Scholar
31 Sakthivel R. and Luo J., Asymptotic stability of nonlinear impulsive stochastic differential equations, Statistics & Probability Letters. (2009) 79, no. 9, 1219–1223, https://doi.org/10.1016/j.spl.2009.01.011, MR2519005, ZBL1166.60316.
10.1016/j.spl.2009.01.011
Web of Science® Google Scholar
32 Luo J., Fixed points and exponential stability for stochastic Volterra-Levin equations, Journal of Computational and Applied Mathematics. (2010) 234, no. 3, 934–940, https://doi.org/10.1016/j.cam.2010.02.013, MR2606678, ZBL1197.60053.
10.1016/j.cam.2010.02.013
Web of Science® Google Scholar
33 Lai X. and Zhang Y., Fixed point and asymptotic analysis of cellular neural networks, Journal of Applied Mathematics. (2012) 12, 689845, MR2965701.
Google Scholar
34 Burton T. A., Stability by Fixed Point Theory for Functional Differential Equations, 2006, Dover, Mineola, NY, USA, MR2281958.
Google Scholar
35 Rao R. F., Zhong S. M., and Wang X. R., Delay-dependent exponential stability for Markovianjumping stochastic Cohen-Grossberg neural networks with p-Laplace diffusion and partiallyknown transition rates via a differential inequality, Advances in Difference Equations. (2013) 2013, article 183, https://doi.org/10.1186/1687-1847-2013-183.
10.1186/1687-1847-2013-183
Google Scholar
36 Burton T. A., Fixed points, stability, and exact linearization, Nonlinear Analysis: Theory, Methods & Applications. (2005) 61, no. 5, 857–870, https://doi.org/10.1016/j.na.2005.01.079, MR2130068, ZBL1067.34077.
10.1016/j.na.2005.01.079
Web of Science® Google Scholar
37 Jin C. and Luo J., Fixed points and stability in neutral differential equations with variable delays, Proceedings of the American Mathematical Society. (2008) 136, no. 3, 909–918, https://doi.org/10.1090/S0002-9939-07-09089-2, MR2361863, ZBL1136.34059.
10.1090/S0002-9939-07-09089-2
Web of Science® Google Scholar
38 Raffoul Y. N., Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Mathematical and Computer Modelling. (2004) 40, no. 7-8, 691–700, https://doi.org/10.1016/j.mcm.2004.10.001, MR2106161, ZBL1083.34536.
10.1016/j.mcm.2004.10.001
Google Scholar
39 Wu M., Huang N. j., and Zhao C. W., Fixed points and stability in neutral stochastic differential equations with variable delays, Fixed Point Theory and Applications. (2008) 2008, 11, 407352, MR2425679, ZBL1255.60102, https://doi.org/10.1155/2008/407352.
10.1155/2008/407352
Google Scholar
40 Zhang B., Fixed points and stability in differential equations with variable delays, Nonlinear Analysis: Theory, Methods & Applications. (2005) 63, no. 5–7, e233–e242, 2-s2.0-28044433663, https://doi.org/10.1016/j.na.2005.02.081, ZBL1159.34348.
10.1016/j.na.2005.02.081
Web of Science® Google Scholar
41 Zhang Y. T. and Luo Q., Global exponential stability of impulsive cellular neural networks withtime-varying delays via fixed point theory, Advances in Difference Equations. (2013) 2013, article 23.
Google Scholar
42 Ding L., Li X., and Li Z., Fixed points and stability in nonlinear equations with variable delays, Fixed Point Theory and Applications. (2010) 2010, 14, 195916, MR2738314, ZBL1214.34061, https://doi.org/10.1155/2010/195916.
10.1155/2010/195916
Google Scholar

Citing Literature

All articles

LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

Abstract

1. Introduction

2. Preliminaries

3. Main Result

4. Numerical Example

5. Conclusion

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

LMI-Based Stability Criterion of Impulsive T-S Fuzzy Dynamic Equations via Fixed Point Theory

Abstract

1. Introduction

2. Preliminaries

3. Main Result

4. Numerical Example

5. Conclusion

Acknowledgments

References

Citing Literature

References

Related

Information