Abstract
Under Neumann or Dirichlet boundary conditions, the stability of a class of delayed impulsive Markovian jumping stochastic fuzzy pLaplace partial differential equations (PDEs) is considered. Thanks to some methods different from those of previous literature, the difficulties brought by fuzzy stochastic mathematical model and impulsive model have been overcome. By way of the LyapunovKrasovskii functional, Itô formula, Dynkin formula and a differential inequality, new LMIbased global stochastic exponential stability criteria for the abovementioned PDEs are established. Some applications of the obtained results improve some existing results on neural networks. And some numerical examples are presented to illustrate the effectiveness of the proposed method due to the significant improvement in the allowable upper bounds of time delays.
MSC: 34D20, 34D23, 34B45, 34B37, 34K20.
Keywords:
differential inequality; Laplace diffusion; Markovian jumping1 Introduction
In this paper, we are concerned with the following delayed impulsive Markovian jumping stochastic fuzzy pLaplace partial differential equations (PDEs):
equipped with the boundary condition
where
where
Remark 1.1 PDEs (1.1) own a wide range of physics and engineering backgrounds. They admit the following three CohenGrossberg neural networks (CGNNs) as their special cases.
where
The stability of pLaplace diffusion stochastic CGNNs (1.2) was discussed by Xiongrui Wang, Ruofeng
Rao and Shouming Zhong in 2012 [2], and the stability of deterministic system (1.3) was investigated by Xinhua Zhang,
Shulin Wu and Kelin Li in 2011 [3]. Impulsive fuzzy CGNNs with nonlinear pLaplace diffusion has never been studied as far as we know, and such a situation
motivates our present study. Both the nonlinear pLaplace diffusion and fuzzy mathematical model bring a great difficulty in setting
up LMI criteria for the stability, and the stochastic functional differential equations
model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs
with diffusion, we have to construct a LyapunovKrasovskii functional in a nonmatrix
form (see, e.g., [4]). But stochastic mathematical formulae are always described in matrix forms. Furthermore,
an impulsive model makes it harder. Recently, some new methods were employed to study
the exponential stability for Markovian jumping, fuzzy neural networks in some related
literature (see, e.g., [514]). Inspired by some methods and the idea of [3,4] and the other abovementioned papers, we overcame the difficulties brought by the
Markovian jumping fuzzy impulsive model. By way of the LyapunovKrasovskii functional,
Itô formula, Dynkin formula, the variational methods in Sobolev space
2 Preliminaries
Throughout this paper, we always assume that the following five conditions hold.
(H1) There exists a positive definite diagonal matrix
(H2) There exist positive definite diagonal matrices
for all
(H3) There exist nonnegative symmetric matrices
where
(H4)
It is obvious from (H4) that system (1.1) admits a zero solution
For convenience’s sake, we introduce the following standard notations similar to those of [2].
In addition, we denote
Next, we give the following lemma, which is completely similar to [[1], Lemma 2.3]. It can be derived by the Gauss formula (see, e.g., [2]).
Lemma 2.1 ([[1], Lemma 2.3], [[11], Lemma 6])
Let
3 Main results
Theorem 3.1Assume that
(C1) there exist a sequence of positive scalars
where matrices
(C2)
(C3) there exists a constant
Then the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1) is globally stochastically exponentially stable in the mean square with the convergence
rate
Proof Consider the LyapunovKrasovskii functional
where
Let ℒ be the weak infinitesimal operator. Then it follows by Lemma 2.1 that
On the other hand, we have
From
So, we can conclude by (H1)(H4)
Completely similar to (2.7)(2.9) in [2], we can get by the Itô formula
Owing to
From (C3), it is not difficult to conclude that
i.e.,
Therefore, we can see by the definition of global stochastic exponential stability
(see, e.g., [15]) that the null solution of impulsive Markovian jumping stochastic fuzzy system (1.1)
is globally stochastically exponentially stable in the mean square with the convergence
rate
Particularly for the case of
Theorem 3.2Let
Proof Indeed, if
Then, by (3.3), we can similarly complete the rest of the proof by way of the methods in (3.4)(3.10). □
4 Applications of main results in neural networks
Let
(H1^{∗}) There exist positive definite diagonal matrices
for all
(H2^{∗}) There exists a positive definite diagonal matrix
(H3^{∗}) There exist positive definite diagonal matrices
for all
(H4^{∗}) There exist nonnegative symmetric matrices
where
(H5^{∗})
Applying our main results to CohenGrossberg neural networks (CGNNs), we can conclude the following corollary from Theorem 3.1 directly.
Corollary 4.1If the following three conditions hold:
(D1) there exist a positive scalar
where matrices
(D2)
(D3) there exists a constant
Then the null solution of impulsive stochastic fuzzy system (1.4) is globally stochastically exponentially stable in the mean square with the convergence
rate
Remark 4.1 Corollary 4.1 not only extends [[4], Theorem 3.1] from nonimpulsive stochastic fuzzy CGNNs to impulsive stochastic fuzzy CGNNs, but also improves the criterion of [[4], Theorem 3.1] from the nonmatrix form to the more condensed matrix form, which can be efficiently tested and verified by computer Matlab LMI toolbox.
If Markovian jumping and fuzzy factors are ignored, we can conclude the following corollary.
Corollary 4.2Assume that
(E1) there exist a sequence of positive scalars
where matrices
(E2)
(E3) there exists a constant
then the null solution of impulsive stochastic system (1.2) is globally stochastically exponentially stable in the mean square with the convergence
rate
Remark 4.2 If letting
Corollary 4.3If the following three conditions hold:
(F1) there exist a positive scalar
where matrices
(F2)
(F3) there exists a constant
then the null solution of impulsive deterministic system (1.3) is globally stochastically exponentially stable in the mean square with the convergence
rate
Remark 4.3 For the same reason as in Remark 4.2, the LMI (4.5) of Corollary 4.3 is more feasible and effective than that of [[3], Theorem 3.1], which may be illustrated by a numerical example below (Example 5.1).
5 Numerical examples
In this section, two examples are given to illustrate that the criteria of Corollary 4.2 and Corollary 4.3 can judge what some existing criteria cannot do. The third numerical example is presented to illustrate the effectiveness of our main results (Theorems 3.13.2).
Example 5.1 Consider impulsive system (1.3) with the following parameters:
In this section, we denote
Assume, in addition,
With the above data, one can use computer Matlab LMI toolbox to solve the LMI (C1)
of [[3], Theorem 3.1], and obtain
However, we can solve LMI (4.5) by Matlab LMI toolbox, and obtain
All the conditions (F1)(F3) of Corollary 4.3 are satisfied, then by Corollary 4.3
the null solution of impulsive deterministic system (1.3) is globally stochastically
exponentially stable in the mean square with the convergence rate 0.0049 and the allowable
upper bound of time delays
Example 5.2 Under the Neumann boundary condition, we consider stochastic system (1.2) with the data (4.1) and the following parameters:
Let
However, we solve LMIs (4.3)(4.4), and obtain
Example 5.3 Consider impulsive stochastic Markovian jumping fuzzy system (1.1) with the following parameters:
In addition,
The two cases of the transition rates matrices are considered as follows:
Now one can use Matlab LMI toolbox to solve the LMI conditions (3.1)(3.2) of Theorem 3.1
for Case (1) and
Moreover, a direct computation derives
By the definition of
Assume that the boundary condition is the Dirichlet boundary one, and then
Similarly, we can calculate and obtain
Let
Obviously in Case (2) we can assume
Let
Similarly, we can solve the corresponding conditions of Theorem 3.2 for Case (1) and
Further computation yields
Table 1 shows that the upper bounds of time delay decrease when there exist unknown elements of a transition rates matrix. This means that unknown elements of transition rates bring a great difficulty in judging the stability.
Table 1. Allowable upper bounds of time delays and the convergence rate
In some related literature [18,19], their impulsive assumption is
Remark 5.1 The parameters of impulsive deterministic system (1.3) do not satisfy the conditions of [[3], Theorem 3.1] so that we are not sure whether system (1.3) is stable, for the conditions of [[3], Theorem 3.1] are only sufficient ones, not necessary for the stability of system (1.3). However, we can conclude the stability by our Corollary 4.3, which implies that Corollary 4.3 allows for more effectiveness and less conservatism than [[3], Theorem 3.1]. By the same token as in Remark 5.2, Corollary 4.2 is better than [[2], Theorem 2.1].
Remark 5.2 Table 1 shows that the diffusion plays a positive role in the criterion of Theorem 3.2, which admits a wider range of time delays. Table 1 also illustrates the effectiveness and less conservatism of Theorems 3.13.2 due to the significant improvement in the allowable upper bounds of time delays.
Remark 5.3 Finding a solution x to the LMI system
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their valuable suggestions. This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2012JY010), and the Scientific Research Fund of Sichuan Provincial Education Department (12ZB349).
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