Advances in Mathematical Physics
Volume 2015 (2015), Article ID 812150, 10 pages
http://dx.doi.org/10.1155/2015/812150

Research Article

Existence of Exponential -Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle

Ruofeng Rao1,2 and Shouming Zhong³

¹Department of Mathematics, Chengdu Normal University, Chengdu, Sichuan 611130, China
²Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China
³School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 18 February 2015; Accepted 14 June 2015

Academic Editor: Klaus Kirsten

Copyright © 2015 Ruofeng Rao and Shouming Zhong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The authors obtained a delay-dependent exponential -stability criterion for a class of Markovian jumping nonlinear diffusion equations by employing the Lyapunov stability theory and some variational methods. As far as we know, it is the first time to apply Ekeland variational principle to obtain the existence of exponential stability equilibrium of -Laplacian dynamic system so that some methods used in this paper are different from those methods of many previous related literatures. In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay.

1. Introduction

Nonlinear diffusion equations have been investigated extensively by many authors owing to their physics and biological engineering backgrounds, population dynamics, and so on (see [1–9] and references therein). In addition, Markovian jumping systems have attracted rapidly growing interest due to the fact that Markovian jumping parameters are useful in modeling abrupt phenomena, such as random failures, operating in different points of a nonlinear plant, and changing in the interconnections of subsystems (see [8, 10–12] and references therein). On the other hand, fuzzy logic theory has shown to be an appealing and efficient approach to deal with the analysis and synthesis problems for complex nonlinear systems. Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear system using some local linear subsystems (see [13–15] and references therein). As pointed out in the above related literature, the Markovian jumping T-S fuzzy mathematical models have always found their extensive applications in the real world. However, almost all the applications are greatly dependent on the stability of systems, which can often come down to the stability of the equilibrium solution for the corresponding mathematical models. So in this paper, we may consider the stability of the nonlinear -Laplace () diffusion fuzzy equations with Markovian jumping. Note that when , the so-called reaction-diffusion equations have been widely investigated (see [16–20] and references therein). For example, under Dirichlet boundary conditions, the existence result of -stability equilibrium solution in the sense of norm for a class of time-delay reaction-diffusion equations was obtained in [16]. Motivated by the abovementioned literature, in this paper, we will synthetically employ Ekeland variational principle, Sobolev imbedding theorem, and the Lyapunov functional method to study the existence of exponential -stability nonconstant equilibrium solution in the sense of norm () for delayed Markovian jumping fuzzy equations with nonlinear -Laplace diffusion ().

Let be a right-continuous Markov process on the complete probability space with a natural filtration and take values in the finite space with generator given by

where is transition probability rate from to    and , , and

Consider the following fuzzy T-S Markovian jumping -Laplace partial dynamic equations.

Fuzzy rule : IF is and … is THEN

where    is the premise variable and    is the fuzzy set that is characterized by membership function. is the number of the IF-THEN rules and is the number of the premise variables. Denote the premise variable vector and , where    is the membership function of the system with respect to the fuzzy rule can be regarded as the normalized weight of each IF-THEN rule, satisfying

Using a singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier, system (2) is inferred as follows:

with the initial condition

where is a bounded subset in with smooth boundary . is a scalar, and is a finite set. The divergence is the -Laplacian (see [21] for details). In mode , we denote and , which imply the connection strengths of the th neuron on the th neuron at the system mode , respectively. For any given , is a corresponding constant, dependent on .

Remark 1. Our methods employed in this paper are different from those of previous related literature. For example, homomorphic mapping theory was employed to obtain the existence of equilibrium of ordinary differential equations in [22]; topological degree theory was used to obtain the existence of equilibrium for fuzzy ordinary differential equations in [23] and of equilibrium for reaction-diffusion partial differential equations in [17]. In this paper, Ekeland variational principle is originally proposed to solve the existence of nonconstant equilibrium for nonlinear diffusion equations. Note that the abovementioned constant equilibrium point can actually be regarded as the special case of our nonconstant equilibrium point with ≡ constant for . In addition, our criterion about existence is only involved in the activation functions while other more parameters need be considered in the proof of the existence of constant equilibrium point in those previous literatures (see Remark 8 below for details).

2. Preparation

Throughout this paper, we assume the following:

(A1)There exists a positive definition matrix such that

(A2)There exist positive definite matrices and such that

Definition 2. is called a nonconstant equilibrium solution of PDEs (4) if and only if satisfies (4). In addition, the nonconstant equilibrium solution of PDEs (4) is called stochastically global exponential -stability about norm, if there are constants and   for every stochastic field solution of PDEs (4) such that

where one denotes by the norm and denotes Usually,

Definition 3. Let be a Banach space, , and . satisfies the condition if any sequence , such that

has a convergent subsequence. By the way, the above sequence with and is called the sequence of for a given

The following lemma originated from the famous Sobolev imbedding theorem.

Lemma 4. Let be a bounded subset in with smooth boundary . For , there exist the corresponding positive constants and such that, for any ,

where the Sobolev space is the completion of with respect to the norm

In 1979, Ekeland proposed the following famous Ekeland variational principle and its proof in [24]. As is well known, Ekeland variational principle has been the most important result in nonlinear analysis and has been applied to optimization theory, control theory, economic equilibrium theory, critical point theory, dynamic systems, and so forth. In this paper, we also need the following Ekeland variational principle.

Lemma 5 (Ekeland variational principle [24, Theorem 1]). Let be a complete metric space and let be a lower semicontinuous function, bounded from below and not identical to Let be given and let be such that

Then there exists such that

and, for each in ,

3. Main Result

Theorem 6. Let Assume that there exists a positive scalar with such that

where is an odd number and so is . Assume, in addition, .
If there exist a sequence of positive scalars    such that

where , for all , , , , , and

then there exists a nonconstant equilibrium solution for PDEs (4), which is stochastically global exponential -stability about norm.

Proof. The following proof may be divided into two big steps.

Step 1. Firstly, we need prove that there exists a nonconstant equilibrium solution for (4).
Consider the functional

where , , , and , where the Sobolev space is the completion of with respect to the norm
It is obvious that , for all If its critical point exists, then must be a nonconstant equilibrium solution of (4). So we only need to prove the existence of the critical point of for all .
Next, it follows from (13) that there exists a large enough such that

Furthermore, we can conclude by the continuity of and that

and hence,

where is a constant.
For , we can derive by and the restrictive conditions on ,

where is a constant.
On the other hand, for , we can get by and the restrictive conditions on ,

Hence,

Similarly, we can also deduce that

Denote and ; then and are constants, independent of . Similarly, we can prove that there exist and such that

where and are the constants, independent of
Thereby, we have

Since , we know that if and if And hence for all Besides, we know from the above analysis and the Sobolev imbedding theorem (Lemma 4) that there exist positive constants and , independent of , such that

Denote Owing to , there exists a large enough constant such that for all And hence

which implies that is bounded below. And the infimum may be defined as , where we denote for convenience and denote by its dual space. Define the operators as follows:

where denotes the adjoint pair for and . It follows by (A1) and (A2) that all , , are continuous. Similarly to (19), we can get

where is a constant. we can conclude from (13) and (29) that there exist positive constants and , independent of , such that

Here, and .
So we know from [25, 26] that both and are continuous, and is compact.
Next, we claim that, for each , the functional must satisfy the condition if only every sequence of is bounded.
Indeed, let be the sequence of for any given . It is obvious that

Owing to , we have If is bounded, we know from the reflexivity of the Banach space that there exists a weakly convergent subsequence (say, ). Since is compact, must own a convergent subsequence, which implies that must own a convergent subsequence. And furthermore, the continuity of yields that owns a convergent subsequence.
Below, we only need to prove that every sequence (say, ) of is bounded.
Indeed, similarly to (26), we have

Owing to the boundedness of , it is not difficult to prove by the application of reduction to absurdity that must be bounded in .
Now we may define the metric for the space as follows:

Then is a complete metric space with the above metric. From the continuity of and the above analysis, we know that is a lower semicontinuous function and bounded from below. Owing to , we can compute and deduce that , which implies , .
According to Ekeland variational principle, for given , there exists such that

So we can deduce from (34)

Here, is Gâteaux derivative of at , and is Fréchet derivative of at . Besides, (35) yields Then we can conclude from the condition that there exist a convergent subsequence and such that Moreover, the continuity of yields . Hence, has a critical point in for all . That is, there exists an equilibrium for (4).

Step 2. Below, we will prove the exponential -stability for the equilibrium point .
Consider the Lyapunov-Krasovskii functional as

where

Let be the weak infinitesimal operator; then for any given mode , taking the derivative of with respect to along the trajectory of (4) yields

Next, we claim that

To verify (40), we have to prove firstly the following proposition by the Yang inequality.

Proposition 7. For , one has

Proof. In fact, the Yang inequality yields

Synthesizing the above two inequalities results in (41).
So we can get by Gauss formula, the Dirichlet zero-boundary value, and Proposition 7

which proves (40).

In addition, we get by (A1)

Further, we can derive by (A2)

In addition, (A2) and the Yang inequality yield

Combining the above analyses results in

On the other hand, we have

Thus, we get by (14)

So we can obtain by the Dynkin formula

Hence, we have

which implies

where , Then we have

where we denote .

Now, we can conclude from Definition 2 that the nonconstant equilibrium solution of (4) is stochastically exponentially -stable about norm. And that completes the proof of Theorem 6.

Remark 8. In [27], existence theorems of stochastic differential equations on were given under some conditions on activation functions, where is a constant. And in [28, 29], existence theorems of stochastic differential equations were presented under some conditions on function . Motivated by [27], we proposed some conditions on activation functions to set up existence criterion for the equilibrium solution of system (4). In [22, 23], the constant equilibrium solution for all was obtained by homomorphic mapping theory and matrix theory, or matrix theory and homotopy invariance theorem, where , and each is a constant. In this paper, we also need to consider the equilibrium solution of (4) defined on . Different from [22, 23], we consider the nonconstant equilibrium solution for all This equilibrium solution is a solution for a nonlinear -Laplacian elliptic partial differential equation whose space frame may be considered as infinite dimension function space . And variational method is always a powerful tool to solve the problem. Although the variational method is more complicated than homomorphic mapping method, -matrix method, or homotopy invariance theorem, our criterion about existence is only involved in the activation functions (remark: condition (14) is not used in the proof of existence) and hence is simpler and more effective than other criteria, such as -matrix criteria and LMI-based criteria, because LMI-based criteria or -matrix criteria always involve the computer MATLAB programming in practical application while our condition (13) is easy to verify. So our existence criterion is actually simpler and more effective than LMI-based criteria and other criteria, which is the main contribution in this paper.

Remark 9. LMI-based stability criteria or -matrix stability criteria are always proposed in many literatures related to the mean square stability (see, e.g., [30–33] and references therein). However, when and , -stability criteria always involve more complicated mathematical method and mathematical deduction. For example, the stability criteria in [34] are not simpler than our stability criterion in Theorem 6. Similar phenomena exposed in many literatures related to -stability (see [15, 34–38]). Besides, the nonlinear -Laplacian () operator produces great difficulties in -stability proof. However, our condition (14) is still a LMI condition, which can be computed and verified by computer MATLAB LMI Toolbox in practical application.

4. Numerical Example

Example 1. Consider the 5-Laplace fuzzy T-S dynamical equations as follows.
Fuzzy Ruler 1. IF is , THEN

Fuzzy Ruler 2. IF is , THEN

equipped with the initial value

where , , ,  ,  , , ,  , , , , , , and , and . Consider , , and , ; let ,  , and ,  ; denote , . Let ,  ,  ,  , and Denote and ; then we can compute by computer MATLAB that , , , , , , , and , and hence , , , , and

which imply that condition (14) is satisfied. In addition, condition (13) is obviously satisfied. Therefore, there exists a nonconstant equilibrium solution for PDEs (54a)-(54b), which is stochastically global exponential -stability about norm (see Figures 1 and 2).

Figure 1: The state variable .

Figure 2: The state variable .

5. Conclusions

The nonlinear -Laplace () brings great difficulties to the proof of the existence of the -stability nonconstant equilibrium solution, for -Laplace () partial differential equations (PDEs) always need be considered in Banach space while the common linear Laplace PDEs can be studied in the setting of the special Hilbert space that can be orthogonally decomposed into the direct sum of the eigenfunction spaces. However, by applying Ekeland variational principle and the Yang inequality and constructing the suitable Lyapunov functional, we overcome those difficulties to obtain the existence of exponential -stability nonconstant equilibrium solution for Markovian jumping -Laplace () partial differential equations (PDEs) (4) under Dirichlet boundary condition. As far as we know, it is the first time to apply the Ekeland variational principle to solve the above problem so that some methods used in this paper are different from those of many previous related literatures (see Remark 1). In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very thankful to the anonymous reviewers for their suggestions, which improved the quality of this paper. This work was supported by the National Basic Research Program of China (2010CB732501), by the Scientific Research Fund of Science Technology Department of Sichuan Province (2012JY010), and by Sichuan Educational Committee Science Foundation (08ZB002, 12ZB349, and 14ZA0274).

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