Robust stability of probabilistic delays fuzzy stochastic p-Laplace dynamic equations
© Wang and Rao; licensee Springer. 2014
Received: 27 August 2014
Accepted: 6 November 2014
Published: 26 November 2014
In this paper, the stability of a class of time-delay Takagi-Sugeno (T-S) fuzzy Markovian jumping partial differential equations (PDEs) with p-Laplace and probabilistic time-varying delays is investigated, and the robust exponential stability criterion is obtained by way of some variational methods in Sobolev space , the Lyapunov functional method and the linear matrix inequalities technique. Moreover, a numerical example shows the effectiveness of the proposed methods due to the large allowable variation range of time delay.
1 Introduction and preparation
Given a complete probability space with a natural filtration , where Ω is a sample space, ℱ is σ-algebra of a subset of the sample space, and ℙ is the probability measure defined on ℱ. Let and the random form process be a homogeneous, finite-state Markovian process with right continuous trajectories with generator and transition probability from mode i at time t to mode j at time , , if , and if , where is transition probability rate from i to j () and , and .
where is a standard one-dimensional Brownian motion defined on the probability space. is a positive scalar, is a bounded domain with a smooth boundary ∂ Ω of class by Ω, . In what follows, is always denoted by u for convenience sake. denotes the Hadamard product of matrix and (see  or ), and satisfies for all j, k, . In mode , we denote and . Denote by the time delay which satisfies for any mode . Functions , , . The boundary condition (1.1a) is called Dirichlet boundary condition if and Neumann boundary condition if . Here, denotes the outward normal derivative on ∂ Ω.
The T-S fuzzy mathematical model with time delay is described as follows.
Fuzzy rule j:
where () is the premise variable, (; ) 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.
where , , () is the membership function of the system with respect to the fuzzy rule j. can be regarded as the normalized weight of each IF-THEN rule, satisfying and .
(A1) Let , , and such that , ;
(A2) Let , there exists a positive definite diagonal matrix such that , , and ;
(A3) There exist constant diagonal matrices , , with , , , such that , , , and .
(A4) There exist positive define symmetric matrices , , such that , .
(A5) for any mode , and .
Here is an unknown matrix function satisfying , and , , are known real constant matrices. Throughout this paper, for a matrix , we denote the matrix . In addition, we denote by I the identity matrix with compatible dimension, and denote .
Lemma 1.1 Let be any given scalar, and ℳ, and be matrices with appropriate dimensions. If , then we have .
Lemma 1.2 ([, Lemma 6])
2 Main result
where ; ; ; ; ; .
Proof Consider the Lyapunov-Krasovskii functional , , where , and = [ + ].
It follows immediately by Lemma 1.2 that .
From (A3), we have , + ⩽ , and + ⩽ .
and , , .
Further, we can apply the Schur complement  to (2.1), and derive by Lemma 1.1. Hence, . Define . From the Dynkin formula, we can derive that . Now, for any and any system mode , the solution of system (1.6) with the initial value ϕ satisfies , , or , , where positive scalars , satisfy and for any mode , scalars , . Therefore, PDEs (1.6) is global stochastic exponential robust stability in the mean square. □
Remark 2.1 As pointed out in , diffusion effect exists really in the neural networks when electrons are moving in asymmetric electromagnetic fields . Strictly speaking, reaction-diffusion terms should be considered in any neural networks model [6–8]. Usually, the diffusion behaviors were simulated by linear Laplace diffusion items [9–16]. But not all diffusion behaviors can be simply considered as the linear reaction-diffusion. Indeed, there are various works related to the nonlinear reaction-diffusion [17–21], and even the nonlinear p-Laplace diffusion [17, 20]. So, in this paper, the stability of p-Laplace PDEs was investigated.
Then Theorem 2.1 derives that PDEs (1.6) is global stochastic exponential robust stability in the mean square with a large allowable variation range of time delay .
Remark 2.2 To the best of our knowledge, it is the first attempt to investigate the robust stability of T-S fuzzy Markovian jumping Itô-type stochastic dynamic equations with p-Laplace and probabilistic time-varying delays (see [1, 2, 20, 22–25]). Example 2.1 shows the effectiveness of the proposed methods due to the large allowable variation range of time delay.
Remark 2.3 As pointed out in , almost all the above related literature did not point out the role that the nonlinear p-Laplace items play, except  and . In fact, when , 2-Laplace is the linear Laplace, and there are many papers (see, e.g., [10–13]) in which the Laplace diffusion item plays its role in their stability criteria, for the linear Laplace PDEs can be considered in the special Hilbert space that can be orthogonally decomposed into the direct sum of infinitely many eigenfunction spaces. However, the nonlinear p-Laplace (, ) brings great difficulties for the nonlinear p-Laplace PDEs should be considered in the frame of the Sobolev space that is only a reflexive Banach space. Indeed, owing to the great difficulties, the authors only provide in  and  the stability criterion in which the nonlinear p-Laplace items play roles in the case of and under the Dirichlet boundary condition. So, a further profound study is very interesting, which may call for some new mathematical methods, and even new mathematical theories. Under the Neumann boundary condition, the problem of the role of the nonlinear p-Laplace () item in the stability criteria for fuzzy stochastic p-Laplace PDEs with probabilistic delays still remains open and challenging.
This work is supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ010, 2012JYZ010), and by the Scientific Research Fund of Sichuan Provincial Education Department (11ZA172, 14ZA0274, 12ZB349).
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