Abstract
The problem of almost sure (a.s.) asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching is researched. Firstly, we proposed a new criterion of a.s. asymptotic stability for a general neutral-type stochastic differential equation which extends the existing results. Secondly, based upon this stability criterion, by making use of Lyapunov functional method and designing an adaptive controller, we obtained a condition of a.s. asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching. The synchronization condition is expressed as linear matrix inequality which can be easily solved by Matlab. Finally, we introduced a numerical example to illustrate the effectiveness of the method and result obtained in this paper.
1. Introduction
As it is well known, the stability and synchronization of neural networks can be applied to create chemical and biological systems, secure communication systems, information science, image processing, and so on. In recent years, different control methods are derived to achieve different synchronization, such as randomly occurring control [1], sampled-data control [2, 3], passivity analysis [4], impulsive control [5–8], and adaptive control [9].
By utilizing adaptive control method, the parameters of the system need to be estimated and the control law needs to be updated when the neural networks evolve. In the past decade, much attention has been devoted to the research of the adaptive synchronization for neural networks. In [9, 10], the adaptive lag synchronization of unknown chaotic neural networks is considered. Adaptive synchronization problem of delayed neural networks with stochastic perturbation is studied in [11]. Besides these, there are many literatures to study adaptive synchronization problems (see, e.g., [12, 13] and the references therein).
Recently, the stability and synchronization of neutral-type systems, specially neutral-type neural networks, which depend on the derivative of the state and the delay state have attracted a lot of attention (see, e.g., [14–19] and the references therein) due to the fact that some physical systems in the real world can be described by neutral-type models (see [20]). However, the adaptive control was not investigated in [14–17], and the neutral term of derivative of the delay state was not taken into account in the neural networks proposed in [9–13]. Zhou et al. in [18] did not study the almost sure (a.s.) synchronization for neutral-type neural networks. Zhu et al. in [19] did not research the synchronization problem for neural networks with Markovian switching parameters. From the authors’ best knowledge, so far the almost surely adaptive synchronization problem for neutral-type neural networks with stochastic perturbation and Markovian switching parameters has not been fully investigated yet. This motivates our current work.
In this paper, the problem of almost sure (a.s.) asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching is researched. By making use of Lyapunov functional method and designing an adaptive controller, we obtained a condition of a.s. asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching. Finally, we introduced a numerical example to illustrate the effectiveness of the method and result obtained in this paper. The main contributions of this paper are as follows.(1)A new model for a class of neutral-type neural networks with stochastic perturbation and Markovian switching is given; it is more general than other models.(2)A new criterion of a.s. asymptotic stability for a general neutral-type stochastic differential equation is proposed which extends the existing results.
The notations are quite standard. Throughout this paper, , , and denote the set of nonnegative real numbers, -dimensional Euclidean space, and the set of all real matrices, respectively. The superscript denotes matrix transposition, denotes the trace of the corresponding matrix, and denotes the identity matrix. stands for the Euclidean norm in . stands for the block diagonal matrix. Let be a complete probability space with a filtration satisfying the usual conditions (i.e., the filtration contains all -null sets and is increasing and right continuous). For , denote by the family of all -measurable, -valued random variables such that the mathematical expectation , where denotes the family of all continuous -valued functions, and the norm . Denote by the family of all -measurable, bounded and -value random variables. If is a continuous -valued stochastic process on , we let for which is regarded as a -valued stochastic process. denotes the set of functions from to which are continuously twice differentiable in and once differentiable in .
2. Problem Formulation and Preliminaries
Let be a right-continuous Markov chain on the probability space taking values in a finite state space with generator given by where and is the transition rate from to if while
Consider the following neutral-type neural networks called drive system and represented by the compact form as follows: where is the state vector associated with neurons, denotes the neuron activation functions, and represents the transmission delay. For , we denote , , , , , and , respectively. In neural network (3), , , , and are the connection weight, the discrete delay connection weight, and distributed delay connection weight matrix, respectively; is a positive diagonal matrix; is called the neutral-type parameter matrix; is the constant external input vector.
The initial condition of system (3) is given in the following form: for any .
For the drive system (3), the response system is where is the state vector of the response system (5), is a control input vector, is an -dimensional Brownian motion defined on the complete probability space with a natural filtration (i.e., is a -algebra) and is independent of the Markovian process , and is the noise intensity matrix. It is known that external random fluctuation and other probabilistic causes often lead to this type of stochastic perturbations.
The initial condition of system (5) is given in the following form: for any .
Let be the synchronization error vector. From the drive system and the response system, the error system can be written as follows: where .
The initial condition of system (7) is given in the following form: with .
The primary object here is to deal with the adaptive synchronization problem of the drive system (3) and the response system (5) and derive sufficient conditions such that the response system (5) synchronizes with the drive system (3).
To prove our main results, the following assumptions are needed.
Assumption 1. The activation functions of the neurons satisfy the Lipschitz condition. That is, there exists a constant such that
Assumption 2. The noise intensity matrix satisfies the bounded condition. That is, there exist two positive constants and , such that for all , and for all .
Assumption 3. For the external input matrix , there exists positive constant , such that
where and is the spectral radius of matrix .
The following concept is necessary in this paper.
Definition 4 (see [21]). The trivial solution of the error system (7) is said to be almost surely asymptotically stable if for any initial value .
If the error system (7) is almost surely asymptotically stable, then the drive system (3) and the response system (5) are said to be almost surely asymptotically synchronization.
Consider the more general neutral-type stochastic delay differential equation (NSDDE) with Markovian switching: where is an -dimensional Brownian motion defined on the probability space but independent of the Markov chain and are all Borel-measurable functions.
For NSDDE (13), the following hypothesis is needed.(H1)Both and satisfy the local Lipschitz condition. That is, for each , there is an such that for all and those with .Given any initial data , (13) has a unique solution denoted by on . Moreover, both and are locally bounded in while uniformly bounded in . That is, for any , there is a , such that for all , and with . (H2)For each , there is a constant such that (H3)For each ,
Then, we present some preliminary lemmas which play an important role in the proof of the main results.
Lemma 5 (see [16]). Let . Then for any .
Lemma 6 (the generalized Ito formula, see [17]). Let and be a solution of neutral-type stochastic delay differential equation (13).
Then for any stopping times a.s.
holds provided that and are bounded on with probability 1, where the operator is defined by
Now we cite the convergence theorem of nonnegative semimartingales (see [22], Theorem 7 on page 139) which is a useful lemma.
Lemma 7 (the convergence theorem of nonnegative semimartingales). Let and be two continuous adapted increasing processes on with a.s. Let be a real-valued continuous local martingale with a.s. Let be a nonnegative -measurable random variable such that . Define If is nonnegative, then where . means . In particular, if ., then, with probability one, we have That is, all of the three processes , and converge to finite random variables.
Lemma 8 (Hölder inequality; see [21]). Let , , and . Then
Lemma 9 (Doob martingale inequality; see [21]). Let be an -martingale. Let be a bounded interval in . If , and (the family of -valued random variables with ), then
Lemma 10 (Chebyshev’s inequality; see [21]). If , , then
3. Main Results
In this section, we give some criteria of adaptive synchronization for the drive system (3) and the response system (5). First, we establish a general result which can be applied widely.
3.1. Almost Surely Asymptotically Stable
Theorem 11. Let (H1), (H2), and (H3) hold. Assume that there are functions , , and such that (C1) (C2)for , and
(C3)
Then for any initial data and , one has the following. (R1)Equation (13) has a unique global solution which is denoted by . (R2)Assume that if and only if . The solution obeys that
That is, is almost surely asymptotically stable.
The proof of this theorem is given in the .
Remark 12. Theorem 11 is an extension of Theorem 11 in [16]; that is, when we take in our theorem, then Theorem 11 is coincident with Theorem 3.1 in [16]. Moreover, Theorem 11 is also an extension of Theorem 2.1 in [23] when we take with .
Remark 13. From the proof of Theorem 11, we can see that if condition (H1) is substituted by , then the conclusion (R2) is also true.
3.2. Almost Sure Asymptotical Synchronization
In this subsection, we give a criterion of adaptive almost sure asymptotical synchronization for the drive system (3) and the response system (5).
Theorem 14. For systems (3) and (5), let Assumptions 1–3 hold, and the error system (7) has a unique solution denoted by on for any initial data with .
Assume also that there exist symmetric matrix , diagonal matrix , and positive scalars , , , , such that
where , with being arbitrary negative constants to be chosen, and .
We choose the feedback control with the update law as and , where are arbitrary constants, is the th diagonal entry of matrix , and is the th element of . Then the error system (7) is almost surely asymptotically stable. Therefore, the drive system (3) and the response system (5) are adaptive synchronized a.s.
Proof . Under Assumptions 1–3 and the existence of , it can be seen that , , and satisfy , (H2), and (H3), where
For each , choosing a nonnegative function
where , and computing along the trajectory of error system (7), we have
While
so
It is easy to get that
By (44), we have
According to Assumption 1 and Lemma 5, we have that
Substituting (45)-(46) into (43) yields
Therefore,
where
with
Now, (35) is equivalent to , (36) is just , (37) is equivalent to , and (38) is equivalent to . So from the conditions of this theorem, we know that conditions (C1), (C2), and (C3) in Theorem 11 are all satisfied. So by Theorem 11, the error system (7) is almost surely asymptotically stable. And hence the drive system (3) and the response system (5) are adaptive synchronized a.s. The proof of Theorem 14 is completed.
Remark 15. In this section, a numerical example will be given to support the main results obtained in this paper.
4. Numerical Examples
In this section, a numerical example will be given to support the main results obtained in this paper.
Letting , which means , we give the parameters concerning the drive system (3), the response system (5), and the error system (7) as follows: We further set , ,. Then we can confirm that Assumptions 1–3 are satisfied with , , , and .
Letting and using LMI toolbox in Matlab, we solve matrix inequalities (33)-(38) and obtain the following results:
So from Theorem 14, the drive system (3) and the response system (5) are adaptive synchronized a.s. when the error system (7) has a unique solution.
To illustrate the effectiveness of the result in this paper, we depict the evolution figures of the systems as Figures 1, 2, 3, and 4. Figure 1 shows the two-state Markov chain in the systems. Figure 2 shows that the drive system (3) synchronizes the response system (5) from the moment of . It can be seen from Figure 3 that the state of the error system (7) tends to zero from , which also describes the synchronization of the drive system (3) and the response system (5). The update law of the adaptive control gain is depicted in Figure 4. Figure 4 shows us that the update law of the control gain no longer varies after the response system (5) synchronizes with the drive system (3).
5. Conclusions
In this paper, we have proposed a new criterion of a.s. asymptotic stability for a general neutral-type stochastic differential equation which extends the existing results. Based upon this new stability criterion, we have obtained a condition of a.s. asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching by making use of Lyapunov functional method and designing an adaptive controller. The synchronization condition is expressed as linear matrix inequality which can be easily solved by Matlab. Finally, we have employed a numerical example to illustrate the effectiveness of the method and result obtained in this paper. In the future, we will consider the condition of a.s. asymptotic adaptive synchronization for neutral-type neural networks with time-varying delay by making use of M-matrix method.
Appendix
Proof. The proof of (R1) is the same as [16] and is omitted here. To prove (R2), we will divide it into five steps. We change into in subsequence for simplicity.
Step 1. We prove that the solution of the system obeys
In fact, let
which is a continuous local martingale with , a.s. By generalized Ito formula (Lemma 6), we have
By the convergence theorem of nonnegative semimartingales (Lemma 7), we have (A.1).
Step 2. We prove
Indeed, from (A.1), we have
which together with (C1) yields
Now, for any , by (H2), we have that if , then
where . This implies
where is the bound for the initial data . Hence
Letting and using (A.6), we obtain (A.4).
Step 3. We prove
In fact, taking the expectations on both sides of (A.3) and letting , we obtain that
where .
This implies
or equivalently
From (A.12), we have
or equivalently
Now we will prove (A.10): a.s. In fact, if (A.10) is false, then
Hence there is a number such that
where .
Recalling (A.4), as well as the boundedness of the initial data , we can find a positive number , which depends on , sufficiently large for
where .
It is easy to see from (A.17) and (A.18) that
We now define a sequence of stopping times as follows:
where, throughout this paper, we set .
From (A.14) and the definition of and , we observe that if , then
Let denote the indication function of set . Noting the fact that , whenever , we can derive from (A.11) that
On the other hand, by (H1), there exists a constant , such that
whenever and .
By the Hlder inequality (Lemma 8) and the Doob martingale inequality (Lemma 9), we compute that, for any and ,
Since is continuous in , there exists a closed ball such that is uniformly continuous in . We can therefore choose so small such that
We furthermore choose sufficiently small for
It then follows from (A.24) and Chebyshev’s inequality (Lemma 10) that
Note that
We hence have
By (A.19) and (A.21), we further compute
By (A.25), we hence obtain that
Set
Note that
We derive from (A.22) and (A.31) that
which is a contradiction. So (A.10) must hold.
Step We prove that and
By (A.10) and (A.6), we see that there is an with such that
Choose any . Then is bounded in so there must be an increasing sequence such that and converges to some . Thus
which implies that whence . From this, we can show that
If this is false, then there is some such that
Hence there is a subsequence of such that
for some . Since is bounded, we can find its subsequence which converges to . Clearly, so . But, by (A.36),
a contradiction. Hence (A.38) must hold and (A.35) holds yet.
Step 5. We prove (R2).
Under the assumption that
we have . It then follows from (A.35) that
But, by (H2),
where has been defined above. Letting , we obtain that
This together with (A.4) yields
which is (32) and the proof is therefore completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China under Grant no. 61203337, the Innovation Program of Shanghai Municipal Education Commission (12zz064), the Specialized Research Fund for the Doctoral Program of Higher Education under Grant no. 20120075120009, the Natural Science Foundation of Shanghai under Grant no. 12ZR1440200, the Fundamental Research Funds for the Central Universities (2232012D3-18), and DHU Distinguished Young Professor Program (B201309).