Abstract
This paper investigates the mean square exponential stability problem of a class of complex-valued neural networks with stochastic disturbance and mixed delays including both time-varying delays and continuously distributed delays. Under different assumption conditions concerning stochastic disturbance term from the existing ones, some sufficient conditions are derived for assuring the mean square exponential stability of the equilibrium point of the system based on the vector Lyapunov function method and differential-integral theorem. The obtained results not only generalize the existing ones, but also reduce the conservatism of the previous stability results about complex-valued neural networks with stochastic disturbances. Two numerical examples with simulation results are given to verify the feasibility of the proposed results.
1. Introduction
In recent years, the dynamical behavior analysis of various neural networks models defined in complex number domain has attracted more and more attention due to their extensive applications in many fields, such as filtering, speech synthesis, remote sensing, and signal processing, which cannot be solved comprehensively with only their counterparts defined in real number domain [1, 2].
It is well known that delays arise because of the processing of information in both biological and man-made neural networks in view of a practical point [3, 4]. Besides, the neural networks usually have a spatial extent due to the presences of a multitude of parallel pathway with a variety of axon sizes and lengths. There will be a distribution of conduction velocities along these pathways and a distribution of propagation be designed with discrete delays. Therefore, the more appropriate way is to incorporate continuously distributed delays [3–7]. In [8–11], the authors have studied several kinds of complex-valued neural networks with continuously distributed delays. Some significant results were obtained for assuring the stability of the proposed systems in [8–11].
In fact, most real models of neural networks are affected by many external and internal perturbations which are of great uncertainty, such as impulsive disturbances [5, 9–15], Markovian jumping parameters [16–19], and parameter uncertainties [20–22]. As Haykin [23] points out, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. One approach to the mathematical incorporation of such effects is to use probabilistic threshold models. In current papers, neural networks model with external random perturbations is to view as nonlinear dynamical systems with white noise (perturbations of Brownian motion) [3, 24]. Therefore, it is of real significance to consider that stochastic effects to the stability of neural networks with continuously distributed delays.
As far as we know, the existing stability results concerning various neural networks are mainly applied to judge the neural networks in real number domain, such as [3, 7, 12–14, 16, 17, 24–31]. Recently, there have been some literatures with regard to the complex-valued neural networks with stochastic disturbances [32–34]. In [32], the authors established a class of stochastic memristor-based complex-valued neural networks with time varying delays. In [33], the robust state estimation problem was investigated for a class of complex-valued neural networks involving parameter uncertainties, constant time delays, and stochastic disturbances by resorting to the sampled data information from the available output measurements. However, the infinitely distributed delays have not been considered in the model of complex-valued neural networks studied in [32, 33]. In [34], the passivity analysis was conducted for a class of stochastic memristor-based complex-valued recurrent neural networks with discrete time delays and continuously distributed delays by applying both the scalar Lyapunov function method and the LMI method. The stochastic weighted coefficients in the model [34] were assumed to be constants in real number domain, which was conservatism and needed to be generalized in the complex number domain further. It is well known that the synchronization problem of chaotic neural networks can be translated into the stability problem of the corresponding error system of driving system and driven system. In [35], the authors concerned the problem of finite-time synchronization for a class of complex-valued neural networks with stochastic disturbances and mixed delays. However, the stochastic terms in the driving and driven systems were only assumed to be bounded directly, which was a shortage of generality. Besides, the results in [35] are unable to judge the stability of the corresponding systems.
Separating the model of complex-valued neural networks into its real and imaginary parts is a routine approach to study the dynamical behavior of the systems, such as [2, 8, 11, 15, 32–35]. The complex-valued activation functions were supposed to be with existence, continuity, and boundedness of the partial derivatives of the activation functions about the real and imaginary parts of the state variables [2, 8, 11, 15, 32]. As pointed in [33, 36–38], the assumption concerning activation functions in [2, 8, 11, 15, 32] were with additional restriction that the partial derivatives of its real parts and imaginary parts were required to satisfy the existence and continuity.
To the best of authors’ knowledge, there is no research concerning the exponential stability in the mean square sense for the complex-valued neural networks with mixed delays (including both time-varying delays and continuously distributed delays) and stochastic disturbances. Therefore, in this paper we will establish some new conditions for assuring the stability of the mentioned systems. The main advantages and contributions can be listed as follows. (a) The stochastic neural networks with mixed delays in complex number domain are proposed. (b) Different assumption conditions concerning stochastic disturbance term from the existing ones are given in this paper, which is of less conservatism. (c) Both stochastic disturbances and interval parameter uncertainties are considered in the addressed systems. (d) Some sufficient conditions with simple matrix forms are obtained for ensuring the mean square exponential stability and robust exponential stability in the mean square sense of the systems, which are easy to be checked in practice.
2. Notations and Model Descriptions
In this paper, we consider a class of complex-valued neural networks as follows: In (1), represents the k-th neuron state, , where denotes a complex number set, and denotes the neuron number. , , and represent the connection weighted matrices, respectively. is the external input vector, where denotes the transpose of the vector. represents the activation function. with denotes the neuron self-feedback coefficient matrix, where denote a real number set. denotes the weighted function of stochastic disturbances and is the Brown motion defined on a complete probability space with a natural filtration generated by .
It is assumed that is the initial condition of system (1), where is continuous function mapping from to , .
Let be the equilibrium point of (1), where () and i denotes the imaginary unit; i.e., .
Let , where and represent the real part and imaginary part of a complex number, respectively. Furthermore, the activation functions are assumed to be expressed as (2) by separating them into their real parts and imaginary parts, where and represent the real part and the imaginary part of , respectively.
Next, some assumptions concerning (1) are given to obtain the stability results.
Assumption 1. Suppose that () is a bounded function with .
Assumption 2. Suppose that : are piecewise continuous functions and satisfy where is continuous on , and .
Let denote the module of a complex number.
Assumption 3. It is assumed that with the form of (2) satisfies the mean value theorem of multivariable functions. That is, for any given , there exist positive constants , , , and such that
Let , , , .
Remark 4. In [2, 8, 11, 15, 32], the activation functions were supposed to satisfy that the partial derivatives of their real parts and imaginary parts were required to be with the existence and continuity. In fact it is an additional restriction on the activation functions in complex number domain. In this paper the mentioned restriction is no longer required.
Assumption 5. Assume that the weighted functions of stochastic disturbances () can be separated into their real parts and imaginary parts with following forms:where . Suppose that there exist nonnegative constants , , , and such that inequalities (5) and (7) hold:
Let , , , .
Next, we will separate (1) into its real part and imaginary part as follows:where . Let and , and , and be the real part and imaginary part of matrices , , and , respectively. Let and be the real part and imaginary part of external input .
For a complex number vector , let and .
Definition 6. The equilibrium point of (1) is said to be exponentially stable in the mean square sense if, for all , there exist constants >0 and such that where
Lemma 7 (see [8]). Let be a real number matrix with , . The following statements are equivalent:
(i) is an M-matrix.
(ii) The real parts of all eigenvalues of are positive.
(iii) There exists a positive vector such that 0.
Lemma 8 (see [39]). Let be an process given by Let be again an process with second derivative with respect to y; then where is computed according to the rules .=.=.=0 and .=.
Remark 9. By substituting (13) into (14) and considering the computation rule in Lemma 8, we can obtain the following equation, which will be used in the process of the proof for theorems follow-up.
3. Main Results on Mean Square Exponential Stability
For simplification, let , , .
In what follows, we will study the mean square exponential stability of the equilibrium point of (1). For analysis convenience, we translate the coordinate of (1). Let , .
By the translation, (1) can be changed as follows, where ,where
Moreover, the initial conditions of (1) are of the forms , .
Obviously, if the zero solution of the system being composed of (16) and (17) is exponentially stable in the mean square sense, then the equilibrium point of the system being composed of (9) and (10) is also exponentially stable in the mean square sense.
Next, we will give some sufficient conditions for assuring the mean square exponential stability of the equilibrium point of (1). First of all, we let
Theorem 10. If Assumptions 1–5 are satisfied and the matrix is an M-matrix, then the equilibrium point of (1) is of mean square exponential stability for any external input .
Proof. Because is an M matrix, according to Lemma 7 there exists a positive vector such that the following inequalities hold, where and , when ; and when : Construct the following functions: Because and are continuous with respect to , and , , there exist constants and , such that and , . Obviously, there exists a positive constant satisfying such that and hold; that is,Choose a candidate vector Lyapunov function in what follows: i.e., where .
Let denote if there is no confusion, .
(i) When , according to formula (15) and considering the real parts (16) of (1), we getNext, we will transform differential form of (26) into integral form. By using stochastic differential equation theory [39], we can integrate on both sides of (26) from to for any and compute the expectation of (26): It is well known that the expectation of process is a continuous function. According to the properties of integral [39] and Dini derivation, (27) can be deduced as follows: Considering Assumptions 1–5, (28) can be deduced furthermore as follows: The detailed deducing of (29) can be referred in the Appendix part of this paper.
Define the curveand the setsIt is obvious that we have as .
In what follows, we will prove the mean square exponential stability of (16). That is, there exist constants and such that hold, where , .
Let , , and , where is a constant; then i.e., when , we have , , .
Furthermore, it can be obtained that , , . If they are not true, then there exist some and such that , ; , , , . Substituting them into (29) and considering (22), we haveThis is a contradiction with the assumption ; i.e., for all , we have , , . That is, , .
(ii) When , according to formula (15) and considering the imaginary parts (17) of (1), we getNext, we will transform differential form of (34) into integral form. By using stochastic differential equation theory [39], we can integrate on both sides of (34) from to for any and compute the expectation of (34), we getNoticing that the expectation of process is a continuous function, according to the properties of integral and Dini derivation, (35) can be deduced for all , as follows: Considering Assumptions 1–5, the following inequality can be obtained by adopting the similar analysis of (29): By similar analysis with part (i) previously, we can conclude that and , where , , .
In what follows, we will prove the mean square exponential stability of system (17). That is to say, there exist constants and such that hold, where , .
We claim that and , . If they are not true, then there exist some and time instant such that , ; , , , . Substituting them into (37) and considering (23), we have This is a contradiction with the assumption . Therefore, for all , we have and . That is, , .
To sum up, it follows from (i) and (ii) that hold for all ; i.e., , , .
Furthermore, for , we get Let . We further haveAccording to Definition 6, the zero solution of the complex-valued neural networks being composed of the real part (16) and the imaginary part (17) is exponentially stable in the mean square sense. Namely, the equilibrium point of the complex-valued neural networks being composed of the real part (9) and the imaginary part (10) is exponentially stable in the mean square sense. The proof is completed.
Remark 11. When there is no stochastic disturbance in the complex-valued neural networks described by (1), model (1) in this paper is the same as model in [8]. Theorem 2 in [8] can be obtained directly by using the same analysis in Theorem 10. The obtained results of this paper include the ones in [8], which means that they are with more generality.
Remark 12. The authors in [32] studied a class of delayed memristor-based complex-valued neural networks with stochastic disturbances. The continuously distributed delays were not considered in [32]. Moreover, it can be seen from Assumption 5 of this paper that the stochastic weighted functions and are with regard to both the real part and the imaginary part of neuro state, respectively. Obviously, Assumption 2 concerning stochastic disturbances in [32] was with more constraint because was supposed to the function only with regard to the real part of neuro state and with regard to imaginary part of neuro state. Therefore, the research of this paper extends the corresponding work in [32].
Motivated by the ideal of Assumption 2 in [32], some other assumption conditions on stochastic disturbances are proposed as follows.
Assumption 13. It is assumed that the weighted function of stochastic disturbances () can be separated into its real part and imaginary part with the following forms:where . Suppose that there exist nonnegative constants and such that the following inequalities hold:Let , , and .
Corollary 14. If Assumptions 1–3 and Assumption 13 are satisfied, and the matrix is an M-matrix, then the equilibrium point of (1) is of mean square exponential stability for any external input .
Remark 15. In order to compare the result to one in [32] from the computational point of view, we assume that and . By computation, the parameters of Assumption 2 in [32] are , , , and , respectively. The parameters in Assumption 13 of this paper are and . Although the expression form of Assumption 2 in [32] is different from Assumption 13 in this paper, it can be seen from the proof of Theorem 10 about values of and that the obtained stability conditions concerning the stochastic terms in [32] are the same as ours. Besides, the matrix for judging stability in Corollary 14 in this paper is of more simple and compact expression, which is easier to calculate by MATLAB software.
Remark 16. There are some errors inevitably between theoretical and practical systems for the unexpected factors, such as sensing error, modeling error, parameter aging, and channel strength. Therefore, it is necessary to suppose that parameter uncertainties of system in given intervals can overcome those problems. We call the system with randomly occurring parameter uncertainties interval system. In [18, 20–22, 33, 40], the authors investigated some interval neural networks in complex number domain. However, the stochastic disturbances were not considered in [18, 20–22] and the continuously distributed delays were not considered in [33, 40].
Next, we will give a theorem for judging the robust exponential stability in the mean square sense for a class of interval complex-valued neural networks with mixed delays and stochastic disturbances.
Assumption 17. The weighted matrices defined in complex number domain of (1) are supposed to be in the following intervals: Let
Definition 18. The equilibrium point of (1) is said to be robustly exponentially stable in the mean square sense if, for all , , and , there exist constants >0 and such that where
Theorem 19. If Assumptions 1–5 and Assumption 17 are satisfied, and the matrix is an M-matrix, then the equilibrium point of (1) is of robust exponential stability in the mean square sense for any external input .
Proof. Because is an M-matrix, according to Lemma 7 there exists positive vector such that the following inequalities hold, where and , when ; and when , The rest of proof is almost similar with the proof of Theorem 10 only by replacing all parameters by , respectively, so it is omitted here. The proof is completed.
When the stochastic disturbances satisfy Assumption 13, it is easy to obtain the following Corollary 20 for (1) with the mentioned parameter uncertainties above.
Corollary 20. If Assumptions 1–3 and Assumptions 13 and 17 are satisfied and the matrix is an M-matrix, then the equilibrium point of (1) is of robust exponential stability in the mean square sense for any external input .
Remark 21. When there is only time-varying delays or continuously distributed delays in (1), some corresponding corollaries can be obtained directly by letting or in the matrices , , , defined in Theorem 10, Corollary 14, Theorem 19, and Corollary 20, respectively.
4. Numerical Examples
In this section, two numerical examples with simulation figures will be given to illustrate the obtained results of Theorems 10 and 19, respectively.
Example 1. Consider the following system: In (48), let ; the interconnected matrices areThe activation functions are and. The real part and imaginary part of stochastic weighted function are and , respectively.
By computing, we have It can be obtained thatThe eigenvalue matrix of the matrix is . It follows from Lemma 7 that the matrix is an M-matrix. According to Theorem 10, it can be concluded that the equilibrium point of (48) is exponentially stable in the mean square sense.
Let the delays in (48) be and , . Let , , , . It is assumed that the initial condition of (48) are and .
The numerical simulations of (48) are shown in Figures 1–5. In order to make comparison, Figures 1 and 3 show the state curves of the real part and the imaginary part of (48) without/with stochastic disturbances, respectively. Figures 2 and 4 show the amplitude curves of neuro states of (48) without/with stochastic disturbances, respectively. Figure 5 shows the norm curves of neuro states of (48) both with stochastic disturbance and without stochastic disturbances. From the simulation results, it can be seen that the equilibrium point of (48) is stable. The simulation results verify the correctness of Theorem 10 in this paper.
Example 2. Consider the following system: In (52), let . The interval interconnected matrices are supposed to be The rest of the assumptions concerning the activation functions and stochastic weighted functions are the same as ones given in Example 1.
By computation, we have It can be obtained that Because the eigenvalue matrix of the matrix is , it follows from Lemma 7 that the matrix is an M-matrix. Therefore, the equilibrium point of (52) is robustly exponentially stable in the mean square sense in accordance with Theorem 19.
In order to finish the simulation, let the delays in (52) be and , . Let , , , . It is assumed that the initial conditions of (52) are and .
LetIt is obvious that the matrices , , and .
Figures 6–8 show the state curves of the real part and the imaginary part, the amplitude curves of neuro states, and the norm curves of neuro states of (52), respectively. From the simulation results, it can be seen that the equilibrium point of (52) is stable. The simulation results verify the correctness of Theorem 19.
5. Conclusions and Future Work
Some sufficient conditions have been proposed for assuring the mean square exponential stability of a class of complex-valued neural networks with stochastic disturbances and mixed delays based on the vector Lyapunov function method and differential-integral theorem. The results established in this paper generalize the existing ones and reduce the conservatism of the previous results about complex-valued neural networks with stochastic disturbances. Two numerical examples are given at last to show the correctness and feasibility of the proposed results. Based on the study method presented in this paper and the idea of driving-response conception [35, 41], we will attempt to study the synchronization problem for a class of chaotic complex-valued neural networks with mixed delays and stochastic disturbances in the near future. Besides, numerous references concerning neural networks with impulsive effect have been emerged in the past two decades; see [5, 9–15, 22, 40, 42]. It will be an interesting research to consider impulsive effect from the view of both disturbance and control when scholars study the synchronization problem of various chaotic complex-valued neural networks.
Appendix
The detailed deducing of inequality (29) in Theorem 10 is as follows.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by the Science & Technology Department Program of Sichuan Province [Grant nos. 2017TD0026, and 2019YFG0367], in part by the Sichuan Province Scientific Research Innovation Team Program [Grant no. 17TD0035], in part by the National Natural Science Foundation of China [Grant nos. 51775448, 11402214, and 11572264], in part by the Scientific Research Foundation of the Education Department of Sichuan Province [Grant no. 17ZA0364], in part by the Key Project from Science & Technology Department of Sichuan Province [Grant no. 2011J00043], in part by the Chui Hui Project from the Ministry of Education of PRC [Grant no. z2012024], in part by the Young Scholars of Xihua University [Grant no. 0220170208], in part by the Open Research Subject of Key Laboratory of Fluid and Power Machinery, Xihua University, Ministry of Education [Grant no. szjj2019-015], and in part by the Science & Technology Major Project of Sichuan Province [Grant no. 2019ZDZX0002].