Abstract
This paper pays close attention to the problem of finite-time stabilization related to stochastic inertial neural networks with or without time-delay. By establishing proper Lyapunov-Krasovskii functional and making use of matrix inequalities, some sufficient conditions on finite-time stabilization are obtained and the stochastic settling-time function is also estimated. Furthermore, in order to achieve the finite-time stabilization, both delayed and nondelayed nonlinear feedback controllers are designed, respectively, in terms of solutions to a set of linear matrix inequalities (LMIs). Finally, a numerical example is provided to demonstrate the correction of the theoretical results and the effectiveness of the proposed control design method.
1. Introduction
In recent years, more and more scholars have been attracted by neural networks due to their successful applications in associative memory [1, 2], pattern recognition [3], signal processing, optimization problems, and so forth [4]. These applications always rely on the dynamic behaviors of neural networks. Therefore, the investigation of dynamic trajectories is necessary for applied designation of neural networks. Hence, a large number of studies on stability [5–8], stabilization [9, 10], passivity [11], dissipativity [12, 13], synchronization [14, 15], and state estimation [16, 17] for neural networks have been reported.
On the other hand, many researchers have studied Hopfield neural networks [18], cell neural networks, recurrent neural networks [9, 19], Cohen-Grossberg neural networks, bidirectional associative memory neural networks, and Lotka-Volterra neural networks, as well as inertial neural networks [12, 14, 15, 20], which are more intricate than all kinds of prementioned neural networks with the standard resistor-capacitor variety [21]. The inertial term is taken as a critical tool to bring complex bifurcation behavior and chaos.
It has been confirmed that stochastic disturbances, which are unavoidable in actual applications of artificial neural networks, are probably one of the main sources leading to undesirable behaviors of dynamical systems, especially when a neural network is implemented for applications. Therefore, it is of great significance to study the stability and stabilization problems of neural networks with stochastic disturbances [22–24]. However, to the best of authors’ knowledge, most of the researchers have either investigated the stability for stochastic neural networks with time-delay [25–28] or studied the stability for inertial neural networks with time-delay [20]. There are rare literatures that considered the finite-time stabilization for stochastic inertial neural networks with time-delay.
Inspired by the above comprehensive analysis, in this paper, we are devoted to investigating the finite-time stabilization for stochastic inertial neural networks with time-delay. First, by utilizing an appropriate variable substitution, a stochastic inertial neural network can be transformed into a first-order stochastic differential system. Then, some sufficient conditions on finite-time stability in probability are derived by means of establishing an appropriate Lyapunov function and applying inequality techniques. Moreover, the stochastic settling-time function is also given.
2. Problem Formulation and Preliminaries
2.1. Systems Description
Firstly, the inertial neural networks (INNs) without time-delay are considered, which is described as follows:where is the state of the -th neuron; the second derivative is the inertial term of INNs (1). , are constants. denotes the connection weight between the -th neuron and the -th neuron. stands for activation function of the -th neuron with . is the external input on the -th neuron.
The initial conditions of INNs (1) arewhere and are real-valued continuous functions.
Suppose that the external input is subject to the environmental noise and is described by , where is known as the control input and is a one-dimensional white noise, which is also called Brown motion defined on a complete probability space and satisfied withand is the intensity function of the noise.
Then INNs (1) can be written the following stochastic inertial neural networks (SINNs):
2.2. Problems Formulation
In general, making use of the variable transformation,then the SINNs (4) can be rewritten asand the initial conditions are given aswhere , .
Moreover, the controller is considered; we have the following SINNs:
Denote
Thus, the SINNs (8) can be written in vector form as
The control inputs to be designed are of the following form:where , , , , and , , are the control gain matrices to be determined. is a positive constant with .
Remark 1. There are three cases for the value of . If , the controllers are continuous functions with respect to and , respectively, which bring about the continuity of SINNs (10) with respect to the systems state [29, 30], but the local Lipschitz condition is dissatisfied. If , turn to be discontinuous ones, which have been studied in [31, 32]. If , then they become the typical stabilization issues which only can realize an asymptotical stabilization in infinite time [33, 34] due to the local Lipschitz conditions.
Remark 2. In fact, the control gain matrices , , , in the controllers and play different roles in ensuring the finite-time stability of the SINNs (10) with (11), where and are used to guarantee the Lyapunov stability of the SINNs (10). And the convergence to zero of the SINNs (10) is determined by and .
To achieve our main results, some assumptions, lemmas, and definitions are necessary to introduce firstly.
Assumption 3. The nonlinear activation function satisfies , and there exist some constants , such thathold for all and , where , , .
Remark 4. If we choose , the inequalities in Assumption 3 can be written aswhere , which has been considerably studied.
Assumption 5. The intensity function is a continuous function and is supposed to satisfy that where is a known matrix with appropriate dimensions.
Definition 6. The SINNs (10) are said to be finite-time stabilizable by the controller (11); that is, the SINNs (10) are finite-time stable if, for any initial state , , there exists a finite-time function such that where is called the stochastic settling time function satisfying .
Lemma 7 (see [35]). Suppose that SINNs (10) admit a unique solution. If there exist a function , class functions and , and positive real constant and , such that for all and ,then the origin of SINNs (10) are stochastically finite-time stable, and .
Lemma 8 (see [9]). If are positive number and , then
3. Main Results
3.1. Finite-Time Stabilization Feedback Controller Design without Time-Delay
Theorem 9. The controlled systems (10) with (11) are finite-time stable, if there exist some positive-definite matrices and some known constant matrices with compatible dimensions, such thatMoreover, the upper bound of the stochastic settling time for stabilization can be estimated as ,
where
Proof. Taking controller (11) into SINNs (10), it follows thatNext, we will prove that system (20) is finite-time stable in the sense of Definition 6.
Construct a Lyapunov function aswhere and are positive definite matrices. Then, calculate the time derivative of along the trajectories of systems (20); we getwhereFrom (13), we haveFrom (14), we haveCombining (23)-(25) and condition (18), one can follow thatDue to , together with Lemma 8, one hasand thenSimilarly, we have So, we haveNow, taking the expectations on both sides of (22), and letting , , we can getFrom Lemma 7, we get that the controlled systems (20) are finite-time stable, and the upper bounded stochastic settling time can be estimated byThis completes the proof.
Summing up the above analysis, some sufficient conditions on finite-time stability for the SINNs (10) with (11) are obtained. In the following, we mainly focus on the design of finite-time stabilizing controllers by transforming the sufficient conditions into solvable linear matrix inequalities.
Theorem 10. If there exist some positive define matrices , , , matrices , with appropriate dimensions, for fixed control gain matrices and , such thatwhere then the finite-time stabilization problem is solvable for the stochastic inertial neural networks (4) and the control gain matrices , .
Proof. Setting , , , , (18) can be written aswhere Then, left- and right-multiplying inequality (35) by the block-diagonal matrix , which followswhere and left- and right-multiplying inequality (37) by the block-diagonal matrix , we can obtainwhere By Schur complement, (33) implies the above inequality (39) holds. This completes the proof.
3.2. Finite-Time Stabilization Feedback Controller Design with Time-Delay
In the above section, we discussed the finite-time stabilization for stochastic inertial neural networks without time-delay. However, when designing a neural network or implementing it, the occurrence of time-delay is unavoidable. It may cause instability and oscillation [36–38]. Therefore, in order to reduce the conservatism, in this section, we will study the finite-time stabilization for stochastic inertial neural networks with time-delay.
Consider the following SINNs with time-delay,where is the time-varying delay of -th neuron with .
Denote Then we have
The nonlinear delay-feedback controller is designed as the following form:where , , , are gain matrices to be determined, and , .
Theorem 11. The SINNs with time-delay (43) with (44) are finite-time stable, if there exist some positive-definite matrices such thatwhere Moreover, the upper bound of the stochastic setting time for stabilization can be estimated as with and
Proof. Construct a Lyapunov function:Calculating the It differential of along with (43), we can obtain We can see that the right of inequality (48) equals (23). Hence, the rest of the proof is the same as that of Theorem 9 and it is omitted here.
Similar to the proof of Theorem 10, we have the following result.
Theorem 12. If there exist some positive define matrices , , matrices , with appropriate dimensions, for fixed control gain matrices and , such thatwhere then the finite-time stabilization problem is solvable for the stochastic inertial neural networks (41) and the control gain matrices , .
4. Illustrative Example
Consider the following stochastic inertial neural networks with time-delay:which are equivalent to the following vector form:where
Setting the initial values , , the state trajectories and phrase trajectories of the open-loop system are shown in Figures 1 and 2, respectively. Moreover, take 10 sets of numbers randomly as the initial values of and and satisfy , . Then the corresponding state trajectories and phrase trajectories of the open-loop system are shown in Figures 3 and 4, respectively. Obviously, the stochastic inertial neural networks with time-delay (51) are not finite-time stabilization.
Hence, we need to design the delay-feedback controller as (44) for system (51), where the parameter is chosen as , and the initial values , ,, . The solution of (49) is derived by resorting to Matlab LMI Control Toolbox:
We can get , , . The state trajectories and phrase trajectories of close-loop system are shown in Figures 5 and 6, respectively.
In order to make the result of the simulation more convincing, we take 100 sets of numbers randomly as the initial values of and and satisfy , . Then the corresponding state trajectories are shown in Figure 7 and the corresponding phrase trajectories are shown in Figure 8. Obviously, the stochastic inertial neural networks with time-delay (51) are finite-time stabilization. Moreover, when , , we have state trajectories in Figure 9 and phrase trajectories in Figure 10, which also figure out that the stochastic inertial neural networks with time-delay (51) are finite-time stabilization.
5. Conclusions
In this work, by constructing a proper Lyapunov function, the finite-time stabilization problem has been addressed for stochastic inertial neural networks with or without time-delay. Provided that a set of LMIs are feasible, a suitable delayed or nondelayed nonlinear feedback controller can be designed such that finite-time stability in probability can be ensured for the system under study. An example has been given to demonstrate the correctness of the theoretical results and the effectiveness of the proposed methods.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by the Nation Natural Science Foundation of China under Grants 61473213 and 61671338.