Less Conservative Stability Criteria for Neutral Type Neural Networks with Mixed Time-Varying Delays
This paper investigates the problem of dependent stability criteria for neutral type neural networks with mixed time-varying delays. Firstly, some new delay-dependent stability results are obtained by employing the more general partitioning approach and generalizing the famous Jensen inequality. Secondly, based on a new type of Lyapunov-Krasovskii functional with the cross terms of variables, less conservative stability criteria are proposed in terms of linear matrix inequalities (LMIs). Furthermore, it is the first time that the idea of second-order convex combination and the property of quadratic convex function applied to the derivation of neutral type neural networks play an important role in reducing the conservatism of the paper. Finally, four numerical examples are given to show the effectiveness and the advantage of the proposed method.
During the last two decades, delayed neural networks have drawn a great deal of attention because of their extensive applications in various scientific and technical areas, such as pattern recognition, power systems, parallel computing, signal processing, finance, associative memories, mechanics of structures, and other scientific areas [1–30]. It is well known that time delay regarded as a major cause of instability and poor performance often appears in many neural networks. Therefore, the stability analysis for delayed neural networks has been investigated extensively in recent few decades. Generally speaking, studying the dynamical behavior of delayed neural networks can be mainly classified into two types: delay-independent stability and delay-dependent stability. As is known to all, delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delay is small.
On the other hand, due to the complicated dynamic properties of the neural cells in the real world, there exist many neural network models such as distributed networks, chemical reactors, and heat exchanges that cannot characterize the properties of a neural reaction process precisely. It is natural and important that these systems will contain some information about the derivative of the past state to further describe and model the dynamics of the complex neural reactions. This new type of neural networks is called neutral neural networks or neural networks of neutral type. However, many researchers have focused on the global stability of neural networks of neutral type only with constant time delay in recent years, which is very restrictive. Hence, described with neutral functional differential equations with discrete and distributed delays, these neural networks called neutral type neural networks with mixed time-varying delays have a lot of research on space. The differential expression not only defines the derivative term of the current state but also explains the derivative term of the past state. Furthermore, it is necessary to have some information about the derivative of the past state in the systems to characterize the dynamics of such complex neural reactions. Practically, neutral type phenomenon always appears in studies of automatic control, chemical reactors, distributed networks, dynamic process including steam and water pipes, population ecology, heat exchanges, microwave oscillators, systems of turbojet engine, lossless transmission lines, vibrating masses attached to an elastic bar, and so on. For this reason, there has been a growing research interest in the study of delayed neural networks of neutral type in the recent years. Therefore, some less conservative stability criteria for neutral type neural networks with mixed time-varying delays have been reported in recently [25, 31–35]. Many methods have been proposed in these results to reduce the conservatism of the stability criteria, such as model transformation method, free-weighting matrix method, the method of constructing novel Lyapunov-Krasovskii functionals, delay decomposition technique, and weighting-matrix decomposition method. In , the authors derived some less conservative stability criteria by considering some useful terms and using free-weighting matrix technique. By considering the relationship between the time-varying delay and its lower and upper bound, the results obtained in  were improved in . By constructing a new Lyapunov-Krasovskii functional and using free-weighting matrix method, some more less conservative criteria than those obtained in  were proposed in . Further, the problems of stability analysis of neutral type neural networks with discrete and distributed delays have been investigated in . By using a delay-partitioning approach, a new type of Lyapunov-Krasovskii functionals was constructed to obtain some less conservative stability criteria. However, time delay in  is not only constant delay, but also the delay-partitioning approach is equational; hence, this method has some limitations.
Motivated by this technique, it is the first attempt to investigate the integral nonuniform partitioning method to be extended for neutral type neural networks with mixed time-varying delays. In the paper, the reduced conservatism of Theorem 6 benefits from the construction of the new Lyapunov-Krasovskii functionals in (17), which contain some integral nonuniform partitioning method and triple-integral terms, which play an important role in the improvement of less conservative results. Secondly, a novel handling method is given to establish the relationship among , and , which play an important role in reducing the conservatism of stability criteria further. Furthermore, compared with previous results by using the first-order convex combination property, our derivation makes full use of the idea of second-order convex combination and the property of quadratic convex function given in the form of a lemma without employing Jensen's inequality. Finally, four numerical examples are given to illustrate the effectiveness and the advantage of the proposed main results.
Notation 1. Notations used in this paper are fairly standard: denotes the -dimensional Euclidean space, is the set of all dimensional matrices; denotes the identity matrix of appropriate dimensions, stands for matrix transposition, the natation (resp., ), for means that the matrix is real symmetric positive definite (resp., positive semidefinite); denotes block diagonal matrix with diagonal elements , , the symbol represents the elements below the main diagonal of a symmetric matrix, and is defined as .
Consider the following neural networks of neutral type with mixed time-varying delays: where is the neural state vector, is the neuron activation function, is an external constant input vector, and , , , , and are the constant matrices of appropriate dimensions.
Assumption A. The time-varying delay is continuous and differential function that satisfies
Assumption B. For the constants , , the nonlinear function in (1) satisfies the following condition: Here, we denote , , , , .
Assumption C. For given positive scalars satisfies: It is clear that under Assumption B, the system (1) has one equilibrium point denoted as . For convenience, we firstly shift the equilibrium point to the origin by letting , ; then the system (1) can be transformed into where is the state vector of transformed system, . It is easy to check that the transformed neuron activation function satisfies The following lemmas are introduced, which will be used in the proof of the main results.
Fact 1 (Boyd et al. , (Schur complement)). For a given symmetric matrix , where , the following conditions are equivalent: (1); (2), ;(3), .
Lemma 1 (see ). For symmetric matrices , , , and a vector , Let with . Then we have and , .
Lemma 2 (see ). Let , and let be an appropriate dimensional vector. Then, we have the following facts for any scalar function :(1); (2); (3), where matrices and a vector independent of the integral variable are appropriate dimensional arbitrary ones.
Lemma 3 (see ). For any constant matrix , a scalar and a vector function such that the integrations concerned are well defined; then
Lemma 4 (see ). For any constant matrix , , a scalar function , and a vector-valued function such that the following integrations are well defined:
Lemma 5. Let has continuous derived function on interval . Then for any matrix , scalar , the following inequality holds:
Proof. From Lemma 3, we can get Notice that Therefore, we get This completes the proof.
3. Main Results
In this section we will give sufficient conditions under which the system (5) is asymptotically stable.
Theorem 6. For given scalars and , the system (5) with the neuron activation function satisfying the condition (6) is asymptotically stable if there exists , , , , , diagonal matrices , , , and , , and with appropriate dimensions such that the following symmetric linear matrix inequality holds: where with
Proof. Consider a novel augmented Lyapunov-Krasovskii functional for the system (5) as follows:
The time derivative of along the trajectory of system (5) is given as
Here, is the sum of all integral terms expressed as
Apply Lemma 2 to ,
From Lemmas 4 and 5, we have
Then combining (29) and (30), we can obtain that From Lemma 3, we get Then combining (32) and (33) we can have that From (6), for any positive diagonal matrices one can easily check Furthermore, for arbitrary matrices , , , , with appropriate dimensions, we have
The combination of (26)–(37) gives that where , are defined in (13) and (14), respectively, and Note that the scalar valued function is quadratic function on the scalar and the coefficient of second order is since . This means that the function is a convex quadratic function for . Finally, apply Fact 1 and Lemma 1 in order, then we get which means the asymptotic stability of the system (19). This completes the proof.
Remark 7. In Theorem 6, the augmented vector has integrating terms of activation function which are and . By these terms, more past history of can be used, which lead to less conservative results.
Remark 8. Compared with those in previous articles, Ours constructed a new type of Lyapunov-Krasovskii functional which has three differences: an independent augmented variable ; the cross terms between entries in , , respectively; quadratic terms multiplied by first, second, and third degrees of a scalar function by 1 means the number increase of the integral by 1.
Remark 9. Compared with traditional approach to deal with term like , Lemma 5 provides a new handling method. This new handling method can establish the relationship among , and , which may significantly reduce the conservatism of stability criteria.