State Feedback Stabilization for Neutral-Type Neural Networks with Time-Varying Discrete and Unbounded Distributed Delays

The problem of stabilization for a class of neutral-type neural networks with discrete and unbounded distributed delays is investigated. By introducing an appropriate Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative, delay-range-dependent and rate-dependent stabilization criteria are presented in the form of LMIs with nonlinear constraints. In order to solve the nonlinear problem, a cone complementarity linearization (CCL) algorithm is offered. In addition, several numerical examples are provided to illustrate the applicability of the proposed approach.


Introduction
In 1943, McCulloch and Pitts [1] proposed the concept of artificial neurons and proved that a single neuron can perform logic functions.This is usually viewed as the beginning of era of artificial neural networks, which is called neural networks for short.Recently, neural networks have received considerable attention due to their wide applications in solving some optimization problems, pattern recognition, image processing, and signal processing.Since time delay is frequently a source of instability and/or oscillation of many practical systems, several approaches have been proposed for analysis and synthesis of delayed neural networks (see [2][3][4][5][6][7][8] and the references therein).
A neutral-type time-delay system contains delays in the state and its derivatives.Such system can be found in population ecology, lossless transmission lines, heat exchangers, and so forth.Because of its wider applications, the stability for the class of neutral-type systems has received considerable attention in the last several decades (see, e.g., [9][10][11][12][13]). Corresponding to the class of neutral-type timedelay systems, we will also get a class of neutral-type neural networks.Stability criteria for neutral-type neural networks have been proposed in [3,[14][15][16][17][18][19][20][21] by constructing appropriate Lyapunov-Krasovskii functional and applying the linear matrix inequality (LMI) approach.Recently, the stability conditions for the neutral-type neural networks with discrete and unbounded distributed delays were provided in [17][18][19][20][21]. Specifically, Rakkiyappan and Balasubramaniam [21] studied the global asymptotic stability for a class of neutraltype neural networks with two unbounded distributed delays by the so-called LMI technique.Lu [22] designed a state feedback controller stabilizing the neutral-type neural networks with discrete and bounded distributed delays.However, there are no results about the stabilization for neutral-type neural networks with unbounded distributed delay(s), this motivates our research.
This paper deals with the stabilization problem of a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays.New sufficient conditions for the existence of the state feedback controller are proposed by introducing a Lyapunov-Krasovskii functional and using Jensen inequality technique to deal with its derivative.Since these sufficient conditions are presented in the form of LMIs with nonlinear constraints, the so-called CCL technique is employed to deal with the nonlinear constraints.This allows us to obtain a state feedback gain by LMI Control Toolbox of MATLAB.Several numerical examples are given to illustrate the effectiveness of the proposed approach.

Problem Formulation
Consider the class of neutral-type neural networks with time-varying discrete and unbounded distributed delays described by the integro differential equation: where i = 1, 2, . . ., n, x i (t) is the state variable of the ith neuron, a i > 0 is a constant, b i j , c i j , d i j , e i j , and f i j are connection weight coefficients of the neurons, f j is the neuron activation function, i is the external bias vector element, I i (t) is the external control input vector, φ i (θ) denotes the initial condition, τ(t) and h(t) are time-varying delays satisfying and z j (t) is a real value nonnegative continuous function on [0, +∞) and satisfies When I i (t) ≡ 0 for all i, the global asymptotic stability of the system (1) has been considered in [21].Here, the control input item I i (t) in ( 1) is borrowed from [22].Throughout this paper, we assume that the neuron activation functions f j ( j = 1, 2, . . ., n) satisfy the following condition: where be an equilibrium point of the system (1).Set y i = x i − x * i , i = 1, . . ., n.Then the system (1) can be written as where The main purpose of this paper is to develop a delayrange-dependent and rate-dependent condition for the existence of a state feedback controller which stabilizes the neutral-type neural network (6) with time-varying discrete and unbounded distributed delays.
A delay-range-dependent and rate-dependent criterion stabilizing (6) is obtained based on a Lyapunov-Krasovskii functional approach accompanied with a CCL technique, a delay-range-dependent and rate-dependent criterion stabilizing ( 6) is obtained.Set y t (θ) = y(t + θ), θ ∈ (−∞, 0].Assume that there exists a scalar V > 0 such that 0 −∞ y t (s) 2 ds ≤ V and 0 −∞ ẏt (s) 2 ds ≤ V .This guarantees that the so-called Lyapunov-Krasovskii functional theory can be applied to the considered stabilization problem.

Main Results
In this section, we will investigate a delay-range-dependent and rate-dependent stabilization criterion for the neutraltype neural network (6).This requires the following several lemmas.
Lemma 1 (Jensen inequality [23]).Given a real symmetry positive-definite matrix P ∈ R n×n , a pair of scalars a and b satisfying b ≥ a ≥ 0. If a vector-valued function Lemma 2 (Schur complementary lemma [24]).For a given matrix S := Based on the above three lemmas, the following delayrange-dependent and rate-dependent stabilization criterion for the neutral-type neural network (6) can be investigated by constructing an appropriate Lyapunov-Krasovskii functional and applying the CCL technique.

Theorem 4. For given scalars h
and τ 3 such that h 2 > h 1 > 0 and τ 2 > τ 1 > 0, the system (6) subject to (3) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices where Furthermore, when the LMIs ( 10)-( 12) with the constraint ( 13) are feasible, a desired state feedback gain is given by Proof.Choose a Lyapunov-Krasovskii functional as follows: with Set y t 2 dθ.By some derivation, we have where , which implies that the functional V (y t ) is a legitimate due to [26].Next we deal with the derivatives of V i (y t ) (i = 1, . . ., 6).Set When the controller ( 7) is applied to the system (6), the resultant closed-loop system is obtained as follows: Due to Lemma 1, (3), the derivatives of V i (y t ) (i = 1, . . ., 4) along the trajectory of ( 21) are V2 y t = y T (t) V4 Journal of Control Science and Engineering By Lemma 3 and (4), we can obtain the derivatives of V 5 (y t ) and V 6 (y t ) as follows: For the functions g j , j = 1, 2, . . ., n, using g where L = diag(L 1 , . . ., L n ).Noting that K = P −1 K, we have The combination of ( 22)-( 28) gives where By Lemma 2 and ( 10)-( 13), we can get that where Thus it is easy to see that V (y t ) < 0, which, together with (7) and [26,Theorem 3.1.6],implies that the system ( 6) is asymptotically stabilizable.The proof is completed.Remark 5. A delay-range-dependent and rate-dependent condition under which the system ( 6) is asymptotically stabilizable is investigated in Theorem 4. In [7,8], the stability of a class of the neural networks with unbounded distributed delays has been studied; however, the neutral term was not considered in their models.In [18], only a unbounded distributed delay is taken into account.But in this paper the neutral-type neural networks with two unbounded distributed delays are studied.Therefore, Theorem 4 in this paper can be viewed as an extension of the corresponding results in [7,8,18].Remark 6.In order to apply [26,Theorem 3.1.6]to conclude that the system ( 6) is asymptotically stabilizable, we denote a new norm • 2 w on a space of functions.Noting that the stabilization criterion proposed in Theorem 4 includes the nonlinear constraint (13), we cannot obtain the desired gain K by using the LMI Controller Toolbox of MATLAB.In order to deal with the nonlinear constraint ( 13), we offer the following CCL algorithm.
The algorithm is available to determine the maximum value of h 2 for given h 1 , h 3 , τ 1 , τ 2 , and τ 3 and the corresponding gain K.
is feasible for the variables N, T i (i = 1, . . ., 6), R l (l = 4, 5), and Q j ( j = 4, 5, 6) and the matrices K and P obtained in Step 3, then set K = P −1 K and stop.If the LMI (36) is infeasible within a specified number of iterations, then stop; otherwise, set k = k + 1 and go to Step 3.
Remark 8.The idea of the above algorithm is taken from [27].Compared with the CCL algorithms proposed in [28,29], the merit of the above algorithm is that more freedoms are provided to test the iteration stop conditions (see [27,Remark 4] for details).
Next, we consider the following neutral-type neural network (37), which is a special case of (6).A stability criterion for the neural network (37) has been presented in [21].The following corollary can be immediately obtained from Theorem 4, which gives a stabilization criterion for the neutral-type neural network (37): where Corollary 9.For given scalars h 2 and h 3 such that h 2 > 0, the neutral-type neural network (37) subject to (38) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices P, P, and N := diag ( n 1 , . . ., n n ), a matrix K and positive scalars ε 1 and ε 2 , such that Journal of Control Science and Engineering where − e T 8 N e 8 − ε 2 e T 6 e 6 , Furthermore, when the LMIs (39) with the constraint (40) are feasible, a desired state feedback gain is given by K = P −1 K.
Proof.If we use the following ξ(t) and V (y t ) instead of ξ(t) and V (y t ) in the proof of Theorem 4, respectively, then the proof can be easily completed: with The following system model is a practical partial element equivalent circuit (PEEC) that is described in [30, Figure 1] by Bellen et al.: where τ is a positive scalar representing the system delay.
It should be emphasized that the matrix A in (44) is not necessarily diagonal, which is different from the one in ( 6).However, one can easily find that the proof of Theorem 4 is always available whether the matrix A is diagonal or not.For this reason, we can derive the following Corollaries 10 and 12 by a method similar to obtaining Theorem 4.
Corollary 10.For a given scalar τ > 0, the neutral-type system (44) is asymptotically stabilizable via the controller (7) if there exist real symmetry positive-definite matrices P, P, Q i (i = 1, . . ., 3), Q 2 , Q 3 , T j , and T j ( j = 1, 2) and a matrix K, such that where Furthermore, when the LMIs (46) and (47) with the nonlinear constraint (48) are feasible, a desired state feedback gain is given by K = P −1 K.  Corollary 12.For a given scalar τ > 0, the unforced system of (44) is asymptotically stable if there exist real symmetry positive-definite matrices P and Q i (i = 1, 2, 3) such that where and ěi is defined as in Corollary 10.

Numerical Examples
In this section, we will illustrate our approach by several numerical examples.
Example 13.Consider the neutral-type neural network (6), where (52) Solving the LMIs ( 10)-( 12) with nonlinear constraint (13) by Algorithm 7 and the LMI Control Toolbox of MATLAB, we can get a desired state feedback gain stabilizing the system as follows: The state response curves of the closed-loop system are given in Figures 1 and 2, where the initial functions in Figures 1 and 2    (58) When I(t) ≡ 0, Bellen et al. [30] gave sufficient conditions for the asymptotic stability of the zero solution to (44) by utilizing a suitable reformulation of the system.By applying Algorithm 7 and the LMI Control Toolbox of MATLAB to solve LMIs (46) and (47) with the nonlinear constraint (48), a required state feedback gain is obtained as follows: (59) The simulation results for the PEEC system are shown by Figures 5 and 6 with the initial value φ(θ) = [sin(t) sin(2t) sin(3t)] T .Figure 5 represents the state responses of the PEEC system when I(t) ≡ 0, and Figure 6 represents the state responses of the resultant closed-loop system.From Figures 5 and 6, we can see that both the openloop and closed-loop systems are stable, however, the speed of convergence towards the null point in the closed-loop system is faster than one in the open-loop system.

Conclusions
In this paper, the delay-range-dependent and rate-dependent stabilization criteria for a class of neutral-type neural networks with time-varying discrete and unbounded distributed delays are established.The criteria are derived by constructing an appropriate Lyapunov-Krasovskii functional and using certain matrix technique.A CCL algorithm is developed to obtain the state feedback gain K. Numerical examples are provided to show that our results are more suitable than some existing ones, which illustrate the merits of the proposed approach.

Remark 11 .
Similar to Algorithm 7, one can solve the stabilization criteria proposed in Corollaries 9 and 10.

Table 1 :
Allowable upper bound of τ M for different β.