An LMI Approach for Dynamics of Switched Cellular Neural Networks with Mixed Delays

Remark 1. We shall point out that the constants l_j and L_j can be positive, negative, or zero, and the boundedness on f_j(·) is no longer needed in this paper. Therefore, the activation function f_j(·) may be unbounded, which is also more general than the form |f_j(u)| ≤ K_j | u | , K_j > 0, j = 1,2, …, n. Different from the bounded case where the existence of an equilibrium point is always guaranteed, under the condition (H₂), in the switched system (3) it may happen that there is no equilibrium point. Thus it is of great interest to investigate the ultimate boundedness solutions and the existence of an attractor by replacing the usual stability property for system (3).

Without loss of generality, let C([−τ^*, 0], Rⁿ) denote the Banach space of continuous mapping from [−τ^*, 0] to Rⁿ equipped with the supremum norm $$. Throughout this paper, we give some notations: A^T denotes the transpose of any square matrix A, A > 0  (<0) denotes a positive (negative) definite matrix A, the symbol “*” within the matrix represents the symmetric term of the matrix, λ_min(A) represents the minimum eigenvalue of matrix A, and λ_max(A) represents the maximum eigenvalue of matrix A.

System (3) is supplemented with initial values of the type

Definition 2 (see [24].)System (3) is uniformly ultimately bounded; if there is $$, for any constant ϱ > 0, there is t^′ = t^′(ϱ) > 0, such that $$ for all t ≥ t₀ + t^′, t₀ > 0, ∥φ∥<ϱ.

Definition 3. The nonempty closed set 𝔸 ⊂ Rⁿ is called an attractor for the solution x(t; φ) of system (3) if the following formula holds:

Definition 4 (see [26].)For any switching signal σ(t) and any finite constants T₁, T₂ satisfying T₂ > T₁ ≥ 0, denote the number of discontinuity of a switching signal σ(t) over the time interval (T₁, T₂) by N_σ(T₁, T₂). If N_σ(T₁, T₂) ≤ N₀ + (T₂ − T₁)/T_α holds for T_α > 0, N₀ > 0, then T_α > 0 is called the average dwell time.

Theorem 5. Assume there is a constant μ, such that $$, and denote g(μ) as

Proof. Choose the following Lyapunov functional:

From assumption (H₂), we have

Then we have

Therefore, we obtain

If one chooses $$, then for any constant ϱ > 0 and ∥φ∥<ϱ, there is t^′ = t^′(ϱ) > 0, such that e^−atV(x(0)) < 1 for all t ≥ t^′. According to Definition 2, we have $$ for all t ≥ t^′. That is to say, system (2) is uniformly ultimately bounded. This completes the proof.

Theorem 6. If all of the conditions of Theorem 5 hold, then there exists an attractor $$ for the solutions of system (2), where $$.

Proof. If one chooses $$, Theorem 5 shows that for any ϕ there is t^′ > 0, such that $$ for all t ≥ t^′. Let $$ be denoted by $$. Clearly, $$ is closed, bounded, and invariant. Furthermore, $$. Therefore, $$ is an attractor for the solutions of system (2). This completes the proof.

Corollary 7. In addition to all of the conditions of Theorem 5 holding, if J = 0 and f_i(0) = 0 for all i = 1,2, …, n, then system (2) has a trivial solution x(t) ≡ 0, and the trivial solution of system (2) is globally exponentially stable.

Proof. If J = 0 and f_i(0) = 0 for all i = 1,2, …, n, then R₁ = 0, and it is obvious that system (2) has a trivial solution x(t) ≡ 0. From Theorem 5, one has

Theorem 8. For a given constant a > 0, if there exist positive-definite matrixes P_i = diag (p_i1, p_i2, …, p_in), Y_i = diag (y_i1, y_i2, …, y_in), i = 1,2, such that the following condition holds:

Proof . Define the Lyapunov functional candidate

If one chooses $$, then for any constant ϱ > 0 and ∥φ∥<ϱ, there is t^′ = t^′(ϱ) > 0, such that $$ for all t ≥ t^′. According to Definition 2, we have $$ for all t ≥ t^′. That is to say, system (3) is uniformly ultimately bounded, and the proof is completed.

Theorem 9. If all of the conditions of Theorem 8 hold, then there exists an attractor $$ for the solutions of system (3), where $$.

Proof. If one chooses $$, Theorem 8 shows that for any ϕ there is t^′ > 0, such that $$ for all t ≥ t^′. Let $$ be denoted by $$. Clearly, $$ is closed, bounded, and invariant. Furthermore, $$. Therefore, $$ is an attractor for the solutions of system (3). This completes the proof.

Corollary 10. In addition to all of the conditions of Theorem 8 holding, if J = 0 and f_i(0) = 0 for all i, then system (2) has a trivial solution x(t) ≡ 0, and the trivial solution of system (3) is globally exponentially stable.

Proof. If J = 0 and f_i(0) = 0 for all i, then it is obvious that system (3) has a trivial solution x(t) ≡ 0. From Theorem 8, one has

Remark 11. Up to now, various dynamical results have been proposed for switched neural networks in the literature. For example, in [15], synchronization control of switched linearly coupled delayed neural networks is investigated; in [16–20], the authors investigated the stability of switched neural networks; in [21, 22], stability and L2-gain analysis for switched delay system have been investigated. To the best of our knowledge, there are few works about the uniformly ultimate boundedness and the existence of an attractor for switched neural networks. Therefore, results of this paper are new.

Remark 12. We notice that Lian and Zhang developed an LMI approach to study the stability of switched Cohen-Grossberg neural networks and obtained some novel results in a very recent paper [20], where the considered model includes both discrete and bounded distributed delays. In [20], the following fundamental assumptions are required: (i) the delay functions τ(t), h(t) are bounded, and $$, $$; (ii) f_i(0) = 0, l_j ≤ (f_j(x) − f_j(y))/(x − y) ≤ L_j, for all i = 1,2, …, n; (iii) the switched system has only one equilibrium point. However, as a defect appearing in [20], just checking the inequality (13) in [20], it is easy to see that the assumed condition on $$ is not correct, which should be revised as $$. On the other hand, just as described by Remark 1 in this paper, for a neural network with unbounded activation functions, the considered system in [20] may have no equilibrium point or have multiple equilibrium points. In this case, it is difficult to deal with the issue of the stability of equilibrium point for switched neural networks. In order to modify this imperfection, after relaxing the conditions $$, $$, and f_i(0) = 0, replacing (i), (ii), and (iii) with assumptions (H₁) and (H₂), we drop out the assumption of the existence of a unique equilibrium point and investigate the issue of the ultimate boundedness and attractor; this modification seems more natural and reasonable.

Remark 13. When investigating the stability, although the adopted Lyapunov function in this paper is similar to those used in [20]; just from Corollaries 7 and 10, the conservatism of the conditions of the delay function in this paper has been further reduced. Hence, the obtained results on stability in this paper are complementary to the corresponding results in [20].

Remark 14. When the uncertainties appear in the system (3), employing the Lyapunov function as (27) in this paper and applying a similar method to the one used in [20], we can get the corresponding dynamical results. Due to the limitation of space, we choose not to give the straightforward but the tedious computations here for the formulas that determine the uniformly ultimate boundedness, the existence of an attractor, and stability.

Example 15. Consider the switched cellular neural networks system (3) with d_i = 1, f_i(x_i(t))  =  0.5tanh(x_i(t))(i = 1,2), τ(t)  =  0.5sin²(t), h(t) = 0.3sin²(t), and the connection weight matrices where

From assumptions (H₁) and (H₂), we can obtain d = 1, l_i = 0, L_i = 0.5, i = 1,2, τ = 0.5, h = 0.3, μ = 1.

Choosing a = 2 and solving LMIs (23), we get