Further analysis on stability of delayed neural networks via inequality technique

In recent years, extensive research has been done in neural networks such as Hopfield neural networks, Cohen-Grossberg neural networks, cellular neural networks, and bidirectional associative memory neural networks, because of their potential applications in pattern recognition, image processing, associative memory, and so on, see [1‐28]. Recently, a new type of neural networks--impulsive neural networks display a combination of characteristics of both the continuous-time and discrete-time systems, which is an appropriate description of the phenomena of abrupt qualitative dynamical changes of essentially continuous-time systems, see [4, 9, 13‐22]. The stability of impulsive delay neural networks has become an important topic of theoretical studies and has been investigated by many researchers via different approaches, see [9, 13‐16, 20‐22] and the references cited therein. For example, Liu et al. [14] obtained some sufficient conditions on global exponential stability by utilizing impulsive delay differential inequality that has been given by Yue et al. [18] for impulsive high-order Hopfield neural networks with time-varying delays as follows:

In [20, 21], Xu and Yang investigated the global exponential stability of impulsive delay neural networks by establishing a delay differential inequality with impulsive initial conditions. The results extend and improve the recent works [23, 24]. More recently, Yang et al. [22] investigated the global exponential stability by Lyapunov function and Halanay inequality for impulsive extended BAM type Cohen-Grossberg neural networks with delays and variable coefficients as follows:

where

Although some stability conditions for impulsive delay neural networks proposed in [9, 14, 15, 18‐22], they have some conservatism to some extent, and there still exists open room for further improvement.

Recently, Li [25] establishes a new impulsive delay differential inequality as follows:

Lemma 1.1. Let α, β, r and τ denote nonnegative constants, and function f ∈ PC(ℝ, ℝ₊) satisfies the scalar impulsive differential inequality

where 0 < σ ≤ + ∞, a _k , b _k , ∈ ℝ₊, k(·) ∈ PC([0, σ], ℝ₊) satisfies∫ 0 σ k ( s ) e η 0 s d s < ∞for some positive constant η₀ > 0 in the case when σ = +∞. Moreover, when σ = +∞, the interval [t - σ, t] is understood to be replaced by (-∞, t].

(i) α > β + r ∫ 0 σ k ( s ) d s .

(ii) There exist constants M > 0, η > 0 such that

where λ ∈ (0, η₀) satisfies

Then,

In particular, it includes the special case:

Lemma 1.2. Let α, β and τ denote nonnegative constants, a _k , b _k ∈ ℝ₊, and function f ∈ PC(ℝ, ℝ₊) satisfies

(i) α > β ≥ 0.

(ii) There exist constants M > 0, η > 0 such that

where λ > 0 satisfies

Then

The purpose of this paper is to improve the results in [9, 14, 15, 18‐22] via the above results in Lemma 1.2, which is a special case of [25]. We will derive some new sufficient conditions to ensure the global exponential stability of equilibrium point for impulsive delay Hopfield neural networks (1.1) and BAM type Cohen-Grossberg neural networks (1.2). The main advantages of the obtained exponential stability conditions include:

In addition, some illustrative examples are also given to demonstrate the effectiveness of the obtained results.

In this section, we will give some new sufficient conditions on the global exponential stability of equilibrium point for the neural network (1.1). The conditions are less restrictive and conservative than that given in [14].

System (1.1) may be rewritten in the following matrices forms:

Remark 2.1. For detail information about (2.1), one may see [14].

Theorem 2.1. Assume that conditions (i), (ii) in Theorem 1 in[14]hold, and

whereη = inf k ∈ ℤ + { t k - t k - 1 } > 0,

and λ > 0 satisfies

Then the equilibrium point of the system (1.1) is globally exponentially stable with the approximate exponential convergence rateλ - ln ρ η.

Remark 2.2. For the proofs of Theorems 2.1, we need only to mention a few points, since the rest is the same as in the proofs of Theorems 1 in [14]. First, similarly one may define V(t) = x ^T (t)Px(t), and it can be deduced that

Then using Lemma 1.2 in this paper (replacing Lemma 1 in [14]), Theorem 2.1 can be obtained.

Similarly we can obtain another stability criterion corresponding to Theorem 2 in [14] as follows:

Theorem 2.2. Assume that conditions (i) in Theorem 2 in[14]hold and

whereη = inf k ∈ ℤ + { t k - t k - 1 } > 0,

and λ > 0 satisfies

Then the equilibrium point of the system (1.1) is globally exponentially stable with the approximate exponential convergence rateλ - ln ρ η.

Remark 2.3. In [14], under the assumption that inf k ∈ ℤ + { t k - t k - 1 } > τ δ , δ > 1 ., Liu et al. obtained some theorems on exponential stability of (1.1). Note that in our theorem 2.1 and 2.2, we only require that inf k ∈ ℤ + { t k - t k - 1 } > 0. Thus, our results improve the previous findings.

Example 2.1 Consider the three-neuron Hopfield neural network (1.1) with g₁(u₁) = tanh(0.63u₁), g₂(u₂) = tanh(0.78u₂), g₃(u₃) = tanh(0.46u₃), h₁(u₁) = tanh(0.09u₁), h₂(u₂) = tanh(0.02u₂), h₃(u₃) = tanh(0.17u₃), C = diag (C₁, C₂, C₃) = diag (0.89, 0.88, 0.53), R = diag (R₁, R₂, R₃) = diag(0.16, 0.12, 0.03), D = diag(d₁, d₂, d₃) = diag(-0.95, -0.84, -0.99), 0 ≤ τ _i (t) ≤ 0.5, i = 1, 2, 3 and

In this example, similar to [14], one may choose P = diag(0.9, 0.7, 0.8), ε₁ = 1, ε₂ = 2 such that Ω < 0 in Theorem 2.1, and that a = 10.2628 > 2.3814 = b. Also, we can compute that ρ = 1. Thus, by Theorem 2.1, the equilibrium point of (2.2) is globally exponentially stable with the approximate convergence rate λ for inf k ∈ ℤ + { t k - t k - 1 } > 0, where λ > 0 satisfies the inequality: λ ≤ 10.2628 - 2.3814e^{λ 0.5}.

Remark 2.4. In [14], Liu et al. obtained that the equilibrium point of (2.2) is globally exponentially stable for inf k ∈ ℤ + { t k - t k - 1 } > 0 . 5 0 5 , which was more restrictive and conservative than that of our result. Therefore, the result in this paper is applicable to more conditions.

In this section, we will reconsider the global exponential stability of impulsive BAM type Cohen-Grossberg neural networks (1.2).

Theorem 3.1. Assume that (H₁) - (H₃) and (i), (ii) in Theorem 2 in[22]hold; moreover, suppose that

whereη = inf k ∈ ℤ + { t k - t k - 1 } > 0,

and λ > 0 satisfies

Then the equilibrium point of the system (1.2) is globally exponentially stable with the approximate exponential convergence rateλ - ln M η.

Proof. Consider Lyapunov function as follows:

Then similar to the proof of Theorem 2 in [22], we arrive at

Then by Lemma 1.2, the result holds. □

Remark 3.1. In [22], Yang et al. obtained a sufficient condition for global asymptotic stability of (1.2), which assumes that inf k ∈ ℤ + { t k - t k - 1 } > τ δ , δ > 1 ., while ours do not impose this restriction.

Example 3.1. Consider the following extended BAM neural networks:

where u _k = w _k = v _k = e _k = 1 + (-1) ^k δ(t - t _k ), the impulse times t _k satisfy 0 ≤ t₀ < t₁ < < t _k < ⋯, lim_{k→ +∞}t _k = +∞ and inf k ∈ ℤ + { t k - t k - 1 } = 1 6. Let τ = 18.

By simple calculation, we can obtain k 1 = 9 1 0, k 2 = 1 2 5, r k = 3 5, R k = 1 1 5, M ≐ max { 1 , a ¯ a ¯ r k - 1 + a ¯ a ¯ R k exp { λ τ } } = 2 0 3 + 4 1 5 exp { 1 8 λ }, where λ > 0 satisfies the inequality:λ ≤ 9 1 0 - 1 2 5 exp { 1 8 λ }. We may choose λ = 0.16, then M ≈ 11.511 < 12.932 = exp {16λ}. By Theorem 3.1, the equilibrium point ( π 2 , π ) of (3.1) is globally exponentially stable with the approximate convergence rate 0.007.

Remark 3.2. It can be easily verified that (iv), (v) in Theorem 2 in [22] are violated in the above example. Thus, our results improve the results in [22].

In this section, we shall give a new inequality that is different from Lemma 1.2 and can be applied to the case not covered in [19, 20].

Theorem 4.1. Suppose that

where λ > 0 satisfies

Then,

Proof. Condition (i) implies that there exists small enough λ > 0 such that the inequality (4.1) holds.

Next, we show

where a₀ = 1, b₀ = 0.

It is clear that f ( t ) ≤ f ¯ ( t 0 ) for t ∈ [t₀ - τ, t₀] by the definition of f ¯.

Take k = 0, we shall show, for t ∈ [t₀, t₁)

Suppose on the contrary, then there exists some t ∈ [t₀, t₁) such that f ( t ) > f ¯ ( t 0 ) exp { - λ ( t - t 0 ) } .

Let

then t^⋆ ∈ [t₀, t₁) and

Since f ¯ ( t ⋆ ) = sup s ∈ [ t ⋆ - τ , t ⋆ ] f ( s ) ,, t^⋆ ∈ [t₀,t₁), we get

Hence, we have

Thus, by the definitions of λ and W₀, we have

which contradicts (3). So we get that (4.2) holds for all t ∈ [t₀, t₁).

Now, we assume that for t ∈ [t_m-1, t _m ), m ∈ ℤ₊

We shall show that for t ∈ [t _m , t_m+1), m ∈ ℤ₊

By (4.3) and the fact that t _m - t_m-1> τ, we know

Hence,

If (4.4) is not true, then there exists some t ∈ [t _m , t_m-1) such that

By (4.5), we define

where

then t* ∈ [t _m , t_m+1) and

Since f ¯ ( t ⋆ ) = sup s ∈ [ t ⋆ - τ , t ⋆ ] f ( s ) ,, t^* ∈ [t _m , t_m+1), we get

In fact, when t* - τ ≥ t _m , from (5), we have

When t* - τ < t _m , note that t _k - t_k-1> τ, we have

This, together with (4), leads to

Hence, we obtain