分数阶忆阻神经网络的O(t-α)与Mittag-Leffler同步研究

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"具有延迟和不连续神经元激活的分数阶忆阻神经网络的O（t-α）同步和Mittag-Leffler同步" 本文是一篇深入研究神经网络同步问题的研究论文，发表在《神经网络》期刊上。作者探讨了分数阶忆阻神经网络（Fractional-order Memristive Neural Networks，FOMNNs）在存在时变延迟和不连续神经元激活函数情况下的O（t-α）同步和Mittag-Leffler同步。忆阻神经网络是一种模拟大脑神经元活动的新型计算模型，其中忆阻器作为关键元件，可以实现动态存储和计算功能。首先，研究中涉及的"分数阶"概念是指系统的动力学特性不仅由整数阶导数决定，还受到非整数阶导数的影响。这使得网络的动态行为更为复杂，但也更接近真实生物神经系统的非线性特性。分数阶系统通常表现出更丰富的行为，包括记忆效应和自相似性。 "时间变延迟"是许多实际系统中的常见现象，例如在神经信号传输过程中，信息传递的时间可能因各种因素而变化。在神经网络模型中，延迟可能导致系统稳定性问题，因此理解和处理这种延迟对于网络设计至关重要。 "O(t-α)同步"是一种非线性同步模式，其中α表示分数阶参数。这种同步意味着网络中的各个节点随着时间的推移以一种特定的O(t-α)速度趋近于彼此的状态，即使在有延迟和不连续性的情况下也能保持网络的一致性。 "Mittag-Leffler同步"则是一种特殊的非线性同步形式，源自Mittag-Leffler函数，该函数在分数阶微积分中广泛应用。这种同步方法考虑了系统的长期行为，特别适合处理具有长期记忆效应的系统，如分数阶系统。论文中，作者提出了针对具有这些复杂特性的FOMNNs的不连续适应控制策略。适应控制是一种动态调整控制器参数的方法，以应对系统不确定性或参数变化，确保网络能够在延迟和非线性激活函数的影响下实现稳定同步。在分析和设计同步策略的过程中，作者可能采用了Lyapunov稳定性理论来证明系统的稳定性，并通过数值模拟验证了所提方法的有效性。这些结果对于理解分数阶忆阻神经网络的行为，以及在神经计算、模式识别、信号处理等领域应用该类网络提供了理论基础。这篇研究论文对分数阶忆阻神经网络的同步问题进行了深入探讨，为理解和优化这类网络提供了新的视角和方法。它不仅丰富了神经网络理论，也为解决实际问题如信息处理和控制系统的优化设计提供了有价值的工具。

12 J. Chen et al. / Neural Networks 100 (2018) 10–24

for i, j = 1, 2, . . . , n and t ≥ 0, where switching jumps T

> 0,

> 0 and T

> 0 are constants,

and

are all

constants and

> 0.

Let τ = max{sup

t≥0

(t); i, j = 1, . . ., n} and let C

C[−τ , 0] denote a Banach space of all continuous functions ϕ :

[−τ , 0] → R. Sometime, for x ∈ R

, we write x ∈ C

means

x(s) ≡ x in [−τ , 0]. For ϕ ∈ C

, let ∥ϕ∥

= sup

s∈[−τ , 0]

∥ϕ(s)∥.

The initial states associated with DFMNN (2.1) are of the form

+ s) = ϕ

(s), s ∈ [−τ , 0], i = 1, . . . , n. (2.5)

Let x

∈ C ([−τ , 0], R

) be defined by x

(s) = x(t + s), −τ ≤ s ≤

0, and the initial states (2.5) can be rewritten as

= ϕ ∈ C

. (2.6)

Now, we introduce the following definitions about set-valued

map and differential inclusion. Suppose E ⊂ R

, then x → F (x)

is called a set-valued map from E to R

, if for each point x ∈ E,

there exists a nonempty set F (x) ⊂ R

. A set-valued map F with

nonempty values is said to be upper semicontinuous at x

∈ E, if

for any open set N containing F (x

), there exists a neighborhood

M of x

such that F (M) ⊂ N. The map F (x) is said to have a closed

(convex, compact)image if for each x ∈ E, F (x) is closed (convex,

compact).

Suppose E ⊂ R

, co[E] denote the closure of the convex hull of

set E. Let f : R × R

→ R

be measurable and essentially locally

bounded. Then the Filippov set-valued map K (f ) : R × R

→ R

defined as follows:

K (f )(t, x) ==



δ>0



µ(N)=0

co[f (t, B(x, δ)/N)]

where B(x, δ) is the ball of center x and radius δ, intersection is

taken over all sets N of measure zero and over all δ > 0, µ(N)

is Lebesgue measure of set N.

Calculus for the map K , the following properties can be ob-

tained (see Wang et al., 2014).

Lemma 2.1. The map K : {f |f : R

→ R

} → {g|g : R

→ 2

}

has the following properties:

(1) Assume that f

: R

, j ∈ {1, 2, . . . , N} are locally bounded,

then

K [



k=1

]⊆



k=1

K [f

]

where



is Kronecker product.

(2) Let g : R

→ R

p×n

be continuous and f : R

→ R

be locally

bounded, then

K [gf ](x) = g(x)K [f ](x).

Definition 2.1. A vector-value function x(t) defined on a non-

degenerate interval I ⊂ R is called a Filippov solution of system

x(t) = f (t, x

), if it is absolutely continuous on any subinterval

, t

] of I and for a.a. t ∈ I, x(t) satisfies the differential inclusion

x(t) ∈ K (f )(t, x

). (2.7)

The global existence results of fractional-order functional dif-

ferential inclusions are given in Wang et al. (2009).

According to Definition 2.1, we can define the Filippov solution

of DFMNN (2.1) as follows. A function x(t) is said to be a (Filippov)

solution of DFMNN (2.1) on [0, +∞) with initial condition (2.5) or

(2.6), if x(t) is absolutely continuous on any compact interval of

[0, +∞) and satisfies differential inclusions

(t) ∈ −K [c

](x

(t))x

(t) +



j=1

K [a

](x

(t))



j=1

K [b

](x

(t − τ

(t))) + I

(2.8)

Or equivalently, there exist ϑ

(·) ∈ K [c

](·), γ

(·) ∈ K [a

](·)

and η

(·) ∈ K [b

](·) satisfy

(t) = −ϑ

(t))x

(t) +



j=1

(t))



j=1

(t − τ

(t))) + I

a.a. t ≥ t

+ θ ) = ϕ

(θ), θ ∈ [−τ , 0],

¯η

(ϕ

(θ)) ∈ K [b

](ϕ

(θ)),

a.a. θ ∈ [−τ , 0].

(2.9)

Consider the DFMNN (2.1) as the driver system, the controlled

response system can be described as follows:

(t) = −c

(t))y

(t) +



j=1

(t))f

(t))



j=1

(t − τ

(t)))g

(t − τ

(t)))

+ I

+ u

(t),

(2.10)

where i = 1, . . . , n, t ≥ t

0 < α < 1. Similarly

(t) ∈ −K [c

(t) +



j=1

K [a

](y

(t))



j=1

K [b

](y

(t − τ

(t)))

+ I

+ u

(t).

(2.11)

Or equivalently, there exist

(·) ∈ K [c

](·), ¯γ

(·) ∈ K [a

](·) and

¯η

(·) ∈ K [b

](·) satisfy

(t) =

(t))y

(t) +



j=1

¯γ

(t))



j=1

¯η

(t − τ

(t))) + I

+ u(t),

a.a. t ≥ t

+ θ ) = ψ

(θ), θ ∈ [−τ , 0],

¯η

(ψ

(θ)) ∈ K [b

](ψ

(θ)),

a.a. θ ∈ [−τ , 0].

(2.12)

From Lemma 2.1, it is obvious that











K [a

] ⊆ K [a

]



K [f

K [b

] ⊆ K [b

]



K [g

K [c

] = K [c

](x

(2.13)

for i, j = 1, 2, . . . , n. According to the definition of the map K we

can get



K [f

](x

) =

co{f

, f

−

}

K [g

](x

) = co{g

, g

−

}

(2.14)

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