忆阻器BAM神经网络周期解：存在性与全局指数稳定性分析

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"该资源是一篇研究论文，探讨了具有时变时滞的忆阻器双向联想记忆（BAM）神经网络周期解的存在性和全局指数稳定性。文章在2016年的《神经网络》期刊（Neural Networks）第75期发表，作者包括Hongfei Li、Haijun Jiang和Cheng Hu，他们来自新疆大学的数学与系统科学学院。论文经历了多次修改，最终于2015年12月18日在线发布。关键词包括忆阻器、周期解、有界性、BAM神经网络和全局指数稳定性。" 正文: 忆阻器是一种非线性电子元件，其电阻特性依赖于通过它的电流历史，这种特性使得忆阻器成为构建模拟神经网络的理想选择，尤其是对于忆阻器基础的BAM神经网络。BAM神经网络是由两个相互连接的单层感知器构成，分别模拟大脑的左半球和右半球，以实现并行信息处理。在本论文中，研究者关注的是时变时滞对忆阻器BAM神经网络动态行为的影响。时滞是许多生物和工程系统中的常见现象，它可以对网络的稳定性产生显著影响。论文采用Filippov解决方案框架，这是一种处理有滑动模式的非光滑动力系统的理论，来分析含有时变时滞的忆阻器BAM神经网络的解的有界性和最终有界性。通过链式规则和不等式技术，证明了解的这些性质。此外，论文引入了一种新的方法，即利用Yoshizawa类似定理来建立周期解的存在性。这种方法可能涉及到集值映射理论，集值映射允许网络的动态不仅取决于当前状态，还取决于过去的状态历史，这在处理时滞问题时尤其重要。通过这样的理论工具，研究者能够证明在网络中存在周期解，即网络的状态会按照一定的周期重复出现。进一步，论文探讨了全局指数稳定性，这是评估系统稳定性的关键指标。全局指数稳定性意味着无论初始条件如何，系统的所有解都将随着时间指数级地接近稳定状态。这对于确保忆阻器BAM神经网络的可靠性和预测性能至关重要。研究者运用相关理论和技巧来证明了这一点，这将有助于理解和设计更稳定的忆阻器神经网络系统。这篇论文为理解和控制具有时变时滞的忆阻器BAM神经网络提供了重要的理论基础，对于优化这类网络的性能，特别是在模拟学习、模式识别和信息处理等领域的应用具有深远意义。

H. Li et al. / Neural Networks 75 (2016) 97–109 99

Now we introduce the concept of Filippov solution (see Kim

et al., 2015). Consider the following non-autonomous time-varying

delayed differential system in vector notation:

= f (t, z(t), z(t − τ (t))), (2)

where z(t) = (z

(t), z

(t), . . . , z

(t))

represents state vector,

z(t − τ (t)) = (z

(t − τ (t)), z

(t − τ (t)), . . . , z

(t − τ (t)))

represents time-varying delayed state vector, f : R × R

× R

→

is discontinuous right-hand sides in z.

Definition 3 (Aubin & Cellina, 1984). Let us consider the set-valued

map F : R × R

× R

→ R

defined as

F(t, z(t), z(t − τ (t)))



>0,



µ(N)=0,

µ(M)=0

co[f (t, B(z, ρ

) \ N, B(z(t − τ (t)), ρ

) \ M)],

where µ(N) (µ(M)) is the Lebesgue measure of set N (M);

B(z, ρ

) is the ball of center z and radius ρ

; B(z(t − τ (t)), ρ

)

is the ball of center z(t − τ (t)) and radius ρ

; co(E) is the closure

of the convex hull of some set E; intersection is taken over all sets

N, M of measure zero and over all ρ

> 0, ρ

> 0. A vector-

value function z(t) in Filippov’s sense (see Kim et al., 2015) of the

Cauchy problem for this system (3) with initial condition z(0) = z

which is absolutely continuous on any compact subinterval [0, T ),

T ∈ (0, ∞], satisfies z(0) = z

and differential inclusion:

∈ F (t, z(t), z(t − τ (t))), for t ∈ [0, T ). (3)

Now, let us consider system (1). Since a

(t)), b

(t)),

(t)) and d

(t)) in system (1) are defined as discontinuous

functions and the classical definition of solutions has been shown

to be invalid for the differential equation with discontinuous right-

hand sides. For this purpose, by applying the above theories of set-

valued maps and functional differential inclusions, we will extend

the concept of the Filippov solution to system (1).

Definition 4 (Filippov Solution). A vector function z(t) =

(t), x

(t), . . . , x

(t), y

(t), . . . , y

(t))

: [−τ , T ) →

n+m

, T ∈ (0, ∞], is a state solution of the delayed and discon-

tinuous system (1) on [−τ , T ), if z(t) is absolutely continuous on

any compact interval of [−τ , T ) and satisfies differential inclusions











(t)

∈ −a

(t)x

(t) +



j=1

co[a

, a

(t))



j=1

co[b

, b

(t − τ (t))) + I

(t),

(t)

∈ −c

(t)y

(t) +



i=1

co[c

, c

(t))



i=1

co[d

, d

(t − τ (t))) + J

(t),

(4)

where

= min{a

∗

, a

∗∗

}, a

= max{a

∗

, a

∗∗

= min{b

∗

, b

∗∗

}, b

= max{b

∗

, b

∗∗

= min{c

∗

, c

∗∗

}, c

= max{c

∗

, c

∗∗

= min{d

∗

, d

∗∗

}, d

= max{d

∗

, d

∗∗

or equivalently, there exist α

∈ co[a

, a

], β

∈ co[b

, b

], γ

∈

co[c

, c

], η

∈ co[d

, d

], such that











(t)

= −a

(t)x

(t) +



j=1

(t))



j=1

(t − τ (t))) + I

(t),

(t)

= −c

(t)y

(t) +



i=1

(t))



i=1

(t − τ (t))) + J

(t),

(5)

for a.e. t ∈ [0, T ), i = 1, 2, . . . , n, j = 1, 2, . . . , m.

Definition 5 (IVP). For any continuous function θ(s) =

(φ

(s), φ

(s), . . . , φ

(s), ψ

(s), . . . , ψ

(s))

: [−τ, 0) →

n+m

, an absolutely continuous function z(t) = z(t, θ) =

(t), x

(t), . . . , x

(t), y

(t), . . . , y

(t))

is said to be a so-

lution of the initial value problem for system (1) on [0, T ) (T might

be to +∞) with initial data θ(s), s ∈ [−τ , 0], if the following con-

dition holds











(t)

= −a

(t)x

(t) +



j=1

(t))



j=1

(t − τ (t))) + I

(t),

(t)

= −c

(t)y

(t) +



i=1

(t))



i=1

(t − τ (t))) + J

(t),

∈ co[a

, a

], β

∈ co[b

, b

∈ co[c

, c

], η

∈ co[d

, d

(s) = φ

(s), y

(s) = ψ

(s), ∀s ∈ [−τ , 0],

for a.e. t ∈ [0, T ), i = 1, 2, . . . , n, j = 1, 2, . . . , m.

Definition 6 (Periodic Solution). A solution z(t) of system (1) on

[0, T ) (T might be to +∞) is a ω-periodic solution with period ω,

if z(t + ω) = z(t), for all t ≥ 0.

Definition 7 (Stability). Let z

∗

(t) = (z

∗

(t), z

∗

(t), . . . , z

∗

n+m

(t))

be a solution of system (1), with initial conditions θ

′

(s) =

(φ

′

(t), φ

′

(t), . . . , φ

′

(t), ψ

′

(t), ψ

′

(t), . . . , ψ

′

(t))

: [−τ , 0) →

n+m

, the solution z

∗

(t) of system (1) is said to be globally

exponentially stable, if there exist constants ε > 0 and M > 0 such

that for any solution z(t) = (z

(t), z

(t), . . . , z

n+m

(t))

of system

(1) with initial conditions θ (s) = (φ

(t), φ

(t), . . . , φ

(t), ψ

(t),

(t), . . . , ψ

(t))

: [−τ , 0) → R

n+m

, satisfies



n+m



k=1

(t) − z

∗

(t)|



≤ M∥θ − θ

′

∥e

−εt

, for t ≥ 0.

Definition 8 (Clarke’s Generalized Gradient Clarke, 1983). Let

V (z) : R

→ R be a locally Lipschitz function, then for any z ∈ R

we can define Clarke’s generalized gradient of V at point z ∈ R

as follows:

∂V (z) = co



lim

k→∞

∇V (z

) : z

→ z, z

∈ N



Ω



where Ω ⊂ R

is the set of points where V is not differentiable

and N ⊂ R

is an arbitrary set with measure zero.

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