西南大学学报 (自然科学版)  2018, Vol. 40 Issue (11): 67-73.  DOI: 10.13718/j.cnki.xdzk.2018.11.011
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  • 具有变时滞的高阶随机Hopfield神经网络在有限时间内的控制同步    [PDF全文]
    蒲浩1, 蒋海军2, 胡成2, 冉杰1, 张转周1     
    1. 遵义师范学院 数学学院, 贵州 遵义 563002;
    2. 新疆大学 数学与系统科学学院, 乌鲁木齐 830046
    摘要:研究了具有变时滞的高阶随机Hopfield神经网络在有限时间内的控制同步.通过李雅普诺夫函数,有限时间内稳定性理论,随机微分方程理论和一些不等式方法,基于p-范数得到了新的有限时间内同步的充分条件.本文结论是对之前相关结论的推广.
    关键词高阶Hopfield神经网络    随机扰动项    变时滞    有限时间内同步    p-范数    

    自从约翰·霍普菲尔德在1982年提出Hopfield神经网络模型[1]以来,其理论在信号与图像处理、联想记忆及组合优化等问题中有着重要的应用[2-3],得到了许多有关Hopfield神经网络稳定性或者同步的结论[4-5].然而,在过去的一些关于Hopfield神经网络稳定性或者同步的文章中,系统是在无限时间内实现稳定或者同步[6-8].

    在神经网络系统中出于高效的目的,需要神经网络系统在有限的时间内实现稳定或者同步,如在工程领域等.自从文献[9]在1991年提出神经网络在有限时间内稳定的理论以来,许多研究者对神经网络在有限时间内的稳定性问题或者同步问题广泛研究,得到了很多有效的理论[10-12].在神经网络系统实现稳定或同步的过程中,由于信号在不同的神经元之间传递的速度是有限的,出现了影响系统稳定或者同步的常见因素时滞.除了时滞对神经网络系统的影响之外,来自系统之外的噪声对系统的稳定性或者同步也有影响,为此在神经网络系统中需要考虑随机扰动对系统的影响[13].在理论研究和工程领域中,相对于一阶神经网络模型,高阶神经网络在网络收敛速度、容错能力、储存水平、逼近能力都具有较强的功能[14].然而对于高阶随机Hopfield神经网络在有限时间内的同步问题研究得较少.

    受此启发,本文研究了具有变时滞的高阶随机Hopfield神经网络在有限时间内的控制同步,在恰当的外部控制输入Ui(t)下,得到了神经网络在有限时间内同步的充分条件.

    1 模型和预备知识

    考虑如下具有变时滞的高阶随机Hopfield神经网络模型

    $ \begin{array}{l} {\rm{d}}{x_i}\left( t \right) = \left[ { - {a_i}{x_i}\left( t \right) + \sum\limits_{j = 1}^n {{b_{ij}}{f_j}\left( {{x_j}\left( t \right)} \right)} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {{c_{ijl}}{g_l}\left( {{x_l}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right){g_l}\left( {{x_l}\left( {t - {\tau _{il}}\left( t \right)} \right)} \right) + {I_i}} } } \right]{\rm{d}}t \end{array} $ (1)

    其中:iI={1,2,…,n};xi(t)表示第i个神经元在t时刻的状态变量;fj(·),gj(·)表示激活函数;bijcijl分别表示突触的一阶和二阶连接强度;ai>0表示第i个神经元的衰减率;τij(t)表示t时刻神经网络中的变时滞且满足0≤τij(t)≤τ${{\dot{\tau }}_{ij}}$(t)≤ρ<1;Ii表示外界对神经网络的输入量.

    系统(1)的初值条件为

    $ {x_i}\left( s \right) = {\varphi _i}\left( s \right),s \in \left[ { - \tau ,0} \right],i \in I $

    其中φ(s)=(φ1(s),φ2(s),…,φn(s))TC=([-τ,0],${\mathbb{R}^{n}}$)指的是把[-τ,0]映射到${\mathbb{R}^{n}}$上的所有连续函数组成的一个集合. p-范数在本文中的形式为

    $ \left\| \mathit{\boldsymbol{\varphi }} \right\| = {\left( {\mathop {\sup }\limits_{s \in \left[ { - \tau ,0} \right]} \sum\limits_{i = 1}^n {{{\left| {{\varphi _i}\left( s \right)} \right|}^p}} } \right)^{\frac{1}{p}}},\left\| \mathit{\boldsymbol{x}} \right\| = {\left( {\sum\limits_{i = 1}^n {{{\left| {{x_i}} \right|}^p}} } \right)^{\frac{1}{p}}} $ (2)

    对于系统(1),我们假设

    (H1)存在实数${{M}_{j}}>0, \text{ }{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}}_{j}}, \text{ }{{\bar{L}}_{j}}$Nj>0,使得|gj(x)|≤Mj${{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{L}}}_{j}}\le \frac{{{f}_{j}}\left( y \right)-{{f}_{j}}\left( x \right)}{y-x}\le {{{\bar{L}}}_{j}}, $ $\left| {{g}_{j}}\left( y \right)-{{g}_{j}}\left( x \right) \right|\le {{N}_{j}}\left| y-x \right|$对任意的xy$\mathbb{R}$,1≤jn成立.

    定义

    $ \begin{array}{l} {L_j} = \mathop {\max }\limits_{1 \le j \le n} \left\{ {\left| {{{\underline L }_j}} \right|,\left| {{{\bar L}_j}} \right|} \right\}\\ {H_i} = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {b_{ii}}{{\bar L}_i}\\ {b_{ii}}{\underline L _i} \end{array}&\begin{array}{l} {b_{ii}} \ge 0\\ {b_{ii}} < 0 \end{array} \end{array}} \right. \end{array} $ (3)

    把系统(1)作为主驱动系统,为了同步,引入如下的响应系统

    $ \begin{array}{l} {\rm{d}}{y_i}\left( t \right) = \left[ { - {a_i}{y_i}\left( t \right) + \sum\limits_{j = 1}^n {{b_{ij}}{f_j}\left( {{y_j}\left( t \right)} \right)} + \\ \sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {{c_{ijl}}{g_j}\left( {{y_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right){g_l}\left( {{y_l}\left( {t - {\tau _{il}}\left( t \right)} \right)} \right) + } } } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{I_i} + {U_i}\left( t \right)} \right]{\rm{d}}t + \sum\limits_{j = 1}^n {{\sigma _{ij}}\left( {t,{e_j}\left( t \right),{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right){\rm{d}}{\omega _j}\left( t \right)} \end{array} $ (4)

    其中ej(t)=yj(t)-xj(t).响应系统(4)的初值条件是yi(s)=ϕi(s),s∈[-τ,0],iI,其中:ϕ(s)=(ϕ1(s),ϕ2(s),…,ϕn(s))TC([-τ,0],${\mathbb{R}^{n}}$),Ui(t)为外部输入控制,ω(t)=[ω1(t),…,ωn(t)]T为定义在完备概率空间(ΩFFtP)上具有自然滤波{Ft}t≥0n维独立布朗运动,且有E{dω(t)}=0,E{[dω(t)]2}=dtσij(tej(t),ej(t-τij(t))dωj(t),ijI表示随机扰动.

    (H2)存在实数dij>0和pij>0,使得σij2(tuv)≤diju2+pijv2,对任意的uvRijI成立.

    为了书写方便记

    $ {\xi _{ij}} = \sum\limits_{l = 1}^n {{{\left[ {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right){M_l}} \right]}^{p{\delta _{ijp}}}}} + \frac{{p - 1}}{2}\sum\limits_{j = 1}^n {\left( {p_{ij}^{p{\vartheta _{ij\left( {p - 1} \right)}}} + p_{ij}^{p{\vartheta _{ijp}}}} \right)} $
    $ {\sigma _i} = \sum\limits_{j = 1}^n {{\xi _{ij}}} ,{\lambda _i} = {a_i} + p{H_i} + {u_i} + \sum\limits_{j = 1,i \ne j}^n {\sum\limits_{k = 1}^{p - 1} {{{\left| {{b_{ij}}} \right|}^{p{\gamma _{ijk}}}}L_j^{p{\gamma _{ijk}}}} } + \sum\limits_{j = 1,i \ne j}^n {{{\left| {{b_{ji}}} \right|}^{p{\gamma _{ijp}}}}L_i^{p{\gamma _{ijp}}}} $
    $ {\mu _i} = \sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {\sum\limits_{k = 1}^{p - 1} {{{\left[ {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right){M_l}} \right]}^{p{\delta _{ijk}}}}} } } $
    $ {\omega _i} = \frac{{\left( {p - 1} \right)}}{2}\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^{p - 2} {\left( {d_{ij}^{p{\zeta _{ijk}}} + p_{ij}^{p{\vartheta _{ijk}}}} \right)} } + \frac{{\left( {p - 1} \right)}}{2}\sum\limits_{j = 1}^n {\left( {d_{ji}^{p{\zeta _{ji\left( {p - 1} \right)}}} + p_{ji}^{p{\zeta _{ijp}}}} \right)} $

    (H3)不等式${{\lambda }_{i}}-{{\mu }_{i}}-\text{ }\frac{{{\sigma }_{i}}}{1-\rho }-{{\omega }_{i}}\ge 0$iI成立.

    本文要通过构造一个恰当的外部输入控制Ui(t),实现驱动系统(1)和响应系统(4)在有限的时间内同步

    $ \begin{array}{l} {U_i}\left( t \right) = - \frac{{{{\rm{e}}^{ - \varepsilon t}}}}{p}\left\{ {\left( {{F_i}\left( \varepsilon \right) + a} \right){{\left| {{e_i}\left( t \right)} \right|}^{1 - p + \theta }}{{\rm{e}}^{\frac{{\theta \varepsilon t}}{p}}} + } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {a{{\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon t}}} } \int_{t - {\tau _{ij}}\left( t \right)}^t {{{\rm{e}}^{\varepsilon s}}{{\left| {{e_j}\left( s \right)} \right|}^p}{\rm{d}}s} } \right)}^{\frac{\theta }{p}}}\frac{1}{{{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}}}} \right\} \end{array} $ (5)

    其中a>0是一个常数,${{F}_{i}}\left( \varepsilon \right)={{\lambda }_{i}}-{{\mu }_{i}}-\text{ }\frac{{{\sigma }_{i}}{{\text{e}}^{\tau \varepsilon }}}{1-\rho }-{{\omega }_{i}}-\varepsilon $iI.

    根据系统(1)和系统(4)知误差系统为

    $ \begin{array}{l} {\rm{d}}{e_i}\left( t \right) = \left[ { - {a_i}\left( {{y_i}\left( t \right) - {x_i}\left( t \right)} \right) + \sum\limits_{j = 1}^n {{b_{ij}}\left( {{f_j}\left( {{y_j}\left( t \right)} \right) - \\ {f_j}\left( {{x_j}\left( t \right)} \right)} \right)} + \sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {{c_{ijl}}\left( {{g_j}\left( {{y_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right) \times } \right.} } } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{g_l}\left( {{y_l}\left( {t - {\tau _{il}}\left( t \right)} \right)} \right) - {g_j}\left( {{x_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right){g_l}\left( {{x_l}\left( {t - {\tau _{il}}\left( t \right)} \right)} \right)} \right) + {U_i}\left( t \right)} \right]{\rm{d}}t + \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^n {{\sigma _{ij}}\left( {t,{e_j}\left( t \right),{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right){\rm{d}}{\omega _i}\left( t \right)} ,i \in I \end{array} $ (6)

    定义1  对于驱动系统(1)和响应系统(4),若存在一个有限的时间常数T>0,使得

    $ \mathop {\lim }\limits_{t \to T} E\left\| {\mathit{\boldsymbol{y}}\left( t \right) - \mathit{\boldsymbol{x}}\left( t \right)} \right\| = 0 $

    同时当tT时‖y(t)-x(t)‖=0,则驱动系统(1)和响应系统(4)以概率1在有限时间(t0T]内同步.

    2 辅助引理

    引理1[15]   (Harder)如果c1c2,…,cn是正实数且0<θp,则

    $ {\left( {\sum\limits_{i = 1}^n {c_i^p} } \right)^{\frac{1}{p}}} \le {\left( {\sum\limits_{i = 1}^n {c_i^\theta } } \right)^{\frac{1}{\theta }}} $

    引理2[16]  若x(t)是一个关于时间t(t≥0)的n${\rm{It\hat o}}'{\rm{s}}$过程,且满足随机微分方程

    $ {\rm{d}}\mathit{\boldsymbol{x}}\left( t \right) = {\mathit{\boldsymbol{g}}_1}\left( t \right){\rm{d}}t + {\mathit{\boldsymbol{g}}_2}\left( t \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right) $

    V(tx(t))∈C1,2(${\mathbb{R}^{+}}\times {\mathbb{R}^{n}};{\mathbb{R}^{+}}$),则V(tx(t))是一个实数值的Itô's过程,且满足随机微分方程

    $ {\rm{d}}V\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right) = LV\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right){\rm{d}}t + {V_\mathit{\boldsymbol{x}}}\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right){\mathit{\boldsymbol{g}}_2}\left( t \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right) $ (7)
    $ LV\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right) = {V_t}\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right) + {V_\mathit{\boldsymbol{x}}}\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right){\mathit{\boldsymbol{g}}_1}\left( t \right) + \frac{1}{2}{\rm{trace}}\left( {\mathit{\boldsymbol{g}}_2^{\rm{T}}{V_{\mathit{\boldsymbol{xx}}}}\left( {t,\mathit{\boldsymbol{x}}\left( t \right)} \right){\mathit{\boldsymbol{g}}_2}\left( t \right)} \right) $ (8)

    其中:${{V}_{t}}\left( t, \text{ }\mathit{\boldsymbol{x}} \right)=\frac{\partial V\left( t, \text{ }\mathit{\boldsymbol{x}} \right)}{\partial t}\text{, }{{V}_{\mathit{\boldsymbol{x}}}}\left( t, \mathit{\boldsymbol{x}} \right)=\left( \frac{\partial V\left( t, \text{ }\mathit{\boldsymbol{x}} \right)}{\partial {{x}_{1}}}, \cdots , \frac{\partial V\left( t, \text{ }\mathit{\boldsymbol{x}} \right)}{\partial {{x}_{n}}} \right), {{V}_{\mathit{\boldsymbol{xx}}}}\left( t, \mathit{\boldsymbol{x}} \right)={{\left( \frac{{{\partial }^{2}}V\left( t, \text{ }\mathit{\boldsymbol{x}} \right)}{\partial {{x}_{i}}\partial {{x}_{j}}} \right)}_{n\times n}}, $ ${{C}^{1, 2}}\left( {\mathbb{R}^{+}}\times {\mathbb{R}^{n}};{\mathbb{R}^{+}} \right)$指的是关于t一阶连续可微和关于x二阶连续可微的所有非负函数V(tx)构成的集合.

    引理3  如果存在实数a>0和0<θ<1,且正定连续函数V(t)满足LV(t)≤-aVθ(t),tt0,则对任意给定的t0V(t)满足(EV(t))1-θV1-θ(t0)-a(1-θ)(t-t0),t0tT,且当tTV(t)=0,其中$T={{t}_{0}}+\frac{{{V}^{1-\theta }}\left( {{t}_{0}} \right)}{a(1-\theta )}$.

      由引理2可知

    $ V\left( t \right) = V\left( {{t_0}} \right) + \int_{{t_0}}^t {LV\left( s \right){\rm{d}}s} + \int_{{t_0}}^t {{V_x}\left( s \right){g_2}\left( s \right){\rm{d}}\omega \left( s \right)} $

    对其两边取数学期望则有

    $ EV\left( t \right) = EV\left( {{t_0}} \right) + \int_{{t_0}}^t {ELV\left( s \right){\rm{d}}s} \le EV\left( {{t_0}} \right) - a\int_{{t_0}}^t {E{V^\theta }\left( s \right){\rm{d}}s} $

    D+EV(t)≤-aEVθ(t).由数学期望Jensen不等式EVθ(t)≥(EV(t))θ,有D+EV(t)≤-a(EV(t))θ,即有(EV(t))1-θV1-θ(t0)-a(1-θ)(t-t0),t0tT,且当tTV(t)=0,其中$T={{t}_{0}}+\frac{{{V}^{1-\theta }}\left( {{t}_{0}} \right)}{a(1-\theta )}$.

    3 主要结果

    定理1  如果(H1)-(H3)成立,则驱动系统(1)和响应系统(4)在恰当的外部输入控制Ui(t)下,在有限时间(t0T]内同步,

    $ T = {t_0} + \frac{{{V^{1 - \theta }}\left( {{t_0}} \right)}}{{a\left( {1 - \theta } \right)}} $

      构造如下形式的Lyapunov泛函

    $ V\left( t \right) = \sum\limits_{i = 1}^n {{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p}} + \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}{{\rm{e}}^{\varepsilon \tau }}}}{{1 - \rho }}\int_{t - {\tau _{ij}}\left( t \right)}^t {{{\rm{e}}^{\varepsilon s}}{{\left| {{e_j}\left( t \right)} \right|}^p}{\rm{d}}s} } } $ (9)

    结合误差系统(6),假设(H1)-(H2)及引理2中的(8)式可以得到下面的式子

    $ \begin{array}{l} LV\left( t \right) = \sum\limits_{i = 1}^n {\left\{ {\varepsilon {{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p} + p{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}\left[ {\left[ { - {a_i}\left( {{y_i}\left( t \right) - {x_i}\left( t \right)} \right) + \\ \sum\limits_{j = 1}^n {{b_{ij}}\left( {{f_j}\left( {{y_j}\left( t \right)} \right) - {f_j}\left( {{x_j}\left( t \right)} \right)} \right)} + } \right.} \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {{c_{ijl}}\left( {{g_j}\left( {{y_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right){g_l}\left( {{y_l}\left( {t - {\tau _{il}}\left( t \right)} \right)} \right) - \\ {g_j}\left( {{x_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right){g_l}\left( {{x_l}\left( {t - {\tau _{il}}\left( t \right)} \right)} \right) + } \right.} } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{U_i}\left( t \right)} \right] + \sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}{{\rm{e}}^{\varepsilon \tau }}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon \tau }}{{\left| {{e_j}\left( t \right)} \right|}^p}} - \sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}{{\rm{e}}^{\varepsilon \tau }}\left( {1 - {{\dot \tau }_{ij}}\left( t \right)} \right)}}{{1 - \rho }}{{\rm{e}}^{\varepsilon \left( {t - {\tau _{ij}}\left( t \right)} \right)}}{{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^p}} } \right\} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{{\rm{e}}^{\varepsilon \tau }}p\left( {p - 1} \right){{\left| {{e_i}\left( t \right)} \right|}^{p - 2}}\sigma _{ij}^2\left( {t,{e_j}\left( t \right),{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right)} } \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{i = 1}^n {\left\{ {\varepsilon {{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p} + {a_i}p{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p} + {H_i}p{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p} + \sum\limits_{j = 1,i \ne j}^n {p{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}} \times } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left| {{b_{ij}}} \right|{L_j}\left| {{e_j}\left( t \right)} \right| + \sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {p{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}\left[ {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right){M_l}} \right]} } \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right| + \sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}{{\rm{e}}^{\varepsilon \tau }}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon \tau }}{{\left| {{e_j}\left( t \right)} \right|}^p}} - \sum\limits_{j = 1}^n {{\xi _{ij}}{{\rm{e}}^{\varepsilon \tau }}{{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^p}} } \right\} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{{\rm{e}}^{\varepsilon t}}p\left( {p - 1} \right){{\left| {{e_i}\left( t \right)} \right|}^{p - 2}}\left( {{d_{ij}}{{\left| {{e_j}\left( t \right)} \right|}^2} + {p_{ij}}{{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^2}} \right)} } \end{array} $

    对于实数ai>0(1≤ip),有不等式$p\prod\limits_{i=1}^{p}{{{a}_{i}}\le }\text{ }\sum\limits_{i=1}^{p}{a_{i}^{p}}$成立,若存在正实数γijkδijkζijk$\vartheta $ijk且满足

    $ \begin{array}{*{20}{c}} {\sum\limits_{k = 1}^p {{\gamma _{ijk}}} = 1}&{\sum\limits_{k = 1}^p {{\delta _{ijk}}} = 1}&{\sum\limits_{k = 1}^p {{\zeta _{ijk}}} = 1}&{\sum\limits_{k = 1}^p {{\vartheta _{ijk}}} = 1} \end{array} $

    $ \begin{array}{l} \sum\limits_{j = 1,i \ne j}^n {\left| {{b_{ij}}} \right|{L_j}\left| {{e_j}\left( t \right)} \right|p{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}} \le \\ \sum\limits_{j = 1,i \ne j}^n {\sum\limits_{k = 1}^{p - 1} {{{\left| {{b_{ij}}} \right|}^{p{\gamma _{ijk}}}}L_j^{p{\gamma _{ijk}}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } + \sum\limits_{j = 1,i \ne j}^n {{{\left| {{b_{ij}}} \right|}^{p{\gamma _{ijp}}}}L_j^{p{\gamma _{ijp}}}{{\left| {{e_j}\left( t \right)} \right|}^p}} \end{array} $ (10)
    $ \begin{array}{l} \sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right)p{M_l}{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right.} \right|} } \le \\ \sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {\sum\limits_{k = 1}^{p - 1} {{{\left[ {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right){M_l}} \right]}^{p{\delta _{ijk}}}}{{\left| {{e_j}\left( t \right)} \right|}^p}} } } + \\ \sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {{{\left[ {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right){M_l}} \right]}^{p{\delta _{ijp}}}}{{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^p}} } \end{array} $ (11)
    $ \begin{array}{l} \sum\limits_{j = 1}^n {p{{\left| {{e_i}\left( t \right)} \right|}^{p - 2}}{d_{ij}}{{\left| {{e_j}\left( t \right)} \right|}^2}} = \\ \sum\limits_{j = 1}^n {p\left[ {\prod\limits_{k = 1}^{p - 2} {d_{ij}^{{\zeta _{ijk}}}\left| {{e_i}\left( t \right)} \right|} } \right]} \left( {d_{ij}^{{\zeta _{ij\left( {p - 1} \right)}}}\left| {{e_j}\left( t \right)} \right|} \right)\left( {d_{ij}^{{\zeta _{ijp}}}\left| {{e_j}\left( t \right)} \right|} \right) \le \\ \sum\limits_{j = 1}^n {\sum\limits_{k = 1}^{p - 2} {d_{ij}^{{\zeta _{ijk}}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } + \sum\limits_{j = 1}^n {\left( {d_{ij}^{p{\zeta _{ij\left( {p - 1} \right)}}} + d_{ij}^{p{\zeta _{ijp}}}} \right){{\left| {{e_j}\left( t \right)} \right|}^p}} \end{array} $ (12)
    $ \begin{array}{l} \sum\limits_{j = 1}^n {p{{\left| {{e_i}\left( t \right)} \right|}^{p - 2}}{p_{ij}}{{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^2}} \le \\ \sum\limits_{j = 1}^n {\sum\limits_{k = 1}^{p - 2} {p_{ij}^{p{\vartheta _{ijk}}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } + \sum\limits_{j = 1}^n {\left( {p_{ij}^{p{\vartheta _{ij\left( {p - 1} \right)}}} + p_{ij}^{p{\vartheta _{ijp}}}} \right){{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^p}} \end{array} $ (13)

    由(10)-(13)式可知

    $ \begin{array}{l} LV\left( t \right) \le \sum\limits_{i = 1}^n {\left\{ {\varepsilon {{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p} + {a_i}p{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p} + {H_i}p{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p} + {{\rm{e}}^{\varepsilon t}} \\ \sum\limits_{j = 1,i \ne j}^n {\sum\limits_{k = 1}^{p - 1} {{{\left| {{b_{ij}}} \right|}^{p{\gamma _{ijk}}}}L_j^{p{\gamma _{ijk}}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } + } \right.} \\ {{\rm{e}}^{\varepsilon t}}\sum\limits_{j = 1,i \ne j}^n {\left| {{b_{ij}}} \right|\left\{ {p{\gamma _{ijp}}} \right\}L_j^{p{\gamma _{ijp}}}{{\left| {{e_j}\left( t \right)} \right|}^p}} + {{\rm{e}}^{\varepsilon t}}\sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {\sum\limits_{k = 1}^{p - 1} {{{\left[ {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right){M_l}} \right]}^{p{\delta _{ijk}}}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } } + \\ {{\rm{e}}^{\varepsilon t}}\sum\limits_{j = 1}^n {\sum\limits_{l = 1}^n {{{\left[ {\left( {\left| {{c_{ijl}}} \right| + \left| {{c_{ilj}}} \right|} \right){M_l}} \right]}^{p{\delta _{ijp}}}}{{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^p}} } + \sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}{{\rm{e}}^{\varepsilon \tau }}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon t}}{{\left| {{e_j}\left( t \right)} \right|}^p}} - \sum\limits_{j = 1}^n {{\xi _{ij}}{{\rm{e}}^{\varepsilon t}}} \times \\ {\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|^p} - p{{\rm{e}}^{\varepsilon t}}{\left| {{e_i}\left( t \right)} \right|^{p - 1}}\frac{{{{\rm{e}}^{ - \varepsilon t}}}}{p}\left[ {\left( {{F_i}\left( \varepsilon \right) + a} \right){{\left| {{e_i}\left( t \right)} \right|}^{1 - p + \theta }}{{\rm{e}}^{\frac{{\theta \varepsilon t}}{p}}} + } \right.\\ \left. {a{{\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon t}}\int_{t - {\tau _{ij}}\left( t \right)}^t {{{\rm{e}}^{\varepsilon s}}{{\left| {{e_j}\left( t \right)} \right|}^p}{\rm{d}}s} } } } \right)}^{\frac{\theta }{p}}}\frac{1}{{{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}}}} \right] + \frac{{\left( {p - 1} \right)}}{2}{{\rm{e}}^{\varepsilon t}}\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^{p - 2} {{{\left| {{e_i}\left( t \right)} \right|}^p}} } \times \\ d_{ij}^{p{\zeta _{ijk}}} + \frac{{\left( {p - 1} \right)}}{2}{{\rm{e}}^{\varepsilon t}}\sum\limits_{j = 1}^n {{{\left| {{e_j}\left( t \right)} \right|}^p}\left( {d_{ij}^{p{\zeta _{ij\left( {p - 1} \right)}}} + d_{ij}^{p{\zeta _{ijp}}}} \right)} + \frac{{\left( {p - 1} \right)}}{2}{{\rm{e}}^{\varepsilon t}}\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^{p - 2} {{{\left| {{e_i}\left( t \right)} \right|}^p}p_{ij}^{p{\vartheta _{ijk}}}} } + \\ \frac{{\left( {p - 1} \right)}}{2}{{\rm{e}}^{\varepsilon t}}\sum\limits_{j - 1}^n {{{\left| {{e_j}\left( {t - {\tau _{ij}}\left( t \right)} \right)} \right|}^p}\left( {p_{ij}^{p{\vartheta _{ij\left( {p - 1} \right)}}} + p_{ij}^{p{\vartheta _{ijp}}}} \right)} \end{array} $ (14)

    定义函数族

    $ {F_i}\left( {{\varepsilon _i}} \right) = {\lambda _i} - {\mu _i} - \frac{{{\sigma _i}{{\rm{e}}^{\tau {\varepsilon _i}}}}}{{1 - \rho }} - {\omega _i} - {\varepsilon _i},i \in I $
    $ {{\dot F}_i}\left( {{\varepsilon _i}} \right) = - \frac{{{\sigma _i}}}{{1 - \rho }}\tau {{\rm{e}}^{{\varepsilon _i}\tau }} - 1 < 0 $

    由假设(H3)知Fi(0)≥0,iI.因此,εi是方程${{F}_{i}}\left( {{\varepsilon }_{i}} \right)={{\lambda }_{i}}-{{\mu }_{i}}-\text{ }\frac{{{\sigma }_{i}}{{\text{e}}^{\tau {{\varepsilon }_{i}}}}}{1-\rho }-{{\omega }_{i}}-{{\varepsilon }_{i}}$iI的唯一正解.记

    $ \begin{array}{*{20}{c}} {\varepsilon = \mathop {\min }\limits_{1 \le i \le n} \left\{ {{\varepsilon _i}} \right\}}&{{F_i}\left( \varepsilon \right) = {\lambda _i} - {\mu _i} - \frac{{{\sigma _i}{{\rm{e}}^{\tau \varepsilon }}}}{{1 - \rho }} - {\omega _i} - \varepsilon \ge 0} \end{array} $

    成立.

    由(14)式和Fi(ε)≥0可知

    $ LV\left( t \right) \le - a\left( {\sum\limits_{i = 1}^n {{{\rm{e}}^{\frac{{\varepsilon t\theta }}{p}}}{{\left| {{e_i}\left( t \right)} \right|}^\theta }} + {{\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon t}}\int_{t - {\tau _{ij}}\left( t \right)}^t {{{\rm{e}}^{\varepsilon s}}{{\left| {{e_j}\left( s \right)} \right|}^p}{\rm{d}}s} } } } \right)}^{\frac{\theta }{p}}}} \right) $ (15)

    根据引理(1)可知

    $ \sum\limits_{i = 1}^n {{{\left| {{e_i}\left( t \right)} \right|}^\theta }} \ge {\left( {\sum\limits_{i = 1}^n {{{\left| {{e_i}\left( t \right)} \right|}^p}} } \right)^{\frac{\theta }{p}}} $ (16)

    则由(15)和(16)式可知

    $ LV\left( t \right) \le - a\left[ {{{\left( {\sum\limits_{i = 1}^n {{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } \right)}^{\frac{\theta }{p}}} + {{\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon t}}\int_{t - {\tau _{ij}}\left( t \right)}^t {{{\rm{e}}^{\varepsilon s}}{{\left| {{e_j}\left( s \right)} \right|}^p}{\rm{d}}s} } } } \right)}^{\frac{\theta }{p}}}} \right] $ (17)

    a≥0,b≥0且0<r<1时,不等式(a+b)rar+br成立,则有

    $ \begin{array}{l} {\left( {\sum\limits_{i = 1}^n {{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } \right)^{\frac{\theta }{p}}} + {\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon t}}\int_{t - {\tau _{ij}}\left( t \right)}^t {{{\rm{e}}^{\varepsilon s}}{{\left| {{e_j}\left( s \right)} \right|}^p}{\rm{d}}s} } } } \right)^{\frac{\theta }{p}}} \ge \\ {\left( {\sum\limits_{i = 1}^n {{{\rm{e}}^{\varepsilon t}}{{\left| {{e_i}\left( t \right)} \right|}^p}} + \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\frac{{{\xi _{ij}}}}{{1 - \rho }}{{\rm{e}}^{\varepsilon t}}\int_{t - {\tau _{ij}}\left( t \right)}^t {{{\rm{e}}^{\varepsilon s}}{{\left| {{e_j}\left( s \right)} \right|}^p}{\rm{d}}s} } } } \right)^{\frac{\theta }{p}}} \end{array} $ (18)

    成立.

    由(17)式和(18)式可知

    $ LV\left( t \right) \le - a{\left( {V\left( t \right)} \right)^{\frac{\theta }{p}}} $

    成立.

    由引理(4)可知

    $ {\left( {EV\left( t \right)} \right)^{1 - \frac{\theta }{p}}} \le {V^{1 - \frac{\theta }{p}}}\left( {{t_0}} \right) - a\left( {1 - \frac{\theta }{p}} \right)\left( {t - {t_0}} \right),t \in \left( {{t_0},T} \right],T = {t_0} + \frac{{{V^{1 - \frac{\theta }{p}}}\left( {{t_0}} \right)}}{{a\left( {1 - \frac{\theta }{p}} \right)}} $ (19)

    由(9)式和(19)式及数学期望的定义可知

    $ {\left( {E\sum\limits_{i = 1}^n {{{\left| {{e_i}\left( t \right)} \right|}^p}} } \right)^{1 - \frac{\theta }{p}}} \le {\left( {EV\left( t \right)} \right)^{1 - \frac{\theta }{p}}} \le {V^{1 - \frac{\theta }{p}}}\left( {{t_0}} \right) - a\left( {1 - \frac{\theta }{p}} \right)\left( {t - {t_0}} \right) $
    $ \mathop {\lim }\limits_{t \to T} \left( {{V^{1 - \frac{\theta }{p}}}\left( {{t_0}} \right) - a\left( {1 - \frac{\theta }{p}} \right)\left( {t - {t_0}} \right)} \right) = 0 $

    可知

    $ \mathop {\lim }\limits_{t \to T} E\left\| {y - x} \right\| = 0 $

    且当tT时‖y-x‖=0.说明驱动系统(1)和响应系统(4)在有限时间(t0T]内是同步的.

    4 数值例子

    下面给出一个例子,说明本文结论是正确的.在驱动系统(1)中,ijl=1,2,g1=g2=f1=f2=tanh(x),a1=a2=1,τij(t)=τil(t)= $\frac{{{\text{e}}^{t}}}{1+{{\text{e}}^{t}}}$,经过计算可知

    $ \begin{array}{*{20}{c}} {\rho = \frac{1}{4}}&{{L_j} = 1}&{{N_j} = 1}&{{M_j} = 1} \end{array} $
    $ {\left( {{b_{ij}}} \right)_{2 \times 2}} = \left( {\begin{array}{*{20}{c}} { - 1.7}&{ - 0.6}\\ {0.5}&{ - 2.5} \end{array}} \right) $
    $ \begin{array}{*{20}{c}} {{c_{111}} = 1.3}&{{c_{121}} = - 0.15}&{{c_{211}} = - 1.48}&{{c_{221}} = 0.2} \end{array} $
    $ \begin{array}{*{20}{c}} {{c_{112}} = - 1.48}&{{c_{122}} = - 0.4}&{{c_{212}} = 0.2}&{{c_{222}} = - 2} \end{array} $

    在响应系统(4)中取

    $ \begin{array}{*{20}{c}} {{{\left( {{p_{ij}}} \right)}_{2 \times 2}} = \left( {\begin{array}{*{20}{c}} {0.5}&{0.4}\\ {0.7}&{0.5} \end{array}} \right)}&{{{\left( {{d_{ij}}} \right)}_{2 \times 2}} = \left( {\begin{array}{*{20}{c}} {0.6}&{0.5}\\ {0.8}&{0.9} \end{array}} \right)} \end{array} $

    p=2时,

    $ \begin{array}{*{20}{c}} {{\lambda _1} - {\mu _1} - \frac{{{\sigma _1}}}{{1 - \rho }} - {\omega _1} = 2{H_1} + {u_1} + \left| {{b_{12}}} \right|{L_2} + \left| {{b_{21}}} \right| - \frac{5}{3}\left( {\left| {{c_{111}}} \right| + \left| {{c_{121}}} \right| + } \right.}\\ {\left. {\left| {{c_{112}}} \right| + \left| {{c_{122}}} \right|} \right) - 2\left( {{p_{12}} + {p_{11}}} \right) - \frac{1}{2}\left( {3{d_{11}} + {d_{12}} + {p_{12}} + {p_{11}} + 2{d_{21}}} \right) = 0.1 > 0} \end{array} $ (20)
    $ \begin{array}{*{20}{c}} {{\lambda _2} - {\mu _2} - \frac{{{\sigma _2}}}{{1 - \rho }} - {\omega _2} = {a_2} + 2{H_2} + {u_2} + \left| {{b_{21}}} \right|{L_1} + \left| {{b_{12}}} \right| - \frac{5}{3}\left( {\left| {{c_{211}}} \right| + \left| {{c_{221}}} \right| + } \right.}\\ {\left. {\left| {{c_{212}}} \right| + \left| {{c_{222}}} \right|} \right) - 2\left( {{p_{21}} + {p_{22}}} \right) - \frac{1}{2}\left( {3{d_{22}} + {d_{21}} + 2{d_{12}} + {p_{21}} + {p_{22}}} \right) = 0.133 > 0} \end{array} $ (21)

    由(20)式和(21)式可知,驱动系统(1)和响应系统(4)在恰当的外部输入控制Ui(t)(i=1,2)下在有限时间内同步.

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    Finite-Time Control Synchronization for High-Order Stochastic Hopfield Neural Networks with Time-Varying Delays
    PU Hao1, JIANG Hai-Jun2, HU Cheng2, RAN Jie1, ZHANG Zhuan-zhou1     
    1. School of Mathematics, Zunyi Normal College, Zunyi Guizhou 563002, China;
    2. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
    Abstract: In this paper, we study the finite-time control synchronization for high-order stochastic Hopfield neural networks with time-varying delays. Through the Lyapunov function, the finite time stability theory, the theory of stochastic differential equation and some inequality methods, some new and useful sufficient conditions on the in finite-time synchronization are obtained based on p-norm. The conclusion of this paper is the generalization of the previous related conclusions.
    Key words: high-order Hopfield neural network    stochastic perturbation term    time-varying delay    finite time synchronization    p-norm    
    X