西南大学学报 (自然科学版)  2019, Vol. 41 Issue (10): 56-61.  DOI: 10.13718/j.cnki.xdzk.2019.10.008
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  • 不动点方法与概率时滞脉冲模糊系统的稳定性    [PDF全文]
    黄家琳1, 李兴贵2     
    1. 四川三河职业学院 基础部, 四川 泸州 646200;
    2. 成都师范学院 数学学院, 成都 611130
    摘要:通过在完备距离空间上定义压缩映射,利用T-S模糊规则、概率时滞的相关性质和压缩映像原理导出了一类T-S模糊概率时滞脉冲双向联想记忆神经网络的稳定性的代数判据.特点在于,导出系统解的存在性的同时给出了该解的稳定性结论.最后,数值实例证实了所述方法的有效性.
    关键词双向联想记忆神经网络    概率时滞    不动点定理    

    以前很多文献用李雅普诺夫函数法导出神经网络的稳定性[1-9],然而每一种方法有其局限性,不动点方法是李雅普诺夫函数法的替代方法之一[10-16].本文作者考虑用压缩映射原理结合线性矩阵不等式方法给出一类T-S模糊概率时滞脉冲双向联想记忆神经网络的稳定性的代数判据(LMI).特别地,LMI判据适合于计算机Matlab LMI工具箱编程运算,符合实际工程中大型计算的要求.由于方法和条件的不同,本文更新了相关文献[14-16]的结果.

    1 预备知识

    本文考虑下述由IF-THEN规则所描述的T-S模糊双向联想记忆神经网络系统模型:

    模糊规则j  令t∈[0,+∞),ttkk=1,2,…,若ω1(t)=μj1,…,ωm(t)=μjm,则

    $ \left\{ {\begin{array}{*{20}{l}} {\frac{{{\rm{d}}x(t)}}{{{\rm{d}}t}} = - \mathit{\boldsymbol{A}}x(t) + {\mathit{\boldsymbol{C}}_j}f(y(t - \tau (t)))}\\ {\frac{{{\rm{d}}y(t)}}{{{\rm{d}}t}} = - \mathit{\boldsymbol{B}}y(t) + {\mathit{\boldsymbol{D}}_j}g(x(t - h(t)))}\\ {x\left( {t_k^ + } \right) - x\left( {{t_k}} \right) = \rho \left( {x\left( {{t_k}} \right)} \right),y\left( {t_k^ + } \right) - y\left( {{t_k}} \right) = \rho \left( {y\left( {{t_k}} \right)} \right)}\\ {x(s) = \xi (s),y(s) = \eta (s)\quad s \in [ - \tau ,0]} \end{array}} \right. $ (1)

    其中

    $ x\left( t \right) = {\left( {{x_1}(t),{x_2}(t), \cdots ,{x_n}(t)} \right)^{\text{T}}} \in {\mathbb{R}^n} $
    $ y\left( t \right) = {\left( {{y_1}(t),{y_2}(t), \cdots ,{y_n}(t)} \right)^{\text{T}}} \in {\mathbb{R}^n} $
    $ \left. {\xi (s),\eta (s) \in C\left( {[ - \tau ,0],{\mathbb{R}^n}} \right]} \right) $

    {μjk}是模糊集(j=1,2,…,Jk=1,2,…,m),ωk(t)是前件变量,m为前件变量的个数,J是IF-THEN规则的个数.激活函数

    $ f(x(t - \tau (t))) = {\left( {{f_1}\left( {{x_1}(t - \tau (t))} \right),{f_2}\left( {{x_2}(t - \tau (t))} \right), \cdots ,{f_n}\left( {{x_n}(t - \tau (t))} \right)} \right)^{\text{T}}} \in {\mathbb{R}^n} $
    $ g(x(t - h(t))) = {\left( {{g_1}\left( {{x_1}(t - h(t))} \right),{g_2}\left( {{x_2}(t - h(t))} \right), \cdots ,{g_n}\left( {{x_n}(t - h(t))} \right)} \right)^{\text{T}}} \in {\mathbb{R}^n} $

    脉冲函数

    $ \rho (x(t)) = {\left( {{\rho _1}\left( {{x_1}(t)} \right),{\rho _2}\left( {{x_2}(t)} \right), \cdots ,{\rho _n}\left( {{x_n}(t)} \right)} \right)^{\text{T}}} \in {\mathbb{R}^n} $

    时滞0≤τ(t),h(t)≤τ,∀i=1,2,…,n.我们简记时滞神经元之间相互联络的权系数矩阵CjDjn维方阵.脉冲时刻tk(k=1,2,…)满足0<t1t2<…,$ \mathop {\lim}\limits_{k \to \infty} {t_k}=\infty $. x(tk+)和x(tk-)分别表示x(t)在tk时刻的右极限和左极限.假设x(tk-)=x(tk)(∀k=1,2,…).令t≥0,ttkk=1,2,…,由单点模糊化、乘积推理和平均加权反模糊化得到模糊系统的整个状态方程为

    $ \left\{ {\begin{array}{*{20}{l}} {\frac{{{\text{d}}x(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{A}}x(t) + \sum\limits_{j = 1}^J {{\rho _j}\left( {\omega \left( t \right)} \right){\mathit{\boldsymbol{C}}_j}f(y(t - \tau (t)))} } \\ {\frac{{{\text{d}}y(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{B}}y(t) + \sum\limits_{j = 1}^J {{\rho _j}\left( {\omega \left( t \right)} \right){\mathit{\boldsymbol{D}}_j}g(x(t - h(t)))} } \\ {x\left( {t_k^ + } \right) - x\left( {{t_k}} \right) = \rho \left( {x\left( {{t_k}} \right)} \right),y\left( {t_k^ + } \right) - y\left( {{t_k}} \right) = \rho \left( {y\left( {{t_k}} \right)} \right)} \\ {x(s) = \xi (s),y(s) = \eta (s)\quad s \in [ - \tau ,0]} \end{array}} \right. $ (2)

    其中

    $ \begin{array}{*{20}{c}} {\omega (t) = \left( {{\omega _1}(t), \cdots ,{\omega _m}(t)} \right)}&{{\rho _j}(\omega (t)) = \frac{{{\mathit{\Upsilon } _j}(\omega (t))}}{{\sum\limits_{i = 1}^J {{\mathit{\Upsilon } _i}} (\omega (t))}}} \end{array} $

    Υj(ω(t))为相应于规则j的隶属度函数,且$ \sum\limits_{j=1}^J {{\rho _j}\left( {\omega \left( t \right)} \right)} =1$ρj(ω(t))≥0.

    由于实际系统中的时滞达到较大值的概率很小,于是我们需要考虑概率时滞

    $ \mathbb{P}\left( {0 \leqslant \tau \left( t \right) \leqslant {\tau _1}} \right) = {c_0}\quad \mathbb{P}\left( {0 \leqslant \tau \left( t \right) \leqslant {\tau _2}} \right) = 1 - {c_0} $

    设实数c0≤1,定义随机变量

    $ \mathscr{C}\left( t \right) = \left\{ {\begin{array}{*{20}{l}} 1&{0 \leqslant \tau (t) \leqslant {\tau _1}} \\ 0&{{\tau _1} < \tau (t) \leqslant {\tau _2}} \end{array}} \right. $

    t≥0,ttkk=1,2,…,考虑概率时滞模糊系统

    $ \left\{ \begin{array}{l} \frac{{{\text{d}}x(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{A}}x(t) + \sum\limits_{i = 1}^j {{\rho _j}} (\omega (t))\left[ {{c_0}{\mathit{\boldsymbol{C}}_j}f\left( {y\left( {t - {\tau _1}(t),x} \right)} \right) + \left( {1 - {c_0}} \right){\mathit{\boldsymbol{C}}_j}f\left( {y\left( {t - {\tau _2}(t),x} \right)} \right) + } \right. \hfill \\ \;\;\;\;\;\;\;\;\left. {\left( {\mathscr{C} - {c_0}} \right)\left( {{\mathit{\boldsymbol{C}}_j}f\left( {y\left( {t - {\tau _1}(t),x} \right)} \right) - {\mathit{\boldsymbol{C}}_j}f\left( {y\left( {t - {\tau _2}(t),x} \right)} \right)} \right)} \right] \hfill \\ \frac{{{\text{d}}y(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{B}}y(t) + \sum\limits_{j = 1}^J {{\rho _j}} (\omega (t))\left[ {{c_0}{\mathit{\boldsymbol{D}}_j}g\left( {x\left( {t - {h_1}(t),x} \right)} \right) + \left( {1 - {c_0}} \right){\mathit{\boldsymbol{D}}_j}g\left( {x\left( {t - {h_2}(t),x} \right)} \right)} \right. +\hfill \\ \;\;\;\;\;\;\;\;\left( {\mathscr{C} - {c_0}} \right)\left( {{\mathit{\boldsymbol{D}}_j}g\left( {x\left( {t - {h_1}(t),x} \right)} \right) - {\mathit{\boldsymbol{D}}_j}g\left( {x\left( {t - {h_2}(t),x} \right)} \right)} \right] \hfill \\ x\left( {t_k^ + } \right) - x\left( {{t_k}} \right) = \rho \left( {x\left( {{t_k}} \right)} \right),y\left( {t_k^ + } \right) - y\left( {{t_k}} \right) = \rho \left( {y\left( {{t_k}} \right)} \right) \hfill \\ x\left( s \right) = \xi \left( s \right),y\left( s \right) = \eta \left( s \right)\quad s \in [ - \tau ,0] \hfill \\ \end{array} \right. $ (3)

    本文假设:f(0)=g(0)=ρ(0)=0∈$ \mathbb{R}^n$;对角矩阵A=diag(a1a2,…,an),B=diag(b1b2,…,bn)正定;对角矩阵FGH分别是向量函数fgρ的利普希茨常数矩阵.

    2 主要结论与证明

    定理1  假设存在常数0<λ<1,使得

    $ \left\{ {\begin{array}{*{20}{l}} {\sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{C}}_j}} \right|\mathit{\boldsymbol{F}} + \frac{1}{\delta }\mathit{\boldsymbol{H}} + \mathit{\boldsymbol{AH}} - \lambda \mathit{\boldsymbol{A}} < 0} } \\ {\sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{D}}_j}} \right|\mathit{\boldsymbol{G}} + \frac{1}{\delta }\mathit{\boldsymbol{H}} + \mathit{\boldsymbol{BH}} - \lambda \mathit{\boldsymbol{B}} < 0} } \end{array}} \right. $ (4)

    则脉冲时滞系统(2)是指数稳定的,其中δ=$ \mathop {{\rm{inf}}}\limits_{k = 1, 2, \cdots} \left( {{t_{k + 1}} - {t_k}} \right) > 0$.

      首先定义空间Ω=Ω1×Ω2.

    Ωi(i=1,2)是这样的函数空间,其函数qi(t):[-τ,∞)→$ \mathbb{R}^n$满足以下4条:

    (a) qi(t)连续于t∈[0,+∞)\{tk}k=1

    (b) q1(t)=ξ(t),q2(t)=η(t)(∀t∈[-τ,0]);

    (c) $ \mathop {\lim}\limits_{t \to t_k^ -} {q_i}\left( t \right) = {q_i}\left( {{t_k}} \right)$,且$ \mathop {\lim}\limits_{t \to t_k^ +} {q_i}\left( t \right)$存在(∀k=1,2,…);

    (d) 当t→∞时eγtqi(t)→0,其中γ>0是常数,满足γ<min{λminAλminB}.

    则易证Ω是下述度量下的完备空间:

    $ {\rm dist}\left( {\bar q,\tilde q} \right) = \mathop {\max }\limits_{i = 1,2, \cdots ,2n - 1,2n} \left( {\mathop {\sup }\limits_{t \geqslant - \tau } \left| {{{\bar q}^{(i)}}\left( t \right) - {{\tilde q}^{(i)}}\left( t \right)} \right|} \right) $

    其中

    $ \bar q = \bar q(t) = \left( {\begin{array}{*{20}{c}} {{{\bar q}_1}\left( t \right)} \\ {{{\bar q}_2}\left( t \right)} \end{array}} \right) = {\left( {{{\bar q}^{(1)}}(t),{{\bar q}^{(2)}}(t), \cdots ,{{\bar q}^{(2n)}}(t)} \right)^{\text{T}}} \in \mathit{\Omega } $
    $ \tilde q = \tilde q(t) = \left( {\begin{array}{*{20}{c}} {{{\tilde q}_1}(t)} \\ {{{\tilde q}_2}(t)} \end{array}} \right) = {\left( {{{\tilde q}^{(1)}}(t), \cdots ,{{\tilde q}^{(2n)}}(t)} \right)^{\text{T}}} \in \mathit{\Omega } $

    这里qiΩi$ {\tilde q_i}$Ωii=1,2.

    现定义压缩映射PΩΩ,这需3步来实现.

    第一步,关于系统(3),我们可以构造如下映射:

    $ \left\{ \begin{array}{l} P\left( {x\left( t \right)} \right) = {{\text{e}}^{ - \mathit{\boldsymbol{A}}t}}\xi \left( 0 \right) + {{\text{e}}^{ - \mathit{\boldsymbol{A}}t}} \\ \left[ {\int_0^t {{{\text{e}}^{\mathit{\boldsymbol{As}}}}}\sum\limits_{j = 1}^J {{\rho _j}} (\omega (t)){\mathit{\boldsymbol{C}}_j} \left( {\mathscr{C} f\left( {y\left( {s - {\tau _1}(s)} \right)} \right) \\ + \left( {1 - \mathscr{C}} \right)f\left( {y\left( {s - {\tau _2}(s)} \right)} \right)} \right){\text{d}}s + } \right. \\ \;\;\;\;\;\;\;\;\;\;\left. {\sum\limits_{0 < {t_k} < t} {{{\text{e}}^{\mathit{\boldsymbol{A}}{t_k}}}} \rho \left( {{x_{{t_k}}}} \right)} \right]\;\;\;\;\;\;t \geqslant 0 \\ P\left( {y\left( t \right)} \right) = {{\text{e}}^{ - \mathit{\boldsymbol{B}}t}}\eta \left( 0 \right) + {{\text{e}}^{ - \mathit{\boldsymbol{B}}t}} \\ \left[ {\int_0^t {{{\text{e}}^{\mathit{\boldsymbol{B}}s}}}\sum\limits_{j = 1}^J {{\rho _j}} \left( {\omega \left( t \right)} \right){\mathit{\boldsymbol{D}}_j}\left( {\mathscr{C} g\left( {x\left( {s - {h_1}(s)} \right)} \right) \\ + \left( {1 - \mathscr{C}} \right)g\left( {x\left( {s - {h_2}(s)} \right)} \right)} \right){\text{d}}s + } \right. \\ \;\;\;\;\;\;\;\;\;\;\left. {\sum\limits_{0 < {t_k} < t} {{{\text{e}}^{\mathit{\boldsymbol{B}}{t_k}}}} \rho \left( {{y_{{t_k}}}} \right)} \right]\;\;\;\;\;\;t \geqslant 0 \\ P\left( {\begin{array}{*{20}{l}} {x(t)} \\ {y(t)} \end{array}} \right) = \left( {\begin{array}{*{20}{l}} {\xi (t)} \\ {\eta (t)} \end{array}} \right)\quad \forall t \in [ - \tau ,0] \\ \end{array} \right. $ (5)

    我们不难证明PΩ上的压缩映射.

    第二步,不难证明$ P\left( \begin{array}{l} x\left( t \right)\\ y\left( t \right) \end{array} \right) \in \mathit{\Omega }, \forall \left( \begin{array}{l} x\left( t \right)\\ y\left( t \right) \end{array} \right) \in \mathit{\Omega }$.换而言之,$ P\left( \begin{array}{l} x\left( t \right)\\ y\left( t \right) \end{array} \right)$满足条件(a)-(d).

    第三步,证明(5)式定义的P是压缩映射.

    对任给$ \left( \begin{array}{l} x\left( t \right)\\ y\left( t \right) \end{array} \right), \left( \begin{array}{l} \bar x\left( t \right)\\ \bar y\left( t \right) \end{array} \right) \in \mathit{\Omega }$,我们有

    $ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left| {P\left( {\begin{array}{*{20}{l}} {x(t)} \\ {y(t)} \end{array}} \right) - P\left( {\begin{array}{*{20}{l}} {\bar x(t)} \\ {\bar y(t)} \end{array}} \right)} \right| \leqslant \hfill \\ \left( {\begin{array}{*{20}{c}} {{{\text{e}}^{ - \mathit{\boldsymbol{A}}t}}\int_0^t {{{\text{e}}^{\mathit{\boldsymbol{A}}s}}} \sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{C}}_j}} \right|} \left( {\mathscr{C}\left| {f\left( {y\left( {s - {\tau _1}(s)} \right)} \right) - f\left( {\bar y\left( {s - {\tau _1}(s)} \right)} \right)} \right| +\\ (1 - \mathscr{C})\left| {f\left( {y\left( {s - {\tau _2}(s)} \right)} \right) - f\left( {\bar y\left( {s - {\tau _2}(s)} \right)} \right)} \right|} \right){\text{d}}s} \\ {{{\text{e}}^{ - \mathit{\boldsymbol{B}}t}}\int_0^t {{{\text{e}}^{\mathit{\boldsymbol{B}}s}}} \sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{D}}_j}} \right|} \left( {\mathscr{C}|g\left( {x\left( {s - {h_1}(s)} \right)} \right) - g\left( {\bar x\left( {s - {h_1}(s)} \right)} \right)| + \\ (1 - \mathscr{C})|g\left( {x\left( {s - {h_2}(s)} \right)} \right) - g\left( {\bar x\left( {s - {h_2}(s)} \right)} \right)|} \right){\text{d}}s} \end{array}} \right) + \hfill \\ \left( {\begin{array}{*{20}{l}} {{{\text{e}}^{ - \mathit{\boldsymbol{A}}t}}\sum\limits_{0 < {t_k} < t} {{{\text{e}}^{\mathit{\boldsymbol{A}}{t_k}}}\left| {\rho \left( {{x_{{t_k}}}} \right) - \rho \left( {{{\bar x}_{{t_k}}}} \right)} \right|} } \\ {{{\text{e}}^{ - \mathit{\boldsymbol{B}}t}}\sum\limits_{0 < {t_k} < t} {{{\text{e}}^{\mathit{\boldsymbol{B}}{t_k}}}\left| {\rho \left( {{y_{{t_k}}}} \right) - \rho \left( {{{\bar y}_{{t_k}}}} \right)} \right|} } \end{array}} \right) \leqslant \hfill \\ \left[ {\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}^{ - 1}}\sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{C}}_j}} \right|} F\mu } \\ {{\mathit{\boldsymbol{B}}^{ - 1}}\sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{D}}_j}} \right|} G\mu } \end{array}} \right) + \frac{1}{\delta }\left( {\begin{array}{*{20}{c}} {{{\text{e}}^{ - \mathit{\boldsymbol{A}}t}}\left( {\int_0^t {{{\text{e}}^{\mathit{\boldsymbol{As}}}}} {\text{d}}s + \delta {{\text{e}}^{\mathit{\boldsymbol{At}}}}} \right){\mathit{\boldsymbol{H}}_\mu }} \\ {{{\text{e}}^{ - \mathit{\boldsymbol{B}}t}}\left( {\int_0^t {{{\text{e}}^{\mathit{\boldsymbol{B}}s}}} {\text{d}}s + \delta {{\text{e}}^{\mathit{\boldsymbol{B}}t}}} \right){\mathit{\boldsymbol{H}}_\mu }} \end{array}} \right)} \right]\\ {\rm dist}\left( {\left( {\begin{array}{*{20}{c}} {x(t)} \\ {y(t)} \end{array}} \right),\left( {\begin{array}{*{20}{c}} {\bar x(t)} \\ {\bar y(t)} \end{array}} \right)} \right) \leqslant \hfill \\ \left[ {\begin{array}{*{20}{c}} {\left( {{\mathit{\boldsymbol{A}}^{ - 1}}\sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{C}}_j}} \right|} \mathit{\boldsymbol{F}} + \frac{1}{\delta }{\mathit{\boldsymbol{A}}^{ - 1}}\mathit{\boldsymbol{H}} + \mathit{\boldsymbol{H}}} \right)\mu } \\ {\left( {{\mathit{\boldsymbol{B}}^{ - 1}}\sum\limits_{j = 1}^J {\left| {{\mathit{\boldsymbol{D}}_j}} \right|} \mathit{\boldsymbol{G}} + \frac{1}{\delta }{\mathit{\boldsymbol{B}}^{ - 1}}\mathit{\boldsymbol{H}} + \mathit{\boldsymbol{H}}} \right)\mu } \end{array}} \right] {\rm dist} \left( {\left( {\begin{array}{*{20}{c}} {x(t)} \\ {y(t)} \end{array}} \right),\left( {\begin{array}{*{20}{c}} {\bar x(t)} \\ {\bar y(t)} \end{array}} \right)} \right) < \hfill \\ \lambda \left( {\begin{array}{*{20}{l}} \mu \\ \mu \end{array}} \right) {\rm dist} \left( {\left( {\begin{array}{*{20}{l}} {x(t)} \\ {y(t)} \end{array}} \right),\left( {\begin{array}{*{20}{l}} {x(t)} \\ {\bar y(t)} \end{array}} \right)} \right) \hfill \\ \end{array} $

    因此

    $ {\rm dist}\left( {P\left( {\begin{array}{*{20}{l}} {x(t)} \\ {y(t)} \end{array}} \right),P\left( {\begin{array}{*{20}{l}} {\bar x(t)} \\ {\bar y(t)} \end{array}} \right)} \right) \leqslant \lambda {\rm dist}\left( {\left( {\begin{array}{*{20}{c}} {x(t)} \\ {y(t)} \end{array}} \right),\left( {\begin{array}{*{20}{c}} {\bar x(t)} \\ {\bar y(t)} \end{array}} \right)} \right) $

    所以PΩΩ是压缩映射,从而存在PΩ上的不动点$ \left( \begin{array}{l} x\left( t \right)\\ y\left( t \right) \end{array} \right)$.即$ \left( \begin{array}{l} x\left( t \right)\\ y\left( t \right) \end{array} \right)$是模糊时滞脉冲系统(2)的解,满足$ {{\rm{e}}^{\gamma t}}\left\| {\left( \begin{array}{l} x\left( t \right)\\ y\left( t \right) \end{array} \right)} \right\|$→0(t→+∞).证毕.

    3 数值实例

    例1  考虑下列模糊BAM神经网络:

    模糊规则1

    t≥0,ttkk=1,2,…,若ω1(t)=$ \frac{1}{{{{\rm{e}}^{{\rm{ - 5}}{\omega _{\rm{1}}}\left( t \right)}}}}$,则

    $ \left\{ {\begin{array}{*{20}{l}} {\frac{{{\text{d}}x(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{A}}x(t) + {\mathit{\boldsymbol{C}}_1}f(y(t - \tau (t)))} \\ {\frac{{{\text{d}}y(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{B}}y(t) + {\mathit{\boldsymbol{D}}_1}g(x(t - h(t)))} \\ {x\left( {t_k^ + } \right) - x\left( {{t_k}} \right) = \rho \left( {x\left( {{t_k}} \right)} \right),y\left( {t_k^ + } \right) - y\left( {{t_k}} \right) = \rho \left( {y\left( {{t_k}} \right)} \right)} \\ {x(s) = \xi (s),y(s) = \eta (s)\;\;\;\;\;s \in [ - \tau ,0]} \end{array}} \right. $ (6)

    模糊规则2

    t≥0,ttkk=1,2,…,若ω2(t)=1-$ \frac{1}{{{{\rm{e}}^{{\rm{ - 5}}{\omega _{\rm{1}}}\left( t \right)}}}}$,则

    $ \left\{ {\begin{array}{*{20}{l}} {\frac{{{\text{d}}x(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{A}}x(t) + {\mathit{\boldsymbol{C}}_2}f(y(t - \tau (t)))} \\ {\frac{{{\text{d}}y(t)}}{{{\text{d}}t}} = - \mathit{\boldsymbol{B}}y(t) + {\mathit{\boldsymbol{D}}_2}g(x(t - h(t)))} \\ {x\left( {t_k^ + } \right) - x\left( {{t_k}} \right) = \rho \left( {x\left( {{t_k}} \right)} \right),y\left( {t_k^ + } \right) - y\left( {{t_k}} \right) = \rho \left( {y\left( {{t_k}} \right)} \right)} \\ {x(s) = \xi (s),y(s) = \eta (s)\;\;\;\;s \in [ - \tau ,0]} \end{array}} \right. $ (7)

    其中τ(t)=h(t)=τ=0.8,t1=0.3,tk=tk-1+0.3kδ=0.5,x(s)=tanh sy(s)=2sin sf(x)=0.1sin xg(x)=0.09sin xρ(x)=0.1xA=(2),B=(1.95),C1=(0.02),C2=(0.03),D1=(0.15),D2=(0.18),F=(0.1),G=(0.09),H=(0.1).

    利用计算机Matlab LMI工具箱解(4)式,得到可行性数据

    $ \lambda = 0.811\;5 $

    则由定理1知,模糊系统(6)-(7)是指数稳定的.

    4 总结

    本文用不动点方法研究了模糊脉冲概率时滞BAM神经网络系统的稳定性,其优点是利用压缩映像原理给出系统解的存在性的同时,也给了该解的全局指数型稳定结论,这点由本文函数空间的构造可以看出.

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    The Fixed Point Theorem and the Stability of Fuzzy Impulsive Systems with Probability Time-Delays
    HUANG Jia-lin1, LI Xing-gui2     
    1. Department of Elementary Education, Sichuan Sanhe College of Professionals, Luzhou Sichuan 646200, China;
    2. Department of Mathematics, Chengdu Normal University, Chengdu 611130, China
    Abstract: By defining a contraction mapping on a complete distance space, the authors employ the T-S fuzzy rule, probabilistic time-delay property and contraction mapping principle to derive an algebraic criterion for the stability of a class of T-S fuzzy probabilistic time-delay impulsive Bidirectional Associative Memory neural networks. Remarkably, the stability of the solution is given as soon as the existence of the solution of the system is derived. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.
    Key words: bi-directional associative memory neural networks    probability time-delay    fixed point theorem    
    X