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 0
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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

1 预备知识

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

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

{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

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

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

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

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

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.

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

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

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)

2 主要结论与证明

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

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}.

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

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

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

 $\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}$

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

3 数值实例

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)

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)

 $\lambda = 0.811\;5$

4 总结

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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