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 0
 Article Options PDF Abstract Figures References 扩展功能 Email Alert RSS 本文作者相关文章 蒲浩 蒋海军 胡成 冉杰 张转周 欢迎关注西南大学期刊社

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

1 模型和预备知识

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

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

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

(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.

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

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

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

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

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

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

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

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

2 辅助引理

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

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

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)

由引理2可知

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

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

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

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

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)

 $\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}$

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

 $\begin{array}{l} \sum\limits_{j = 1,i \ne j}^n {\left| {{b_{ij}}} \right|{L_j}\left| {{e_j}\left( t \right)} \right|p{{\left| {{e_i}\left( t \right)} \right|}^{p - 1}}} \le \\ \sum\limits_{j = 1,i \ne j}^n {\sum\limits_{k = 1}^{p - 1} {{{\left| {{b_{ij}}} \right|}^{p{\gamma _{ijk}}}}L_j^{p{\gamma _{ijk}}}{{\left| {{e_i}\left( t \right)} \right|}^p}} } + \sum\limits_{j = 1,i \ne j}^n {{{\left| {{b_{ij}}} \right|}^{p{\gamma _{ijp}}}}L_j^{p{\gamma _{ijp}}}{{\left| {{e_j}\left( t \right)} \right|}^p}} \end{array}$ (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)

 $\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)

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

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

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

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

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

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

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

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

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

4 数值例子

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

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

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)

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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