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时滞递归神经网络(DRNNs)基于在联想记忆、优化控制、图像处理等领域的应用而被广泛研究。在这些应用中大多数都需要DRNNs是稳定的，指数稳定性作为DRNNs的一种重要的动力学性质[1-4]，它的特征之一是Lyapunov指数不等于0。然而大多数情况下，DRNNs的最大Lyapunov指数是等于零的，其状态轨迹渐近趋于平衡点，即此时DRNNs是渐近稳定的[5-7]。
多项式稳定性是一种特殊的稳定性，它同指数稳定性一样蕴含着渐近稳定性，但其收敛速度比指数稳定性慢一些。它的特征之一就是其最大的Lyapunov指数等于零。目前，这种系统的多项式稳定性研究较少，只有某些系统在某些特殊情况下才具有多项式稳定性，如波动方程[8-9]和随机微分方程[10-14]。需要说明的是，这里的多项式稳定性不是指对于一个多项式来研究这个多项式的稳定性，而是因为某些系统的解的估计式中含有
${t^{ - \lambda }}\;(t \geqslant {t_0},\;\lambda > 0)$ ，类似于多项式，故称这种稳定性为多项式稳定性，具体见下文的定义。由上所述，某种DRNNs是否具有多项式稳定性？文献[15]回答了这个问题，文中给出了比例时滞递归神经网络(RNNs)多项式稳定性的定义，并研究了几类比例时滞RNNs的多项式稳定性和多项式周期性。文献[16]将比例时滞引入细胞神经网络，提出比例时滞细胞神经网络模型。此后各种比例时滞RNNs基于在二次规划问题和QoS路由决策等方面的潜在应用得到了国内外学者的关注，并取得一些研究成果[17-26]，但关于比例时滞RNNs的多项式稳定性的研究还很有限。
除了时滞效应外，脉冲也是影响神经网络动力学性质的重要因素之一。脉冲效应是指在网络运行过程中，系统的状态发生突变，导致网络的行为更加复杂，因此研究脉冲作用下系统的稳定性是非常必要的。目前，已有许多关于脉冲DRNNs的动力学行为的研究成果[24, 27-32]。文献[24]应用Lyapunov稳定性理论结合线性矩阵不等式的方法研究一类比例时滞脉冲RNNs (IRNNs) 的无源性。然而，比例时滞IRNNs的动力学行为的研究还很少。基于此，本文通过构造Lyapunov泛函和LMI的方法对一类比例时滞IRNNs的全局多项式稳定性进行探讨。
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设
$C([\tilde q{t_0},{t_0}],{\mathbb{R}^n})$ 和$C([\log \tilde q{t_0},\log {t_0}],{\mathbb{R}^n})$ 分别表示具一致收敛拓扑的从$[\tilde q{t_0},{t_0}] \to {\mathbb{R}^n}$ 和$[\log \tilde q{t_0},\log {t_0}] \to {\mathbb{R}^n}$ 的所有连续函数构成的Banach空间，这里${t_0} \geqslant 1$ 。对于${{\varphi}} \in C ([\tilde q{t_0},{t_0}],{\mathbb{R}^n})$ ，$||{{\varphi}} ||_{\tilde q}^2 = {\sup _{\tilde q{t_0} \leqslant s \leqslant {t_0}}}||{{\varphi}} (s)|{|^2}$ ，${{\eta}} \in C ([\log \tilde q{t_0},\log {t_0}],{\mathbb{R}^n})$ ，$||{{\eta}} (s)||_{\tilde \tau }^2 = {\sup _{\log \tilde q{t_0} \leqslant s \leqslant \log {t_0}}}||{{\eta}} (s)|{|^2}$ 。对于${{A}} \in {\mathbb{R}^{n \times n}}$ ，${{A}} > 0$ 和${{A}} < 0$ 分别${{A}}$ 为正定和负定，${\lambda _M}({{A}})$ 和${\lambda _m}({{A}})$ 分别表示${{A}}$ 的最大和最小特征值。对于${{x}} = {({x_1},{x_2}, \cdots {x_n})^{\rm{T}}} \in {\mathbb{R}^n}$ ，$||{{x}}|| = {\left(\displaystyle\sum\limits_{i = 1}^n {x_i^2} \right)^{{1 / 2}}},{\mathbb{N}^ * } = \{ 1,2, \cdots \} $ 。考虑如下脉冲RNNs：
$$\left\{ \begin{aligned} &\dot {{x}}(t) = - {{Cx}}(t) + {{Af}}({{x}}(t)) + {{Bf}}({{x}}(qt)) + {{u}} \\ & \quad \quad \;\;\;t \geqslant {t_0},\;t \ne {t_k} \\ &\Delta {{x}}({t_k}) = {{{E}}_k}({{x}}({t_k}^ - ))\qquad{k \in {\mathbb{N}^ * }} \\ &{{x}}(s) = {{\varphi}} (s)\qquad{s \in [\tilde q{t_0},{t_0}]} \end{aligned} \right.$$ (1) 式中，
${{x}}(t) = {({x_1}(t),{x_2}(t), \cdots ,{x_n}(t))^{\rm{T}}}$ 为神经元的状态向量，$n$ 为神经元的个数；${{C}} = {\rm{diag}} {({c_1},{c_2}, \cdots ,{c_n})^{\rm{T}}}$ ，${c_i} > 0$ ；矩阵${{A}} = {({a_{ij}})_{n \times n}}$ 和${{B}} = {({b_{ij}})_{n \times n}}$ 分别表示反馈和延时反馈矩阵；${{f}}({{x}}(t)) = ({f_1}({x_1}(t)),{f_2}({x_2}(t)),$ $ \cdots ,{f_n}({x_n}(t)){)^{\rm{T}}}$ 表示激活向量函数，${{f}}({{x}}(qt)) = ({f_1}({x_1}({q_1}t)), {f_2}({x_2}({q_2}t)), \cdots , {f_n}({x_n}({q_n}t)))^{\rm{T}}$ ；${q_i}t = t - (1 - {q_i})t$ ，这里${q_i} \in (0,1)$ ，$i = 1,2, \cdots ,n$ 。当$t \to + \infty $ 时，$(1 - {q_i})t \to + \infty $ 是无界时滞函数，$\tilde q = \mathop {\min }\limits_{1 \leqslant i \leqslant n} \{ {q_i}\} $ ；${{u}} = {({u_1},{u_2}, \cdots ,{u_n})^{\rm{T}}}$ 是外部输入常向量；${{\varphi}} (s) \in C([\tilde q{t_0},{t_0}],{\mathbb{R}^n})$ 表示${{x}}(s)$ $(s \in [\tilde q{t_0},{t_0}])$ 的初值函数；时间序列${\rm{\{ }}{t_k}{\rm{\} }}$ 满足${t_0} < {t_1}$ $ < \cdots < {t_k} < {t_{k + 1}} < \cdots $ ，且${\lim _{k \to + \infty }}{t_k} = + \infty $ ；$\Delta {{x}}({t_k}) = (\Delta {x_1}({t_k}), \Delta {x_2}({t_k}), \cdots , \Delta {x_n}({t_k}))^{\rm{T}}$ ，$\Delta {x_i}({t_k}) = {x_i}({t_k}^ + ) - {x_i}({t_k}^ - )$ ，${x_i}({t_k}^ + ) = {x_i}({t_k})$ ，且${x_i}({t_k}^ - ) = {\lim _{t \to {t_k}^ - }}{x_i}(t)$ ；${{{E}}_{k}}( \cdot )$ 表示状态变量在时刻${t_k}$ 处的增量变化。本文对激活函数作如下假设：
(H) 假设
${f_i}( \cdot )\;(i = 1,2, \cdots ,n)$ 满足：$$0 \leqslant \frac{{{f_i}(u) - {f_i}(v)}}{{u - v}} \leqslant {l_i}\begin{array}{*{20}{c}} {}&{u,v \in \mathbb{R}} \end{array}\;$$ 记
${{L}} = {\rm{diag}}({l_1},{l_2}, \cdots ,{l_n})$ 。令
${{z}}(t) = {{x}}({{\rm{e}}^t})$ ，则式(1)等价地变换为：$$\left\{ \begin{split} & \dot {{z}}(t) = {{\rm{e}}^t}[ - {{Cz}}(t) + {{Af}}({{z}}(t)) + {{Bf}}({{z}}(t - \tau )) + {{u}}] \\ & \quad \quad \;\;\;t \geqslant \log {t_0},\;t \ne {t_k} \\ & \Delta {{z}}({t_k}) = {{{E}}_k}({{z}}({t_k}^ - ))\qquad \;k \in {\mathbb{N}^ * } \\ & {{z}}(s) = {{\psi}} (s)\qquad s \in [\log \tilde q{t_0},\log {t_0}] \end{split} \right.$$ (2) 式中，
${{z}}(t) = {({z_1}(t),{z_2}(t), \cdots ,{z_n}(t))^{\rm{T}}}$ ；${{f}}({{z}}(t)) = ({f_1}({z_1}(t)),$ ${f_2}({z_2}(t)), \cdots ,{f_n}({z_n}(t)){)^{\rm{T}}}$ ；${{f}}({{z}}(t - \tau )) = ({f_1}({z_1}(t - {\tau _1})),$ ${f_2}({z_2} (t - {\tau _2})), \cdots ,{f_n}({z_n}(t - {\tau _n})){)^{\rm{T}}}$ ，${\tau _i} = - \log {q_i} \geqslant 0$ ，$\tilde \tau = \mathop {\max }\limits_{1 \leqslant i \leqslant n} \{ {\tau _i}\} = - \log \tilde q$ ；${{\psi}} (s) = {{\varphi}} ({{\rm{e}}^s}) \in C([\log \tilde q{t_0},\log {t_0}],{\mathbb{R}^n})$ 。由假设
$(H)$ 可知，式(1)和式(2)的平衡点必存在。设${{{x}}^ * } = {(x_1^ * ,x_2^ * , \cdots ,x_n^ * )^{\rm{T}}}$ 和${{{z}}^ * } = {(z_1^ * ,z_2^ * , \cdots ,z_n^ * )^{\rm{T}}}$ 为式(1)和式(2)的平衡点，由平衡点的定义，可算得${{{x}}^ * } = {{{z}}^ * }$ 。因此，可通过探讨式(2)的平衡点${{{z}}^ * }$ 的稳定性来间接讨论式(1)的平衡点${{{x}}^ * }$ 的稳定性情况。令
${{y}}(t) = {{z}}(t) - {{{z}}^ * }\;$ ，${y_i}(t) = {z_i}(t) - z_i^ * ,\;i = 1,2, \cdots ,n$ ，则式(2)改写为：$$\left\{\; \begin{split} &\dot {{y}}(t) = {{\rm{e}}^t}[ - {{Cy}}(t) + {{Ag}}({{y}}(t))+ \\ &{{Bg}}({{y}}(t - \tau ))]\quad \;\;\; t \geqslant \log {t_0},\;t \ne {t_k} \\ &\Delta {{y}}({t_k}) = {{{E}}_k}({{y}}({t_k}^ - ))\qquad\;k \in {\mathbb{N}^ * } \\ &{{y}}(s) = {{\eta}} (s)\qquad\;s \in [\log \tilde q{t_0},\log {t_0}] \end{split} \right.$$ (3) 式中，
${{g}}({{y}}(t - \tau )) = ({g_1}({y_1}(t - {\tau _1})),{g_2}({y_2}(t - {\tau _2})), \cdots ,$ ${g_n}({y_n}(t - {\tau _n})){)^{\rm{T}}}$ ，${g_i}({y_i}(t)) = {f_i}({y_i}(t) + z_i^ * ) - {f_i}(z_i^ * )$ ，且${g_i}(0) = 0$ ；${{\eta}} (s) = {{\psi}} (s) - {{{z}}^ * }$ 。由(H)和
${g_i}({y_i}(t)) = {f_i}({y_i}(t) + z_i^ * ) - {f_i}( {z_i^ * } )$ ，可得：$$0 \leqslant \frac{{{g_i}({y_i}(t))}}{{{y_i}(t)}} \leqslant {l_i}$$ (4) 由积分中值定理，可得：
$$ 0 \leqslant \int_0^{{y_i}(t)} {{g_i}(s)} {\rm{d}}s \leqslant {g_i}({y_i}(t)){y_i}(t) $$ (5) 由式(4)和式(5)，可得：
$${l_i}{y_i}(t) \geqslant {g_i}({y_i}(t)),\;\;{g_i}({y_i}(t)){y_i}(t) \geqslant 0$$ 从而得：
$${l_i}{g_i}({y_i}(t)){y_i}(t) \geqslant g_i^2({y_i}(t))\quad \forall {y_i}(t) \in \mathbb{R},\;i = 1,2, \cdots ,n$$ (6) 同时，由式(6)可得：
$$\begin{split} & {{{g}}^{\rm{T}}}({{y}}(t)){{DCy}}(t) = \sum\limits_{i = 1}^n {{g_i}({y_i}(t)){d_i}{c_i}{y_i}(t)} \geqslant \\ & \sum\limits_{i = 1}^n {\frac{{{d_i}{c_i}}}{{{l_i}}}{g_i}^2({y_i}(t))} = {{{g}}^{\rm{T}}}({{y}}(t)){{{DCL}}^{ - 1}}{{g}}({{y}}(t))\; \end{split} $$ (7) 定义1 称系统(1)的平衡点
${{{x}}^ * }$ 是全局多项式稳定的(GPS)。若存在$\lambda > 0$ 和$\,\beta \geqslant 1$ ，使得：$$ ||{{x}}(t) - {{{x}}^ * }|{|^2} \leqslant \beta ||{{\varphi}} (s) - {{{x}}^ * }||_{\tilde q}^2{\left( {\frac{t}{{{t_0}}}} \right)^{ - \lambda }}\begin{array}{*{20}{c}} {}&{t \geqslant {t_0}} \end{array} $$ 式中，
$ ||{{\varphi}} (s) - {{{x}}^ * }||_{\tilde q}^2 = \mathop {\sup }\limits_{\tilde q{t_0} \leqslant s \leqslant {t_0}} ||{{\varphi}} (s) - {{{x}}^ * }|{|^2}$ 。定义2 称系统(2)的平衡点
${{{z}}^ * }$ 是全局指数稳定的(GES)。若存在$\lambda > 0$ 和$\beta \geqslant 1$ ，使得：$$||{{z}}(t) - {{{z}}^ * }|{|^2} \leqslant \beta ||{{\psi}} (s) - {{{z}}^ * }||_{\tilde \tau }^2{{\rm{e}}^{ - \lambda (t - \log {t_0})}}\begin{array}{*{20}{c}} {}&{t \geqslant {t_0}} \end{array}$$ 式中，
$ ||{{\psi}} (s) - {{{z}}^ * }||_{\tilde \tau }^2 = \mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant s \leqslant \log {t_0}} ||{{\psi}} (s) - {{{z}}^ * }|{|^2}$ 。 -
定理1 假设(H)成立，
$ {x_i}({t_k})\! =\! {\lambda _{ik}}{x_i}({t_k}^ -\! )$ ，$ \lambda _{ik}^2 \!\leqslant\!\! 1$ 。若存在矩阵$ {{P}} = {({p_{ij}})_{n \times n}} > 0$ ，$ {{M}} = {\rm{diag}}({m_1},{m_2}, \cdots ,{m_n}) > 0$ ，$ {{D}} = {\rm{diag}}({d_1},{d_2}, \cdots ,{d_n}) > 0$ ，常数$ \sigma $ 满足$ 0 < (\sigma - 1)$ $ < 2\min \{ {c_i}\} $ ，使得：$$ \begin{split} & \qquad\qquad\qquad\qquad\qquad\qquad{{\varXi }} = \\ & \left(\!\!\!\! \begin{array}{l} (\sigma - 1){{P}} - {{{C}}^{\rm{T}}}{{P}} - {{{P}}^{\rm{T}}}{{C}} \;\;{{{P}}^{\rm{T}}}{{A}} + \dfrac{1}{2}(\sigma - 1){{D}}\quad {{{B}}^{\rm{T}}}{{P}}\\ {{{A}}^{\rm{T}}}{{P}} + \dfrac{1}{2}(\sigma - 1){{D}}\quad \quad \;- {{DC}}{{{L}}^{ - 1}} \!\!+\!\! {{M}} \!\!+\!\! {{DA}} \;\; \dfrac{1}{2}{{{D}}^{\rm{T}}}{{B}}\\ \quad {{{P}}^{\rm{T}}}{{B}}\quad \quad \quad \quad \quad \quad\;\; \dfrac{1}{2}{{{B}}^{\rm{T}}}{{D}}\quad \quad \quad \quad \quad\; - {{\tilde q}^{ - \sigma }}{{M}} \end{array}\!\!\!\! \right)< 0 \end{split} $$ 则系统(1)的平衡点
${{{x}}^ * }$ 是GPS。证明：考虑如下Lyapunov泛函：
$$\begin{split} & V(t) = {{\rm{e}}^{(\sigma - 1)t}}{{{y}}^{\rm{T}}}(t){{Py}}(t) + \sum\limits_{i = 1}^n {{{\rm{e}}^{(\sigma - 1)t}}{d_i}} \int_0^{{y_i}(t)} {{g_i}(s)} {\rm{d}}s + \\ &\qquad \;\; \sum\limits_{i = 1}^n {\int_{t - {\tau _i}}^t {{{\rm{e}}^{\sigma \theta }}{m_i}g_i^2({y_i}(\theta ))} {\rm{d}}\theta } \end{split} $$ (8) 式中，
$ {{P}} = {({p_{ij}})_{n \times n}} > 0,\;\sigma > 1,\;{d_i} > 0,\;{m_i} > 0,\;i = 1,2, \cdots ,n$ 。当
$t \ne {t_k}$ 时，将式(8)沿系统(3)对$t$ 进行求导，得：$$\begin{split} &\quad\dot V(t) = {{\rm{e}}^{(\sigma - 1)t}}[(\sigma - 1){{{y}}^{\rm{T}}}(t){{Py}}(t) + 2{{{y}}^{\rm{T}}}(t){{P}}\dot {{y}}(t) + \\ &\sum\limits_{i = 1}^n {(\sigma - 1){d_i}\int_0^{{y_i}(t)} {{g_i}(s)} } {\rm{d}}s]{\rm{ + }}{{\rm{e}}^{(\sigma - 1)t}}\sum\limits_{i = 1}^n {{g_i}({y_i}(t))} {d_i}{{\dot y}_i}(t) + \\ &\;\;\;\sum\limits_{i = 1}^n {[{{\rm{e}}^{\sigma t}}{m_i}{g_i}^2({y_i}(t))} - {{\rm{e}}^{\sigma (t - {\tau _i})}}{m_i}{g_i}^2({y_i}(t - {\tau _i}))] \leqslant \end{split}$$ $$\begin{split} &\qquad\;{{\rm{e}}^{(\sigma - 1)t}}[(\sigma - 1){{{y}}^{\rm{T}}}(t){{Py}}(t) + 2{{{y}}^{\rm{T}}}(t){{P}}\dot {{y}}(t) + \\ &\qquad\qquad\sum\limits_{i = 1}^n {(\sigma - 1){g_i}({y_i}(t)){d_i}{y_i}(t)} ] + \\ &\;\;{{\rm{e}}^{(\sigma - 1)t}}\sum\limits_{i = 1}^n {{g_i}({y_i}(t))} {d_i}{{\dot y}_i}(t) + {{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t)){{Mg}}({{y}}(t)) - \\ &\qquad\;\;{{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t - \tau )){{\rm{e}}^{ - \sigma \bar \tau }}{{Mg}}({{y}}(t - \tau )) = \\ &{{\rm{e}}^{(\sigma - 1)t}}{{{y}}^{\rm{T}}}(t)(\sigma - 1){{Py}}(t) + 2{{\rm{e}}^{\sigma t}}{{{y}}^{\rm{T}}}(t){{P}}( - {{Cy}}(t) + {{Ag}}({{y}}(t)) + \\ &\;\;\;\;{{Bg}}({{y}}(t - \tau )))\;\; + (\sigma - 1){{\rm{e}}^{(\sigma - 1)t}}{{{g}}^{\rm{T}}}({{y}}(t)){{Dy}}(t) + \\ &\;\;\;{{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t)){{D}}( - {{Cy}}(t) + {{Ag}}({{y}}(t)) + {{Bg}}({{y}}(t - \tau ))) + \\ &{{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t)){{Mg}}({{y}}(t)) - {{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t - \tau )){{\rm{e}}^{ - \sigma \bar \tau }}{{Mg}}({{y}}(t - \tau )) \leqslant \\ &\;\;\;\;\;{{\rm{e}}^{\sigma t}}\left\{ {{{{y}}^{\rm{T}}}(t)[(\sigma - 1){{P}} - {{{P}}^{\rm{T}}}{{C}} - {{{C}}^{\rm{T}}}{{P}}]{{y}}(t)} \right. + \\ & \qquad2{{{y}}^{\rm{T}}}(t){{PAg}}({{y}}(t)) + 2{{{y}}^{\rm{T}}}(t){{PBg}}({{y}}(t - \tau )) + \\ &\qquad {{{g}}^{\rm{T}}}({{y}}(t))(\sigma - 1){{Dy}}(t) - {{{g}}^{\rm{T}}}({{y}}(t)){{DCy}}(t) + \\ &\qquad {{{g}}^{\rm{T}}}({{y}}(t)){{DA}}{{{g}}^{\rm{T}}}({{y}}(t)) + {{{g}}^{\rm{T}}}({{y}}(t)){{DBg}}({{y}}(t - \tau )) + \quad \\ &\;\;\;\left. {{{{g}}^{\rm{T}}}({{y}}(t)){{Mg}}({{y}}(t)) - {{{g}}^{\rm{T}}}({{y}}(t - \tau )){{\rm{e}}^{ - \sigma \bar \tau }}{{Mg}}({{y}}(t - \tau ))} \right\} \end{split}$$ (9) 由式(9)和式(7)，得：
$$ \begin{split} &\;\;\;\;\;\; \dot V(t) \leqslant {{\rm{e}}^{\sigma t}}\left\{ {{{{y}}^{\rm{T}}}(t)[(\sigma - 1){{P}} - {{{P}}^{\rm{T}}}{{C}} - {{{C}}^{\rm{T}}}{{P}}]{{y}}(t)} \right. + \\ & \;\;\;\;\quad2{{{y}}^{\rm{T}}}(t){{PAg}}({{y}}(t)) + 2{{{y}}^{\rm{T}}}(t){{PBg}}({{y}}(t - \tau )) + \\ & \;\;\;\;{{{g}}^{\rm{T}}}({{y}}(t))(\sigma - 1){{Dy}}(t) - {{{g}}^{\rm{T}}}({{y}}(t)){{DC}}{{{L}}^{ - 1}}{{g}}({{y}}(t)) + \\ & \;\;\;\;\;\;{{{g}}^{\rm{T}}}({{y}}(t)){{DAg}}({{y}}(t)) + {{{g}}^{\rm{T}}}({{y}}(t)){{DBg}}({{y}}(t - \tau ))) + \\ & \;\;{{{g}}^{\rm{T}}}({{y}}(t)){{Mg}}({{y}}(t)) - \left. {{{{g}}^{\rm{T}}}({{y}}(t - \tau ))({{\rm{e}}^{ - \sigma \tilde \tau }}{{M}}){{g}}({{y}}(t - \tau ))} \right\} = \\ &\;\;\;\;\;\;\; {{\rm{e}}^{\sigma t}}\left\{ {{{{y}}^{\rm{T}}}(t)[(\sigma - 1){{P}} - {{{P}}^{\rm{T}}}{{C}} - {{{C}}^{\rm{T}}}{{P}}]{{y}}(t)} \right. + \\ & \;\;\;\;\;\;2{{{y}}^{\rm{T}}}(t){{PAg}}({{y}}(t)) + 2{{{y}}^{\rm{T}}}(t){{PBg}}({{y}}(t - \tau )) + \\ & \;\;\;\;{{{g}}^{\rm{T}}}({{y}}(t))(\sigma - 1){{Dy}}(t) - {{{g}}^{\rm{T}}}({{y}}(t)){{DC}}{{{L}}^{ - 1}}{{g}}({{y}}(t)) + \\ &\;\; \;\;\;{{{g}}^{\rm{T}}}({{y}}(t)){{DAg}}({{y}}(t)) + {{{g}}^{\rm{T}}}({{y}}(t)){{DBg}}({{y}}(t - \tau ))) + \\ & \;\;\;\;{{{g}}^{\rm{T}}}({{y}}(t)){{Mg}}({{y}}(t)) - \left. {{{{g}}^{\rm{T}}}({{y}}(t - \tau ))({{\rm{e}}^{ - \sigma \tilde \tau }}{{M}}){{g}}({{y}}(t - \tau ))} \right\} = \\ &\qquad\qquad\qquad\qquad\;\;\; {{\rm{e}}^{\sigma t}}{{{\xi }}^{\rm{T}}}(t){{\varXi \xi }}(t) \end{split} $$ 式中，
${{\xi}=\left(y^{\rm{T}}(t)\;g^{\rm{T}}(y(t))\;g^{\rm{T }}(y(t-\tau))\right)^{\rm{T}}}$ 。由已知$\varXi < 0$ ，可得：$$\dot V(t) \leqslant 0\quad\;t \ne t{}_k$$ (10) 另一方面，当
$t \ne {t_k}$ 时，由式(8)，得：$$ \begin{gathered} V({t_k}) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {} {{\rm{e}}^{(\sigma - 1){t_k}}}} {y_i}({t_k}){p_{ij}}{y_j}({t_k})\; + \\ \;\sum\limits_{i = 1}^n {{{\rm{e}}^{(\sigma - 1){t_k}}}{d_i}} \int_0^{{y_i}({t_k})} {{g_i}(s)} {\rm{d}}s + \sum\limits_{i = 1}^n {} \int_{{t_k} - {\tau _i}}^{{t_k}} {{{\rm{e}}^{\sigma \theta }}{m_i}g_i^2({y_i}(\theta ))} {\rm{d}}\theta = \\ \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {} {{\rm{e}}^{(\sigma - 1){t_k}^ - }}} \lambda _{ik}^2{y_i}(t_k^ - ){p_{ij}}{y_j}({t_k}^ - ) + \\ \;\;\sum\limits_{i = 1}^n {{{\rm{e}}^{(\sigma - 1){t_k}^ - }}{d_i}} \int_0^{{\lambda _{ik}}{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s + \end{gathered} $$ $$ \sum\limits_{i = 1}^n {} \int_{{t_k}^ - - {\tau _i}}^{{t_k}^ - } {{{\rm{e}}^{\sigma \theta }}{m_i}g_i^2({y_i}(\theta ))} {\rm{d}}\theta $$ 由于
$\lambda _{ik}^2 \leqslant 1$ ，则$\displaystyle\int_0^{{\lambda _{ik}}{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s \leqslant \displaystyle\int_0^{{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s$ 。于是：$$ \begin{split} &\qquad\;\; V({t_k}) \leqslant V({t_k}^ - ) = {{\rm{e}}^{(\sigma - 1){t_k}^ - }}{{{y}}^{\rm{T}}}({t_k}^ - ){{Py}}({t_k}^ - ) + \\ & \sum\limits_{i = 1}^n {{{\rm{e}}^{(\sigma - 1){t_k}^ - }}{d_i}} \int_0^{{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s + \sum\limits_{i = 1}^n {\int_{{t_k}^ - - {\tau _i}}^{{t_k}^ - } {{{\rm{e}}^{\sigma \theta }}{m_i}g_i^2({y_i}(\theta ))} {\rm{d}}\theta } \end{split} $$ (11) 由(10)和(11)，当
$t \geqslant \log {t_0}$ 时，得：$$V(t) \leqslant V(\log {t_0}),\;t \in ({t_{k - 1}},{t_k}],\;k \in {\mathbb{N}^ * }$$ 当且仅当
${{y}}(t) \!=\! {{g}}({{y}}(t)) \!=\! {{g}}({{y}}(t\! -\! \tau ))$ 时，$\dot V(t) \!=\! 0$ 。且：$$ \begin{gathered} V(\log {t_0}) = {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{{{y}}^{\rm{T}}}(\log {t_0}){{Py}}(\log {t_0}) + \\ \sum\limits_{i = 1}^n {{{\rm{e}}^{(\sigma - 1)\log {t_0}}}{d_i}\int_0^{{y_i}(\log {t_0})} {{g_i}(s)} {\rm{d}}s} + \\ \sum\limits_{i = 1}^n {\int_{\log {q_i}{t_0}}^{\log {t_0}} {{{\rm{e}}^{\sigma \theta }}{m_i}g_i^2({y_i}(\theta ))} {\rm{d}}\theta } \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{y^{\rm{T}}}(\log {t_0}){{Py}}(\log {t_0}) + \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}\sum\limits_{i = 1}^n {{d_{^i}}{g_i}({y_i}(\log {t_0})){y_i}(\log {t_0})} + \\ \sum\limits_{i = 1}^n {\int_{\log {q_i}{t_0}}^{\log {t_0}} {{{\rm{e}}^{\sigma \theta }}{m_i}l_i^2y_i^2(\theta )} {\rm{d}}\theta } \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}\left( {{{{y}}^{\rm{T}}}(\log {t_0}){{Py}}(\log {t_0}) + {{{y}}^{\rm{T}}}(\log {t_0}){{LDy}}(\log {t_0})} \right) + \\ \mathop {\max }\limits_{1 \leqslant i \leqslant n} \left\{ {{m_i}l_i^2t_0^\sigma {\sigma ^{ - 1}}} \right\}\sum\limits_{i = 1}^n {\mathop {\sup }\limits_{\log {q_i}{t_0} \leqslant \theta \leqslant \log {t_0}} y_i^2(\theta )} \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{\lambda _M}({{P}} + {{LD}})||{{y}}(\log {t_0})|{|^2} + \\ \mathop {\max }\limits_{1 \leqslant i \leqslant n} \left\{ {{m_i}l_i^2t_0^\sigma {\sigma ^{ - 1}}} \right\}\sum\limits_{i = 1}^n {\mathop {\sup }\limits_{\log {q_i}{t_0} \leqslant \theta \leqslant \log {t_0}} y_i^2(\theta )} \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{\lambda _M}({{P}} + {{LD}})||{{y}}(\log {t_0})|{|^2} + \\ \mathop {\max }\limits_{1 \leqslant i \leqslant n} \left\{ {{m_i}l_i^2t_0^\sigma {\sigma ^{ - 1}}} \right\}\mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} \sum\limits_{i = 1}^n {y_i^2(\theta )} \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}\bigg\{ {{\lambda _M}({{P}} + {{LD}})} + \\ {\mathop {\max }\limits_{1 \leqslant i \leqslant n} [{m_i}l_i^2t_0^\sigma {\sigma ^{ - 1}}]} \bigg\}\mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{y}}(\theta )|{|^2} \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}\bigg\{ {{\lambda _M}({{P}} + {{LD}})} + \\ {\mathop {\max }\limits_{1 \leqslant i \leqslant n} [{m_i}l_i^2t_0^\sigma {\sigma ^{ - 1}}]} \bigg\}\mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\eta}} (\theta )|{|^2} \end{gathered} $$ 又因为：
$${{\rm{e}}^{(\sigma - 1)t}}{{{y}}^{\rm{T}}}(t){{Py}}(t) \leqslant V(t) \leqslant V(\log {t_0})$$ 于是，得：
$$\begin{gathered} {\lambda _m}{\rm{(}}{{P}}{\rm{)}}{{\rm{e}}^{(\sigma - 1)t}}||{{y}}(t)|{|^2} \leqslant {{\rm{e}}^{(\sigma - 1)t}}{{{y}}^{\rm{T}}}(t){{Py}}(t) \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}\bigg\{ {{\lambda _M}({{P}} + {{LD}})} + \\ \; {\mathop {\max }\limits_{1 \leqslant i \leqslant n} [{m_i}l_i^2t_0^\sigma {\sigma ^{ - 1}}]} \bigg\}\mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\eta}} (\theta )|{|^2} \end{gathered} $$ 所以：
$$||{{y}}(t)|{|^2} \leqslant \beta \mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\eta}} (\theta )||_{}^2{{\rm{e}}^{ - \lambda (t - \log {t_0})}}\;$$ (12) 式中，
$\lambda = \sigma - 1 > 0$ ；$\,\beta = \dfrac{1}{{{\lambda _m}({{P}})}}\{ {\lambda _M}({{P}} + {{LD}}) + $ $ \mathop {\max }\limits_{1 \leqslant i \leqslant n} \{ {m_i}l_i^2t_0^\sigma {\sigma ^{ - 1}}\} \geqslant 1$ 。从而，得：
$$||{{z}}(t) - {{{z}}^ * }|{|^2} \leqslant \beta \mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\psi}} (\theta ) - {{{z}}^ * }||_{}^2{{\rm{e}}^{ - \lambda (t - \log {t_0})}}\;$$ (13) 再将
${{{x}}^ * } = {{{z}}^ * }$ 和${{z}}(t) = {{x}}({{\rm{e}}^t})$ 代入式(13)，得：$$||{{x}}({{\rm{e}}^t}) - {{{x}}^ * }|{|^2} \leqslant \beta \mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\varphi}} ({{\rm{e}}^\theta }) - {{{x}}^ * }||_{}^2{{\rm{e}}^{ - \lambda (t - \log {t_0})}}\;$$ (14) 在式(14)中，令
${{\rm{e}}^t} = \gamma $ ，则$t = \log \gamma \geqslant 0,\;t \geqslant {t_0}$ 。再令${{\rm{e}}^\theta } = \xi $ ，则$ \theta \in \left[ {\log \;\tilde q{t_0},\log {t_0}} \right]$ ，$ \xi \in \left[ {q{t_0},\;{t_0}} \right]$ ，有：$$||{{x}}(\gamma ) - {{{x}}^ * }|{|^2} \leqslant \beta \mathop {\sup }\limits_{\tilde q{t_0} \leqslant \xi \leqslant {t_0}} ||{{\varphi}} (\xi ) - {{{x}}^ * }|{|^2}{{\rm{e}}^{ - \lambda (\log \gamma - \log {t_0})}}$$ 再取
$\gamma = t$ ，有：$$||{{x}}(t) - {{{x}}^ * }|{|^2} \leqslant \beta \mathop {\sup }\limits_{\tilde q{t_0} \leqslant \xi \leqslant {t_0}} ||{{\varphi}} (\xi ) - {{{x}}^ * }|{|^2}{\left( {\frac{t}{{{t_0}}}} \right)^{ - \lambda }}\quad t \geqslant {t_0}$$ 根据定义1，可知系统(1)的平衡点
${x^ * }$ 是GPS。由式(13)和定义2，可得下面结果。
定理2 假设(H)成立，
${x_i}({t_k}) \!=\! {\lambda _{ik}}{x_i}({t_k}^ - \!)$ ，$\lambda _{ik}^2 \!\leqslant\!\! 1$ 。若存在矩阵${{P}} = {({p_{ij}})_{n \times n}} > 0$ ，${{M}} = {\rm{diag}}({m_1},{m_2}, \cdots ,{m_n}) > 0$ ，${{D}} = {\rm{diag}}({d_1},{d_2}, \cdots ,{d_n}) > 0$ ，和常数$\sigma $ 满足$0 < (\sigma - 1)$ $ < 2\min \{ {c_i}\} $ ，使得：$$ \begin{split} &\qquad\qquad\qquad\qquad\qquad\qquad {{\varXi }} =\\ &\left( \begin{array}{l} (\sigma - 1){{P}}\!\! -\!\! {{{C}}^{\rm{T}}}{{P}} \!\!-\!\! {{{P}}^{\rm{T}}}{{C}}\;\;{{{P}}^{\rm{T}}}{{A}} + \dfrac{1}{2}(\sigma - 1){{D}}\;\;\; {{{B}}^{\rm{T}}}{{P}}\\ {{{A}}^{\rm{T}}}{{P}} + \dfrac{1}{2}(\sigma - 1){{D}}\quad \; - {{DC}}{{{L}}^{ - 1}} \!\!+ \!\!{{M}}\!\! +\!\! {{DA}}\;\; \dfrac{1}{2}{{{D}}^{\rm{T}}}{{B}}\\ \quad \quad{{{P}}^{\rm{T}}}{{B}}\quad \quad \quad \quad \;\; \dfrac{1}{2}{{{B}}^{\rm{T}}}{{D}} \quad \quad \quad \quad \quad \; - \;{{\rm{e}}^{\sigma \tilde \tau }}{{M}} \end{array} \right)< 0 \\ \end{split} $$ 则系统(2)的平衡点
${{{z}}^ * }$ 是GES。定理3 假设(H)成立，
${x_i}({t_k}) \!=\! {\lambda _{ik}}{x_i}({t_k}^ - \!)$ ，$\lambda _{ik}^2 \!\leqslant\!\! 1$ 。当${q_i} = q,\;i = 1,2, \cdots ,n$ 时，若存在矩阵${{P}} = {({p_{ij}})_{n \times n}} > 0$ ，${{M}} > 0$ ，${{D}} = {\rm{diag}}({d_1},{d_2}, \cdots ,{d_n}) > 0$ ，和常数$\sigma $ 满足$0 < (\sigma - 1) < 2\min \{ {c_i}\} $ 使得：$$ \begin{split} &\qquad\qquad\qquad\qquad\qquad\qquad {{\varXi}} = \\ & \left(\!\!\!\! \begin{array}{l} (\sigma - 1){{P}}\!\! -\!\! {{{C}}^{\rm{T}}}{{P}} \!\!- \!\!{{{P}}^{\rm{T}}}{{C}} \;\;\;{{{P}}^{\rm{T}}}{{A}} + \dfrac{1}{2}(\sigma - 1){{D}}\quad \;{{{B}}^{\rm{T}}}{{P}} \\ {{{A}}^{\rm{T}}}{{P}} + \dfrac{1}{2}(\sigma - 1){{D}} \quad \;\;- {{{D}}{{CL}}^{ - 1}} \!\!+\!\! {{M}} \!\!+\!\! {{D}}{{A}}\quad \dfrac{1}{2}{{{D}}^{\rm{T}}}{{B}} \\ \qquad {{{P}}^{\rm{T}}}{{B}}\quad \quad \quad \quad \;\; \; \dfrac{1}{2}{{{B}}^{\rm{T}}}{{D}}\quad \quad \quad \quad\quad \;\;\; - {q^{ - \sigma }}{{M}} \\ \end{array} \!\!\!\!\right)< 0\\ \end{split} $$ 则系统(1)的平衡点
${{{x}}^ * }$ 是GPS。证明：当
${q_i} = q,\;i = 1,2, \cdots ,n$ 时，则有系统(2)和(3)中的${\tau _i} = \tau ,\;i = 1,2, \cdots ,n$ 。 考虑如下Lyapunov泛函：$$\begin{split} & V(t) = {{\rm{e}}^{(\sigma - 1)t}}{{{y}}^{\rm{T}}}(t){{Py}}(t) + \mathop \sum \limits_{i = 1}^n {{\rm{e}}^{(\sigma - 1)t}}{d_i}\int_0^{{y_i}(t)} {{g_i}(s)} {\rm{d}}s + \!\!\!\\ & \qquad \quad \;\; \int_{t - \tau }^t {{{\rm{e}}^{\sigma \theta }}{{{g}}^{\rm{T}}}({{y}}(\theta ))} {{Mg}}({{y}}(\theta )){\rm{d}}\theta \!\!\!\!\!\!\!\!\!\!\!\! \end{split} $$ (15) 式中，
${{P}} = {({p_{ij}})_{n \times n}} > 0,\;{{M}} > 0,\;\sigma > 1,\;{d_i} > 0,\;\;i = 1, 2, \cdots ,n$ 。当
$t \ne {t_k}$ 时，将式(14)沿系统(3)对$t$ 进行求导，得：$$ \begin{gathered} \dot V(t) = {{\rm{e}}^{(\sigma - 1)t}}[(\sigma - 1){{{y}}^{\rm{T}}}(t){{Py}}(t) + \\ \;\;2{{{y}}^{\rm{T}}}(t){{P}}\dot {{y}}(t) + \sum\limits_{i = 1}^n {(\sigma - 1){d_i}} \int_0^{{y_i}(t)} {{g_i}(s)} {\rm{d}}s] + \\ \;\;{{\rm{e}}^{(\sigma - 1)t}}\mathop \sum \limits_{i = 1}^n {g_i}({y_i}(t)){d_i}{{\dot y}_i}(t) + {{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t)){{Mg}}({{y}}(t)) - \\ \;\;{{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t - \tau )){{\rm{e}}^{ - \sigma \tau }}{{Mg}}({{y}}(t - \tau )) \leqslant \\ {{\rm{e}}^{(\sigma - 1)t}}[(\sigma - 1){{{y}}^{\rm{T}}}(t){{Py}}(t) + 2{{{y}}^{\rm{T}}}(t){{P}}\dot {{y}}(t) + \\ \sum\limits_{i = 1}^n {(\sigma - 1){g_i}({y_i}(t)){d_i}{y_i}(t)} ] + \\ \;\;{{\rm{e}}^{(\sigma - 1)t}}\sum\limits_{i = 1}^n {{g_i}({y_i}(t)){d_i}{{\dot y}_i}(t)} + {{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t)){{Mg}}({{y}}(t)) - \\ \;\;{{\rm{e}}^{\sigma t}}{{{g}}^{\rm{T}}}({{y}}(t - \tau )){{\rm{e}}^{ - \sigma \tau }}{{Mg}}({{y}}(t - \tau )) \leqslant {{\rm{e}}^{\sigma t}}{{{\xi }}^{\rm{T}}}(t){{\varXi \xi }}(t) \end{gathered} $$ 中间过程与定理1的证明类似，这里省略。其中
${{\xi}} (t) = ({{{y}}^{\rm{T}}}(t),{{{g}}^{\rm{T}}}({{y}}(t)),{{{g}}^{\rm{T}}}({{y}}(t - \tau )))$ 。由定理3的条件${{\varXi}} < 0$ ，可得：$$\dot V(t) \leqslant 0\quad t \ne t{}_k$$ (16) 另一方面，当
$t \ne {t_k}$ 时，由式(15)，得：$$ \begin{gathered} V({t_k}) = {{\rm{e}}^{(\sigma - 1){t_k}}}(\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{y_i}({t_k}){P_{ij}}{y_j}({t_k})} } + \\ \sum\limits_{i = 1}^n {{d_i}} \int_0^{{y_i}({t_k})} {{g_i}(s)} {\rm{d}}s) + \int_{{t_k} - \tau }^{{t_k}} {{{\rm{e}}^{\sigma \theta }}{{{g}}^{\rm{T}}}({{y}}(\theta ))} {{Mg}}({{y}}(\theta )){\rm{d}}\theta =\\ \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{{\rm{e}}^{(\sigma - 1){t_k}^ - }}} } \lambda _{ik}^2{y_i}({t_k}^ - ){P_{ij}}{y_j}({t_k}^ - ) + \\ \end{gathered} $$ $$ \begin{gathered} \sum\limits_{i = 1}^n {{{\rm{e}}^{(\sigma - 1){t_k}^ - }}{d_i}} \int_0^{{\lambda _{ik}}{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s + \\ \int_{{t_k}^ - - \tau }^{{t_k}^ - } {{{\rm{e}}^{\sigma \theta }}{{{ g}}^{\rm{T}}}({{y}}(\theta ))}{{ Mg}}(y(\theta )){\rm{d}}\theta \end{gathered} $$ 由于
$\lambda _{ik}^2 \leqslant 1$ ，则$\int_0^{{\lambda _{ik}}{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s \leqslant \int_0^{{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s$ 。于是，得：$$\begin{split} &\qquad\qquad\qquad\quad V({t_k}) \leqslant V({t_k}^ - ) = \\ & {{\rm{e}}^{(\sigma - 1){t_k}^ - }}{{{y}}^{\rm{T}}}({t_k}^ - ){{Py}}({t_k}^ - )\; + \sum\limits_{i = 1}^n {{{\rm{e}}^{(\sigma - 1){t_k}^ - }}{d_i}} \int_0^{{y_i}({t_k}^ - )} {{g_i}(s)} {\rm{d}}s + \\ &\qquad\qquad\quad \;\;\int_{{t_k}^ - - \tau }^{{t_k}^ - } {{{\rm{e}}^{\sigma \theta }}{{{g}}^{\rm{T}}}({{y}}(\theta ))} {{Mg}}({{y}}(\theta )){\rm{d}}\theta \end{split} $$ (17) 由式(16)和式(17)可知，当
$t \geqslant \log {t_0}$ 时，得：$$V(t) \leqslant V(\log {t_0})\quad t \in ({t_{k - 1}},{t_k}],\;k \in {\mathbb{N}^ * }$$ 当且仅当
${{y}}(t) \!=\! {{g}}({{y}}(t)) \!=\! {{g}}({{y}}(t\! -\! \tau ))$ 时，$\dot V(t)\! =\! 0$ 。且：$$ \begin{gathered} V(\log {t_0}) = {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{{{y}}^{\rm{T}}}(\log {t_0}){{Py}}(\log {t_0}) + \\ \mathop \sum \limits_{i = 1}^n {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{d_i}\int_0^{{y_i}(\log {t_0})} {{g_i}(s)} {\rm{d}}s + \\ \int_{\log \tilde q{t_0}}^{\log {t_0}} {{{\rm{e}}^{\sigma \theta }}{{{g}}^{\rm{T}}}({{y}}(\theta ))} {{Mg}}({{y}}(\theta )){\rm{d}}\theta \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{{{y}}^{\rm{T}}}(\log {t_0}){{Py}}(\log {t_0})+\\ \mathop \sum \limits_{i = 1}^n {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{d_{^i}}{g_i}({y_i}(\log {t_0})){y_i}(\log {t_0}) + \\ \int_{{\rm{log}}\tilde q{t_0}}^{\log {t_0}} {{{\rm{e}}^{\sigma \theta }}{{{y}}^{\rm{T}}}(\theta ))} {{LMLy}}(\theta ){\rm{d}}\theta \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}\left( {{{{y}}^{\rm{T}}}(\log {t_0}){{Py}}(\log {t_0}) + {{{y}}^{\rm{T}}}(\log {t_0}){{LDy}}(\log {t_0})} \right) + \\ \left[ {{\lambda _M}({{LML}})t_0^\sigma {\sigma ^{ - 1}}} \right]\mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{y}}(\theta )|{|^2} \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{\lambda _M}({{P}} + {{LD}})||{{y}}(\log {t_0})|{|^2} + \\ \left[ {{\lambda _M}({{LML}})t_0^\sigma {\sigma ^{ - 1}}} \right]\mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{y}}(\theta )|{|^2} \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}\{ {\lambda _M}({{P}} + {{LD}}) + \\ {\lambda _M}\left( {{{LML}}} \right)t_0^\sigma {\sigma ^{ - 1}}\} \mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\eta }}(\theta )|{|^2} \end{gathered} $$ 又因为：
$${{\rm{e}}^{(\sigma - 1)t}}{{{y}}^{\rm{T}}}(t){{Py}}(t) \leqslant V(t) \leqslant V(\log {t_0})$$ 于是，有：
$$\begin{gathered} {\lambda _m}{\rm{(}}{{P}}{\rm{)}}{{\rm{e}}^{(\sigma - 1)t}}||{{y}}(t)|{|^2} \leqslant {{\rm{e}}^{(\sigma - 1)t}}{{{y}}^{\rm{T}}}(t){{Py}}(t) \leqslant \\ {{\rm{e}}^{(\sigma - 1)\log {t_0}}}{\rm{\{ }}{\lambda _M}({{P}} + {{LD}}) + \\ \;\;{\lambda _M}\left( {{{LQL}}} \right)t_0^\sigma {\sigma ^{ - 1}}{\rm{\} }}\mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\eta}} (\theta )|{|^2} \end{gathered} $$ 所以：
$$\begin{split} &\;\;\qquad{\left\| {{{y}}(t)} \right\|^2} \leqslant \frac{1}{{{\lambda _m}({{P}})}}\{ {\lambda _M}({{P}} + {{LD}}) + \\ &{\lambda _M}({{LQL}})t_0^\sigma {\sigma ^{ - 1}}\} \mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\eta}} (\theta )|{|^2}{{\rm{e}}^{ - \lambda (t - \log {t_0})}} \end{split} $$ (18) 其中
$ \lambda = \sigma - 1 > 0$ 。取：$$\beta = \frac{1}{{{\lambda _m}({{P}})}}{\rm{\{ }}{\lambda _M}({{P}} + {{LD}}) + {\lambda _M}({{LQL}})(1 - {{\rm{e}}^{ - \sigma \tau }}){\sigma ^{ - 1}}{\rm{\} }} \geqslant 1$$ 因此，由(18)，得：
$$||{{y}}(t)|{|^2} \leqslant \beta \mathop {\sup }\limits_{\log \tilde q{t_0} \leqslant \theta \leqslant \log {t_0}} ||{{\eta}} (\theta )||_{}^2{{\rm{e}}^{ - \lambda (t - \log {t_0})}}\;$$ 余下部分与定理1相同。
在定理1、2和3中，当
$\sigma = 1$ 时，则全局多项式稳定性和指数稳定性都退化为全局渐近稳定性。推论1 假设
$(H)$ 成立，${x_i}({t_k}) = {\lambda _{ik}}{x_i}({t_k}^ - )$ ，$\lambda _{ik}^2 \leqslant 1$ 。若存在矩阵${{P}} > 0$ ，${{M}} = {\rm{diag}}({m_1},{m_2}, \cdots ,{m_n}) > 0$ ，${{D}} = {\rm{diag}}({d_1},{d_2}, \cdots ,{d_n}) > 0$ ，使得：$${{\varXi}} = \left( {\begin{array}{*{20}{c}} { - {{{C}}^{\rm{T}}}{{P}} - {{{P}}^{\rm{T}}}{{C}}}&{{{{P}}^{\rm{T}}}{{A}}}&{{{{B}}^{\rm{T}}}{{P}}} \\ {{{{A}}^{\rm{T}}}{{P}}}&{ - {{D}}{{C}}{{{L}}^{ - 1}} + {{M}} + {{D}}{{A}}}&{\dfrac{1}{2}{{{D}}^{\rm{T}}}{{B}}} \\ {{{{P}}^{\rm{T}}}{{B}}}&{\dfrac{1}{2}{{{B}}^{\rm{T}}}{{D}}}&{ - {{\tilde q}^{ - {\rm{l}} }}{{M}}} \end{array}} \right) < 0$$ 则系统(1)和(2)的平衡点
${{{x}}^ * }$ 和${{{z}}^ * }$ 是全局渐近稳定的。推论2 假设(H)成立，
${x_i}({t_k}) = {\lambda _{ik}}{x_i}({t_k}^ - )$ ，$\lambda _{ik}^2 \leqslant 1$ 。当${q_i} = q,i = 1,2, \cdots ,n$ 时，若存在矩阵${{P}} > 0$ ，${{M}} > 0$ 和${{D}} = {\rm{diag}}({d_1},{d_2}, \cdots ,{d_n}) > 0$ ，使得：$${{\varXi}} = \left( {\begin{array}{*{20}{c}} { - {{{C}}^{\rm{T}}}{{P}} - {{{P}}^{\rm{T}}}{{C}}}&{{{{P}}^{\rm{T}}}{{A}}}&{{{{B}}^{\rm{T}}}{{P}}} \\ {{{{A}}^{\rm{T}}}{{P}}}&{ - {{D}}{{C}}{{{L}}^{ - 1}} + {{M}} + {{D}}{{A}}}&{\dfrac{1}{2}{{{D}}^{\rm{T}}}{{B}}} \\ {{{{P}}^{\rm{T}}}{{B}}}&{\dfrac{1}{2}{{{B}}^{\rm{T}}}{{D}}}&{ - {q^{ - 1}}{{M}}} \end{array}} \right) < 0$$ 则系统(1)和(2)的平衡点
${{{x}}^ * }$ 和${{{z}}^ * }$ 都是全局渐近稳定的(GAS)。虽然系统(1)和(2)等价，且具有相同的平衡点
${{{x}}^ * } = {{{z}}^ * }$ ，但是他们的稳定性却不同，系统(2)的平衡点${{{z}}^ * }$ 是GES，系统(1)的平衡点${{{x}}^ * }$ 却是GPS。当
$\lambda = 0.5\text{、}1\text{、}1.5\text{、}3$ 时，函数$y = {t^{ - \lambda }}$ 和$y = {{\rm{e}}^{ - \lambda t}}$ 的函数图像，如图1所示，可以直观地看到多项式稳定性的收敛速度比指数稳定性的慢一些。当系统(1)无脉冲影响时，本文所得结论仍然成立。
图 1 函数
$y = {t^{ - \lambda }}$ 和$y = {{\rm{e}}^{ - \lambda t}}$ 的图像 -
例1 考虑二维如下RNNs：
$$\left\{ \begin{aligned} & {{\dot x}_1}(t) = - 4.6{x_1}(t) - 2{f_1}({x_1}(t)) + {f_2}({x_2}(t)) + \\ & \quad 1.5{f_1}({x_1}(0.4t)) + 0.2{f_2}({x_2}(0.8t)) + {u_1}\quad t \ne {t_k} \\ & {{\dot x}_2}(t) = - 3{x_2}(t) + {f_1}({x_1}(t)) - 3{f_2}({x_2}(t)) + \\ & \quad 1.4{f_1}({x_1}(0,4t)) + 2.3{f_2}({x_2}(0.8t)) + {u_2}\quad t \ne {t_k} \\ &{x_1}({t_k}) = {E_{1k}}{x_1}({t_k}^ - )\quad k \in {\mathbb{N}^ * } \\ & {x_2}({t_k}) = {E_{2k}}{x_2}({t_k}^ - )\quad k \in {\mathbb{N}^ * } \end{aligned} \right.$$ (19) 式中，
${f_1}({x_1}(t)) = 0.25(|{x_1}(t) + 1| - |{x_1}(t) - 1|)$ ；${f_2}({x_2}(t))=$ $ \sin (0.25{x_2}(t)) + 0.25{x_2}(t)$ ；$ q_1=0.4$ ；$ q_2=0.8$ ；${{u}} = {(0,0)^{\rm{T}}}$ ；${l_i} = 0.5,\;i = 1,2$ ；${{L}} = {\rm{diag}}(0.5,0.5)$ ；${E_{1k}} = - 0.23$ ；${E_{2k}} = 0.38$ 。由式(19)，可知：$${{C}} = \left( {\begin{array}{*{20}{c}} \begin{array}{l} 4.6 \\ 0 \\ \end{array} &\begin{array}{l} 0 \\ 3 \\ \end{array} \end{array}} \right),{{A}} = \left( {\begin{array}{*{20}{c}} { - 2}&1 \\ 1&{ - 3} \end{array}} \right),{{B}} = \left( {\begin{array}{*{20}{c}} {1.5}&{0.2} \\ {1.4}&{2.3} \end{array}} \right)$$ 取
${{P}} = {{Q}} = {{D}} = {\rm{diag}}(1,1)$ ，$\sigma = 1.1$ ，通过应用Matlab计算，得：$$ {{\varXi}} = \left( {\begin{array}{*{20}{c}} { - 9.100\;0}&0&{ - 1.950\;0}&{1.000\;0}&{1.500\;0}&{1.400\;0}\\ 0&{ - 5.900\;0}&{1.000\;0}&{ - 2.950\;0}&{0.200\;0}&{2.300\;0}\\ { - 1.950\;0}&{1.000\;0}&{ - 10.200\;0}&{1.000\;0}&{0.750\;0}&{0.100\;0}\\ {1.000\;0}&{ - 2.950\;0}&{1.000\;0}&{ - 8.000\;0}&{0.700\;0}&{{\rm{1}}.150\;0}\\ {1.500\;0}&{1.400\;0}&{ 0.750\;0}&{0.700\;0}&{ - 2.739\;9}&0\\ {0.200\;0}&{2.300\;0}&{0.100\;0}&{1.150\;0}&0&{ - 2.739\;9} \end{array}} \right) $$ 且
${\lambda _\varXi } $ = −12.756 7, −10.255 2, −7.726 6, −4.182 5, −2.375 5,$ - 1.383\;2$ ，即${{\varXi}} < 0$ 。由定理1可知，系统(19)是GPS。系统(19)的平衡点为${{{x}}^ * } = {(0,0)^{\rm{T}}}$ 。时间响应轨线如图2～图5所示。
图 2 系统(19)无脉冲时，
${x_1}(t)$ 的时间响应轨线
图 3 系统(19)带脉冲时，
${x_1}(t)$ 的时间响应轨线
图 4 系统(19)无脉冲时，
${x_2}(t)$ 的时间响应轨线
图 5 系统(19)带脉冲时，
${x_2}(t)$ 的时间响应轨线例2 考虑三维具比例时滞RNNs：
$$\left\{ \begin{aligned} & {{\dot x}_1}(t) = - 3.5{x_1}(t) - 2{f_1}({x_1}(t)) + 1.2{f_2}({x_2}(t)) + \\ & \quad {f_3}({x_3}(t)) + 0.5{f_1}({x_1}(0.3t)) + 0.3{f_2}({x_2}(0.5t) + {u_1} \\ & {{\dot x}_2}(t) = - 5.6{x_2}(t) + 2{f_1}({x_1}(t)) - {f_2}({x_2}(t))) + \\ & \quad 3.2{f_1}({x_1}(0.3t) + 2.4{f_2}({x_2}(0.5t) + {u_2} \\ &{{\dot x}_3}(t) = - 3.7{x_3}(t) + 3{f_1}({x_1}(t)) + 2{f_3}({x_3}(t)) + \\ & \quad 3{f_2}({x_2}(0.5t) + 1.5{f_3}({x_3}(0.8t)) + {u_3} \\ & {x_1}({t_k}) = {E_{1k}}{x_1}({t_k}^ - )\quad k \in {\mathbb{N}^ * } \\ &{x_2}({t_k}) = {E_{2k}}{x_2}({t_k}^ - )\quad k \in {\mathbb{N}^ * } \\ & {x_3}({t_k}) = {E_{3k}}{x_3}({t_k}^ - )\quad k \in {\mathbb{N}^ * } \end{aligned} \right.$$ (20) 式中，
${f_i}({x_i}(t)) = \tanh (0.5{x_i}(t)),\;i = 1,2$ ；${f_3}({x_3}(t)) = $ $\cos ({x_3}(0.2t))$ ，$ q_1=0.3,q_2=0.5,q_3=0.8$ ，$\tilde q = 0.3$ ，${{u}} = {(0, - 3,4)^{\rm{T}}}$ ，${E_{1k}} = 0.63$ ，${E_{2k}} = - 0.28$ ，${E_{3k}} = 0.18$ ，$k \in {\mathbb{N}^ * }$ 。Lipschitz常数为${l_i} = 0.5,\;i = 1,2$ ，$ l_3=0.2$ ，${{L}} = {\rm{diag}}(0.5,0.5,0.2)$ 。由式(20)，可知：
$$ \begin{split} &{{C}} = \left( \begin{array}{l} \begin{array}{*{20}{c}} {3.5}&0&0 \end{array} \\ \begin{array}{*{20}{c}} 0&{5.6}&0 \end{array} \\ \begin{array}{*{20}{c}} 0&{\;\;0}&{\;3.7} \end{array} \\ \end{array} \right),\;{{A}} = \left( {\begin{array}{*{20}{c}} { - 2}&{1.2}&1 \\ 2&{ - 1}&0 \\ 3&0&2 \end{array}} \right),\\ &\qquad\quad\;\; \;{{B}} = \left( {\begin{array}{*{20}{c}} {1.5}&{1.3}&0 \\ {3.2}&{2.4}&0 \\ 0&3&{1.5} \end{array}} \right) \end{split} $$ 利用Matlab找到
$\sigma = 1.236\;7$ 和以下矩阵：$$ \begin{aligned} & {{{P}} = \left( {\begin{array}{*{20}{c}} {1.989\;7}&{0.002\;3}&{0.104\;5} \\ {0.002\;3}&{1.002\;3}&{0.119\;8} \\ {0.104\;5}&{0.119\;8}&{0.998\;7} \end{array}} \right)}\\ & {{{M}} = \left( {\begin{array}{*{20}{c}} {1.567\;8}&0&0 \\ 0&{2.089\;7}&0 \\ 0&0&{3.045\;7} \end{array}} \right)}\\ & {{{D}} = \left( {\begin{array}{*{20}{c}} {2.012\;3}&0&0 \\ 0&{1.478\;5}&0 \\ 0&0&{3.027\;83} \end{array}} \right)} \end{aligned} $$ 得到矩阵
${{\varXi}} $ ，且${\lambda _\varXi } $ = −48.370 1, −21.586 4, −2.085 5, −5.302 1, −17.523 0, −8.011 9, −10.002 8, −14.865 7, −12.968 0, 由此可知$\varXi < 0$ ，由定理1，可知系统(20)是GPS。由Matlab计算，得系统(20)的平衡点为${{{x}}^ * } = (0.154\;5,$ $- 0.539\;5,1.880\;5{)^{\rm{T}}}$ 。时间响应曲线如图6～11所示。
图 6 系统(20)无脉冲时，
${x_1}(t)$ 的时间响应轨线
图 7 系统(20)带脉冲时，
${x_1}(t)$ 的时间响应轨线
图 8 系统(20)无脉冲时，
${x_2}(t)$ 的时间响应轨线
图 9 系统(20)带脉冲时，
${x_2}(t)$ 的时间响应轨线
图 10 系统(20)无脉冲时，
${{{x}}_3}(t)$ 的时间响应轨线
图 11 系统(20)带脉冲时，
${{{x}}_3}(t)$ 的时间响应轨线 -
本文通过构造合适的Lyapunov泛函，研究了一类具比例时滞IRNNs的全局多项式稳定性，所得准则是以LMI形式给出的，便于应用Matlab验证。可调参数的引入使得所得条件的适用范围扩大了。激活函数较广泛，可以是无界的，也可以是不可微的。本文的研究方法也适合于具比例时滞RNNs的多项式周期性、多项式同步性和多项式耗散性等动力学行为的研究。
Global Polynomial Stability of a Class of Impulsive Recurrent Neural Networks with Proportional Delays
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摘要: 该文对一类具比例时滞脉冲递归神经网络给出全局多项式稳定性定义。通过引入可调参数，构造适合的Lyapunov泛函和运用线性矩阵不等式 (LMI) 的方法，对该系统的全局多项式稳定性进行讨论，得到了保证该系统全局多项式稳定的判定准则，且这些准则以LMI的形式给出的，方便应用Matlab工具箱进行验证。该文还揭示了多项式稳定性与指数稳定性之间的关系，最后通过数值算例验证了所得准则的准确性。
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关键词:
- 脉冲效应 /
- Lyapunov泛函 /
- 比例时滞 /
- 多项式稳定性 /
- 递归神经网络
Abstract: The definition of global polynomial stability for a class of impulsive recurrent neural networks (IRNNs) with proportional delays is given. By introducing adjustable parameters, several suitable Lyapunov functionals are constructed and the method of linear matrix inequality (LMI) is used to discuss the global polynomial stability of the system. Several criteria for guaranteeing the global polynomial stability of the system are obtained. And these criteria are given in the form of LMI, which is convenient to use Matlab toolbox for verification. The relationship between polynomial stability and exponential stability is revealed. The criteria are verified by numerical examples. -
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