神经网络是近几年的一个热门话题, 它已被广泛应用到科学和工程的各个领域, 如信号处理、模型识别、关联存储器、电路设计和并行计算^[1-5].众所周知, 由于放大器的有限切换速度, 时间延迟不可避免地出现在神经网络中, 并且可能导致系统振荡和不稳定性.因此, 时间延迟神经网络的稳定性分析问题是神经网络领域中的重点.最近, 许多研究结果已经提出了稳定性问题的处理, 并且带有时间延迟的神经网络的性能已经得到改进^{[6-12, 14-15]}.

时滞分割作为将延迟区间划分为许多子区间的延迟分解方法可以获得较低保守性的稳定性条件.前面的方法是Lyapunov-Krasovskii函数在整个区间的导数, 其使用时滞分割法将范围向下到每个子区间.在2001年, 文献[9]提出了第一个有效延迟划分的理念, 它让我们知道可以降低稳定性标准的保守性方法.当分区越来越细时, 矩阵公式和计算负担变得越来越复杂, 但它可以显示最大延迟边界的改善.

最近, 用于神经网络稳定性分析的时滞分割法已经得到了大量的研究结果, 其以线性矩阵不等式的形式求导而得并且通过基于优化的技术可以容易地求解.在文献[13, 17, 19-20]中, 提出了广义的时滞分割方法, 它只是划分了时间延迟区间的一部分并且分解成l个等效段, 并且使用一些常规方式来处理Lyapunov-Krasovski函数, 诸如在每个延迟子区间中利用不同的自由加权矩阵.文献[25]中提出了一种用于中性型神经网络的延迟相关稳定性分析的延迟分割方法, 这使得时间延迟区间[0, r]被分解为l个等效段, 但时间延迟不是变量.与其他发展相比, 文献[18]中提出的方法表现出显著的优势, 它首先假设在某个特定子区间的时间延迟, 然后延伸到任意子区间.然而, 文献[18]中提出的Lyapunov-Krasovskii函数仅包括一些简单的积分和双重积分项, 并不包含三重积分项.增加一些三重积分的Lyapunov-Krasovskii函数对于降低保守性是非常有用的.使用延迟分解方法和互逆凸技术, 对于伴有时变延迟的神经网络, 一些改进的稳定性标准在文献[23-24]中已经得到了发展.当分区变得更精细时, 保守性显著降低.为了进一步加强研究结果, 本文采用新不等时滞分割法, 并通过引入变量$\rho_{k}(t)$ ($\rho_{k}(t)=\frac{\tau(t)-\tau^{-}}{r \times 2^{r+1-k}}$) 以确定$\tau(t)$作为子区间的时间延迟.

参考上面提到的讨论, 第一次尝试研究新不等时滞分割法以处理具有离散和分布式延迟的神经网络稳定性分析的问题.在本文中, 通过构造一个新的Lyapunov-Krasovskii函数, 其中一些项包含三重积分, 比如

$ \int_{-\tau^{-}-\frac{j_{i}}{2^{i}}\delta-(i-1)\delta}^{-\tau^{-} -\frac{j_{i}-1}{2^{i}}\delta-(i-1)\delta}\int_{\eta}^{-\tau^{-} -\frac{j_{i}-1}{2^{i}}\delta-(i-1)\delta} \int_{t+\theta}^{t}\dot{x}^T(s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta, $

$ \int_{-\tau^{-}-\frac{j_{i}}{2^{i}}\delta-(i-1)\delta}^{-\tau^{-}-\frac{j_{i}-1}{2^{i}} \delta-(i-1)\delta}\int_{-\tau^{-}-\frac{j_{i}}{2^{i}}\delta -(i-1)\delta}^{\eta}\int_{t+\theta}^{t}\dot{x}^T(s)W_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta, $

这在改进低保守性的结果中起到重要作用.此外, 新不等时滞分割法还用来分析神经网络全局渐近稳定性的问题.首先, 通过使用新不等时滞分割法, 将时间延迟间隔$[\tau^{-}, \tau^{+}]$分为$r$段, 然后我们将每个子区间$[\tau^{-}+(k-1)\delta, \tau^{-}+k\delta]$划分为$2^{r+1-k}$段.其次, 与文献[25]相比, 没有解释子区间中的时变延迟$\tau(t)$.为了得到准确的结果, 本文通过引入时间变量$\rho_{k}(t)$来细分时间延迟区间$[\tau^{-}, \tau^{+}]$.在不等时滞分割法中, 每个子区间中存在不同的$\rho_{k}(t)$.这是本文的亮点, 在之前的文献中从来没有提出过.第三, 延迟区间$[\tau^{-}, \tau^{+}]$被分解为$2^{r+3}-2$个子区间, 每个子区间不相等, 在每个子区间中应用牛顿莱布尼兹公式, 并且选择不同的无权矩阵, 这将导致较低保守性的结果.与文献[18, 23-25]相比, 新不等时滞分割法是本文的亮点.此外, 与以前的结果相比, 为了降低保守性的目的, 一个新积分不等式被应用在互逆凸不等式中.通过使用线性矩阵不等式来评估低保守性的稳定性标准.最后, 给出了一些示例以显示所提方法的有效性.注:该文中, $\mathbb{R} ^n$表示$n$维欧几里德空间, $\mathbb{R} ^{n \times n}$是所有$n \times n$实矩阵的集合.对于对称矩阵$X$, 符号$X> 0$ ($X\ge 0$) 意味着是实对称正定矩阵 (正半定), 对于对称矩阵$X$和$Y$, 符号$X> Y$ ($X\ge Y$) 意味着矩阵$X-Y$是正定的 (非负的).矩阵$A^T$是矩阵$A$的转置, 符号$*$被用作包含对称的项的省略, $I$表示具有适当维度的单位矩阵, $O_{m \times n}$表示具有$m \times n$维的零矩阵. $col[x_{1}, x_{2}, \cdots, x_{n}]$表示$[x_{1}^{T}, x_{2}^{T}, \cdots, x_{n}^{T}]^{T}$.如果没有明确说明, 则假设矩阵具有兼容的维度.

对于一个离散分布式延迟神经网络模型

$ \begin{eqnarray} \dot{x}(t)=-Ax(t)+Bf(x(t))+Cf(x(t-\tau(t)))+D\int_{t-d(t)}^{t}f(x(s)){\rm d}s, \end{eqnarray} $

(2.1)

其中$x(t)=[x_1(t), \cdots, x_{n}(t)]^T\in \Re^n$是神经元状态向量; $f(x(\cdot))=[f_{1}(x_1(\cdot)), \cdots, f_{n}(x_n(\cdot))]^T \in \Re^n$是神经元激活函数向量; $A=diag(a_{1}, \cdots, a_{n})$且$a_{i}>0, i=1, 2, \cdots, n$. $B\in \mathbb{R}^{n\times n}$是互连权重矩阵, $C, D\in \mathbb{R}^{n\times n}$是延迟互连权重矩阵.在全文中, 我们有以下假设:

假设2.1   在 (2.1) 式中的时间延迟$\tau(t)$, $d(t)$是连续时变函数并满足

$ \left\{ {\begin{array}{*{20}{l}} {0 \le {\tau ^-} \le \tau (t) \le {\tau ^ + }, \dot \tau (t) \le \mu, }\\ {0 \le d(t) \le \;d_3^ +, } \end{array}} \right. $

(2.2)

对于任意整数$r\geq 1$, $k\geq 1$, $i\geq 1$, $i=1, 2, \cdots, 2^{r+1-k}$, $k=1, 2, \cdots, r$, 令$\delta=\frac{\tau^{+}-\tau^{-}}{r}$, $\rho_{k}(t)=\frac{\tau(t)-\tau^{-}}{r \times 2^{r+1-k}}$, 区间$[\tau^{-}, \tau^{+}]$可分解为$r$段.然后将每个子区间$[\tau^{-}+(k-1)\delta, \tau^{-}+k\delta]$分解为$2^{r+1-k}$段.对于每个子区间$[\tau^{-}+(k-1)\delta+\frac{i-1}{2^{r+1-k}}\delta, \tau^{-}+(k-1)\delta+\frac{i}{2^{r+1-k}}\delta]$=$[\tau^{-}+(k-1)\delta+\frac{i-1}{2^{r+1-k}}\delta, \tau^{-}+(k-1)\delta+\frac{i-1}{2^{r+1-k}}\delta+\rho_{k}(t)]\bigcup[\tau^{-}+(k-1)\delta+\frac{i-1}{2^{r+1-k}}\delta+\rho_{k}(t), \tau^{-}+(k-1)\delta+\frac{i}{2^{r+1-k}}\delta]$.另一方面, 对任意$t\geq 0$, 存在整数$i=1, 2, \cdots, 2^{r+1-k}$, $k=1, 2, \cdots, r$, 使$\tau(t)\in[\tau^{-}+(k-1)\delta+\frac{i-1}{2^{r+1-k}}\delta, \tau^{-}+(k-1)\delta+\frac{i}{2^{r+1-k}}\delta]$.

假设2.2  系统 (2.2) 中的任意激活函数$f_{i}(\cdot)$是连续有界的, 并满足以下不等式

$ \begin{eqnarray} \sigma_{i}^-\leq\frac{f_{i}(a)-f_{i}(b)}{a-b}\leq \sigma_{i}^+, k=1, 2, \cdots, n, \quad\mbox{且}\quad f_{i}(0)=0, \end{eqnarray} $

(2.3)

其中$a$, $b \in{\mathbb{R}}$, $a\neq b$, $\sigma_{i}^-$, $\sigma_{i}^+$是已知常量.

注2.2   $\sigma_{i}^-$, $\sigma_{i}^+(i=1, 2, \cdots, n)$是一些常量, 在假设2.2中可以是正的, 负的和零.因此, 这种类型的激活函数比通常的激活函数和分段线性函数$f_{i}(u)=\frac{1}{2}(|u_{i+1}|-|u_{i}|)$显然更加普遍, 这个分段线性函数有利于获得低保守性结果.

在得出主要结论之前, 我们将引用之后要使用的几个引理:

引理2.1^[21]对于任意常矩阵$V$, $W\in{\mathbb{R}^{n\times n}}$和$M>0$, 标量$b>a$, 矢量函数$V:[a, b]\rightarrow{\mathbb{R} ^m}$, 特别是以下集合有明确的定义,

$ \begin{eqnarray} (b-a)\int_{a}^{b}{V^T(s)MV(s)}{\rm d}s\geq \bigg(\int_{a}^{b}V(s){\rm d}s\bigg)^TM\int_{a}^{b}V(s){\rm d}s, \end{eqnarray} $

(2.4)

$ \begin{eqnarray} \frac{\tau^{2}}{2}\int_{-\tau}^{0}\int_{t+\theta}^{t}{W^T(s) MW(s){\rm d}s{\rm d}\theta}\geq \bigg(\int_{-\tau}^{0}\int_{t+\theta}^{t}W(s){\rm d}s{\rm d}\theta \bigg)^TM\int_{-\tau}^{0}\int_{t+\theta}^{t}W(s){\rm d}s{\rm d}\theta. \end{eqnarray} $

(2.5)

引理2.2^[22]   令$f_{1}$, $f_{2}, \cdots, f_{n}: \mathbb{R}^{m}\rightarrow \mathbb{R} $在$\mathbb{R}^{m}$公开的子集合D中有正值.那么, 在D之上$f_{{i}}$的互逆凸组合满足

$ \begin{eqnarray*} {\min_{\{\alpha_i|\alpha_i>0, {\sum\limits_i}\alpha_i=1\}}}\sum\limits_i\frac{1}{\alpha_i}f_i(t)=\sum\limits_if_i(t) +{\max_{g_{i, j}(t)}}\sum\limits_{i\neq j}g_{i, j}(t), \end{eqnarray*} $

服从

$ \begin{eqnarray*} \{g_{i, j}:\mathbb{R}^m\mapsto \mathbb{R}, g_{j, i}(t)\triangleq g_{i, j}(t), \left[ \begin{array}{cc} f_i(t) ~~&g_{i, j}(t)\\ g_{j, i}(t) ~~&f_j(t) \end{array} \right]\geq 0\}. \end{eqnarray*} $

在此部分, 我们将通过LMI方法来得到主要结论, 为了方便陈述, 在下文中, 我们定义

$ \Sigma^+=diag[\sigma_{1}^{+}, \cdots, \sigma_{n}^{+}] \quad \mbox{和}\quad \Sigma^-=diag[\sigma_{1}^{-}, \cdots, \sigma_{n}^{-}], $

$ \begin{eqnarray*} \eta_{1}^T(t)& = &\Big[ x^T(t-\frac{1}{2}\delta) ~~ x^T(t-\frac{2}{2^1}\delta) ~~ x^T(t-\delta-\frac{1}{2^2}\delta) ~~ \cdots ~ ~ x^T(t-\delta-\frac{2^2}{2^2}\delta)~~ \cdots \\ && x^T(t-(r-1)\delta-\frac{1}{2^r}\delta) ~~ \cdots ~~ x^T(t-(r-1)\delta-\frac{2^{r}-1}{2^r}\delta) ~~ x^T(t-r\delta)\Big], \end{eqnarray*} $

$ \begin{eqnarray*} \eta_{2}^T(t)& =& \Big[ x^T(t-\frac{1}{2}\delta-\rho_{1}(t))~~ x^T(t-\delta-\rho_{1}(t)) ~~ x^T(t-\delta-\frac{1}{2^2}\delta-\rho_{2}(t))\\ && \cdots ~~ x^T(t-(r-1)\delta-\frac{1}{2^r}\delta-\rho_{r}(t)) ~ ~ x^T(t-(r-1)\delta-\frac{2}{2^r}\delta-\rho_{r}(t)) \\ && \cdots~~ x^T(t-(r-1)\delta-\frac{2^{r}-1}{2^r}\delta-\rho_{r}(t)) ~~ x^T(t-r\delta-\rho_{r}(t))\Big], \end{eqnarray*} $

$ \begin{eqnarray*} \eta_{3}^T(t)& =&\Big[ \frac{1}{\rho_{1}(t)}\int_{t-\tau^{-}-\frac{1}{2}\delta}^{t-\tau^{-} -\frac{1}{2}\delta+\rho_{1}(t)}x^T(s){\rm d}s~~ \frac{1}{\rho_{1}(t)}\int_{t-\tau^{-}-\delta}^{t-\tau^{-} -\delta+\rho_{1}(t)}x^T(s){\rm d}s\\ && \frac{1}{\rho_{2}(t)}\int_{t-\tau^{-}-\delta-\frac{1}{2^2} \delta}^{t-\tau^{-}-\delta-\frac{1}{2^2}\delta+\rho_{2}(t)}~~ x^T(s){\rm d}s ~~ \cdots ~~ \frac{1}{\rho_{2}(t)}\int_{t-\tau^{-}-\delta- \frac{2^2}{2^2}\delta}^{t-\tau^{-}-\delta-\frac{2^2}{2^2}\delta +\rho_{2}(t)}x^T(s){\rm d}s \\ && \cdots ~~ \frac{1}{\rho_{r}(t)}\int_{t-\tau^{-}-(r-1)\delta- \frac{1}{2^r}\delta}^{t-\tau^{-}-(r-1)\delta-\frac{1} {2^r}\delta+\rho_{r}(t)} ~~ x^T(s){\rm d}s~~ \cdots \\ &&\frac{1}{\rho_{r}(t)}\int_{t-\tau^{-}-(r-1)\delta- \frac{2^r-1}{2^r}\delta}^{t-\tau^{-}-(r-1)\delta-\frac{2^r-1}{2^r}\delta +\rho_{r}(t)}x^T(s){\rm d}s ~~ \frac{1}{\rho_{r}(t)}\int_{t-\tau^{-}-r\delta}^{t-\tau^{-}-r \delta+\rho_{r}(t)}x^T(s){\rm d}s\Big], \end{eqnarray*} $

$ \begin{eqnarray*} \eta_{4}^T(t) &=& \Big[ \frac{1}{\frac{1}{2}\delta-\rho_{1}(t)}\int_{t-\tau^{-}- \frac{1}{2}\delta+\rho_{1}(t)}^{t-\tau^{-}}x^T(s){\rm d}s ~~ \frac{1}{\frac{1}{2}\delta-\rho_{1}(t)}\int_{t-\tau^{-} -\delta+\rho_{1}(t)}^{t-\tau^{-}-\frac{1}{2}\delta}x^T(s){\rm d}s \\ && \frac{1}{\frac{1}{2^2}\delta-\rho_{2}(t)}\int_{t-\tau^{-}-\delta- \frac{1}{2^2}\delta+\rho_{2}(t)}^{t-\tau^{-}-\delta} x^T(s){\rm d}s ~ ~ \cdots ~~ \\ && \frac{1}{\frac{1}{2^2}\delta-\rho_{2}(t)}\int_{t-\tau^{-} -\delta-\frac{2^2}{2^2}\delta+\rho_{2}(t)}^{t-\tau^{-}-\delta- \frac{2^2-1}{2^2}}x^T(s){\rm d}s ~~ \cdots ~~ \\ && \frac{1}{\frac{1}{2^r}\delta-\rho_{r}(t)}\int_{t-\tau^{-}-(r-1) \delta-\frac{1}{2^r}\delta+\rho_{r}(t)}^{t-\tau^{-} -(r-1)\delta}x^T(s){\rm d}s ~~\cdots \\ && \frac{1}{\frac{1}{2^r}\delta-\rho_{r}(t)}\int_{t-\tau^{-}-(r-1) \delta-\frac{2^r-1}{2^r}\delta+\rho_{r}(t)}^{t-\tau^{-}-(r-1) \delta-\frac{2^r-2}{2^r}}x^T(s){\rm d}s \\ && \frac{1}{\frac{1}{2^r}\delta-\rho_{r}(t)}\int_{t-\tau^{-} -r\delta+\rho_{r}(t)}^{t-\tau^{-}-(r-1)\delta-\frac{2^r-1} {2^r}\delta}x^T(s){\rm d}s\Big], \end{eqnarray*} $

$ \begin{eqnarray*} \eta^T(t) &=&\Big[ x^T(t) ~~ x^T(t-\tau^{-}) ~~ \eta_{1}^T(t-\tau^{-}) ~~ \eta_{2}^T(t-\tau^{-}) ~ ~\eta_{3}(t) ~~ \eta_{4}^T(t) \\ && f^T(x(t))~~ f^T(x(t-\tau(t))) ~~ x^T(t-\tau(t)) ~~ \int_{t-d(t)}^{t}f^T(x(s)){\rm d}s \Big], \end{eqnarray*} $

$ e_{k}=\left[ O_{n\times(k-1)n} ~~I_n~~ O_{n\times(2^{r+3}-2-k)n} \right], $

我们所提的主要结论如下:

定理3.1  根据假设2.1, 2.2, 神经网络 (2.1) 是全局渐近稳定的, 当存在矩阵$V_{ij_{i}}>0$, $U_{ij_{i}}>0$, $W_{ij_{i}}>0$, $Q_{ij_{i}}>0$, $M_{ij_{i}}>0$, $F_{ij_{i}}>0$, $Q_{i}^{\ast}$, $R_{i}>0$, $\Lambda_{ij_{i}, p}$, $i=1, 2, \cdots, r$, $j_{i}=1, 2, \cdots, 2^i$, $p=1, 2, \cdots, 2^{r+3}-2$, $M>0$, 和正定对角矩阵$K_1, K_2, T, T_1, T_2$使下面的LMIs持有

$ \begin{eqnarray} \Phi_{ij_{i}}=\left[ \begin{array}{cc} Q_{ij_{i}} & Q^{\ast} \\ {} * & Q_{ij_{i}} \\ \end{array} \right]>0 \quad\mbox{, }\quad j_{i}=1, 2, \cdots, 2^i, i=1, 2, \cdots, r, \end{eqnarray} $

(3.1)

$ \begin{eqnarray} \Phi_{2}= \left[\begin{array}{ccccccccc} \Xi & \Lambda_{11} & \Lambda_{12} & \Lambda_{21} & \cdots & \Lambda_{22^{2}} & \cdots & \Lambda_{r1} & \Lambda_{r2^{r}}\\ {}* &-R_{1} & 0 & 0 & \cdots & 0 & \cdots & 0 & 0\\ {}* & * &-R_{1} & 0 & \cdots & 0 & \cdots & 0 & 0\\ {}* & * & * &-R_{2} & \cdots & 0 & \cdots & 0 & 0\\ %\hdotsfor{9}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ {}* & * & * & \cdots & * & 0 & 0 & -R_{r} & 0\\ {}* & * & * & \cdots & * & * & 0 & 0 & -R_{r}\\ \end{array}\right] <0, \end{eqnarray} $

(3.2)

其中

$ \begin{array}{*{20}{c}} {{K_1} = diag({k_{11}}, {k_{12}}, \cdots, {k_{1n}}), \quad {K_2} = diag({k_{21}}, {k_{22}}, \cdots, {k_{2n}}), }\\ {{T_1} = diag({t_{11}}, {t_{12}}, \cdots, {t_{1n}}), \quad {T_2} = diag({t_{21}}, {t_{22}}, \cdots, {t_{2n}}), }\\ {T = diag({t_1}, {t_2}, \cdots, {t_n}), \quad {\Lambda _{i{j_i}}} = col[{\Lambda _{i{j_i}, 1}}, {\Lambda _{i{j_i}, 2}}, \cdots, {\Lambda _{i{j_i}, {2^{r + 3}}-2}}], }\\ {{W_{11}} = {e_{{2^{r + 3}} -5}} -{\Sigma ^ -}{e_1}, \quad {W_{12}} = {\Sigma ^ + }{e_1} - {e_{{2^{r + 3}} - 5}}, }\\ {{W_{13}} = - A{e_1} + B{e_{{2^{r + 3}} - 5}} + C{e_{{2^{r + 3}} - 4}} + D{e_{{2^{r + 3}} - 2}}, }\\ {{W_1} = W_{11}^T{K_1}{W_{13}} + W_{12}^T{K_2}{W_{13}}, } \end{array} $

$ \begin{array}{*{20}{c}} {{W_2} = \sum\limits_{i = 1}^r {\sum\limits_{{j_i} = 1}^{{2^i}} \{ } \frac{\delta }{{{2^i}}}W_{13}^T{V_{i{j_i}}}{W_{13}}- \frac{2}{{{\delta ^i}}}{{({e_{{2^i} + {j_i}- 1}}- {e_{{2^i} + {j_i}}})}^T}{V_{i{j_i}}}({e_{{2^i} + {j_i} - 1}} - {e_{{2^i} + {j_i}}})\}, }\\ {{W_{31}} = col[{e_{{2^{r + 1}} + {2^i} + {j_i}-2}}-{e_{{2^{r + 2}} + {2^i} + {j_i}-4}}], }\\ {{W_{32}} = col[{e_{{2^i} + {j_i}-1}}-{e_{{2^{r + 2}} + {2^{r + 1}} + {2^i} + {j_i}-6}}], }\\ {{W_{33}} = col[{e_{{2^{r + 2}} + {2^{r + 1}} + {2^i} + {j_i}-6}}-{e_{{2^{r + 1}} + {2^i} + {j_i}-2}}], }\\ {{W_{34}} = col[{e_{{2^{r + 2}} + {2^i} + {j_i}-4}}-{e_{{2^i} + {j_i}}}], } \end{array} $

$ \begin{eqnarray*} W_{3}&=&\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}} \bigg[\frac{\delta^{2}}{2}W_{13}^T (\frac{1}{2^i})^{2}(U_{ij_{i}}+W_{ij_{i}})W_{13}-2W_{31}^TU_{ij_i}W_{31}\\ &&-2W_{32}^T U_{ij_i}W_{32}-2W_{33}^TW_{ij_i}W_{33}-2W_{34}^T W_{ij_i}W_{34}\bigg], \end{eqnarray*} $

$ \begin{array}{*{20}{c}} {{W_4} = \sum\limits_{i = 1}^r {\sum\limits_{{j_i} = 1}^{{2^i}} {(e_{{2^i} + {j_i}- 1}^T{M_{i{j_i}}}{e_{{2^i} + {j_i}- 1}}- \frac{\mu }{{{2^i}r}}e_{{2^{r + 1}} + {2^i} + {j_i} - 2}^T{M_{i{j_i}}}{e_{{2^{r + 1}} + {2^i} + {j_i} - 2}})} }, }\\ {{W_5} = \sum\limits_{i = 1}^r {\sum\limits_{{j_i} = 1}^{{2^i}} {\frac{\delta }{{{2^i}}}} } (e_{{2^{r + 1}} + {2^i} + {j_i} - 2}^T{F_{i{j_i}}}{e_{{2^{r + 1}} + {2^i} + {j_i} - 2}} + e_{{2^i} + {j_i} - 1}^T{Q_{i{j_i}}}{e_{{2^i} + {j_i} - 1}}), }\\ {W_1^ * = col[{e_{{2^{r + 1}} + {2^i} + {j_i}-2}}-{e_{{2^i} + {j_i}}}], \quad W_2^ * = col[{e_{{2^i} + {j_i}-1}}-{e_{{2^{r + 1}} + {2^i} + {j_i}-2}}], }\\ {{W^ * } = W{{_1^ * }^T}{Q_{i{j_i}}}W_1^ * + W{{_1^ * }^T}Q_i^ * W_2^ * + W{{_2^ * }^T}Q{{_i^ * }^T}W_2^ * + W{{_2^ * }^T}{Q_{i{j_i}}}W_2^ *, } \end{array} $

$ \begin{array}{*{20}{c}} {{W_6} = e_1^T{\Sigma ^ + }{e_1}-e_{{2^{r + 3}}-5}^TT{e_1}-(1 - \mu )e_{{2^{r + 3}} - 3}^T{\Sigma ^ + }T{e_{{2^{r + 3}} - 3}} + (1 - \mu )e_{{2^{r + 3}} - 4}^TT{e_{{2^{r + 3}} - 3}}, }\\ {{W_7} = {{({d^ + })}^2}e_{{2^{r + 3}} - 5}^TM{e_{{2^{r + 3}} - 5}} - e_{{2^{r + 3}} - 2}^TM{e_{{2^{r + 3}} - 2}}, } \end{array} $

$ \begin{array}{l} {W_8} =-2e_{{2^{r + 3}}-5}^T{T_1}{e_{{2^{r + 3}}-5}} + e_1^T{T_1}({\Sigma ^ + } + {\Sigma ^ - }){e_{{2^{r + 3}} - 5}} + e_{{2^{r + 3}} - 5}^T({\Sigma ^ + } + {\Sigma ^ - }){T_1}{e_1}\\ \;\;\;\;\;\;\; - 2e_1^T{\Sigma ^ + }{T_1}{\Sigma ^ - }{e_1}, \end{array} $

$ \begin{eqnarray*} W_9&=&-2e_{2^{r+3}-4}^TT_{1}e_{2^{r+3}-4}+e_{2^{r+3}-3}^TT_{1} (\Sigma^{+}+\Sigma^{-})e_{2^{r+3}-4} \\ &&+e_{2^{r+3}-4}^T(\Sigma^{+}+\Sigma^{-})T_{1}e_{2^{r+3}-3} -2e_{2^{r+3}-3}^T\Sigma^{+}T_{1}\Sigma^{-}e_{2^{r+3}-3}, \end{eqnarray*} $

$ W_{10}=(e_{2^i+j_{i}-1}-e_{2^i+j_{i}})^TR_{i}(e_{2^i+j_{i}-1}-e_{2^i+j_{i}}), $

$ \begin{eqnarray*} \Xi&=&W_{1}+W_{1}^T+\sum\limits_{i=2}^{5}W_{i}+\sum\limits_{i=7}^{10}W_{i}+W_{6}+ W_{6}^T-W^{\ast}\\ &&+\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}} (\Lambda_{ij_{i}}e_{2^i+j_{i}-1} +e_{2^i+j_{i}-1}^T\Lambda_{ij_{i}}^T) - \sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}}(\Lambda_{ij_{i}}e_{2^i+j_{i}}+e_{2^i+j_{i}}^T\Lambda_{ij_{i}}^T), \end{eqnarray*} $

证  针对以下候选Lyapunov-Krasovskii函数

$ V(x_{t})=\sum\limits_{i=1}^{7}V_{i}(x_{t}), $

其中

$ V_{1}(x_{t})=2\sum\limits_{j=1}^{n}k_{1j}\int_{0}^{x_{j}(t)}(f_{j}(s)-\sigma_{j}^{-}s){\rm d}s+2\sum\limits_{j=1}^{n}k_{2j}\int_{0}^{x_{j}(t)}(\sigma_{j}^{+}s-f_{j}(s)){\rm d}s, $

$ \begin{eqnarray*} V_{2}(x_{t})&=&\sum\limits_{j_{1}=1}^{2}\int_{-\tau^{-}-\frac{j_{1}}{2}\delta}^{-\tau^{-}-\frac{j_{1}-1}{2}\delta}\int_{t+\theta}^{t}\dot{x}^T(s)V_{1j_{1}}\dot{x}(s){\rm d}s{\rm d}\theta \\ && +\sum\limits_{j_{2}=1}^{2^2}\int_{-\tau^{-}-\frac{j_{2}}{2^2}\delta-\delta}^{-\tau^{-}-\frac{j_{2}-1}{2^2}\delta-\delta}\int_{t+\theta}^{t}\dot{x}^T(s)V_{2j_{2}} \dot{x}(s){\rm d}s{\rm d}\theta+\cdots \\ &&+\sum\limits_{j_{r}=1}^{2^r}\int_{-\tau^{-}-\frac{j_{r}}{2^r}\delta-(r-1)\delta}^{-\tau^{-}-\frac{j_{r}-1}{2^r}\delta-(r-1)\delta}\int_{t+\theta}^{t}\dot{x}^T(s)V_{rj_{r}}\dot{x}(s){\rm d}s{\rm d}\theta, \end{eqnarray*} $

$ \begin{eqnarray*} V_{3}(x_{t})&=&\sum\limits_{j_{1}=1}^{2}\int_{-\tau^{-}-\frac{j_{1}}{2}\delta}^{-\tau^{-}-\frac{j_{1}-1}{2}\delta}\int_{\eta}^{-\tau^{-}-\frac{j_{1}-1}{2}\delta}\int_{t+\theta}^{t}\dot{x}^T(s)U_{1j_{1}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta \\&& +\sum\limits_{j_{1}=1}^{2}\int_{-\tau^{-}-\frac{j_{1}}{2}\delta}^{-\tau^{-}-\frac{j_{1}-1}{2}\delta}\int_{-\tau^{-}-\frac{j_{1}}{2}\delta}^{\eta}\int_{t+\theta}^{t}\dot{x}^T(s)W_{1j_{1}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta \\&& +\sum\limits_{j_{2}=1}^{2^2}\int_{-\tau^{-}-\frac{j_{2}}{2^2}\delta-\delta}^{-\tau^{-}-\frac{j_{2}-1}{2^2}\delta-\delta}\int_{\eta}^{-\tau^{-}-\frac{j_{2}-1}{2^2}\delta-\delta}\int_{t+\theta}^{t}\dot{x}^T(s)U_{2j_{1}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta\\&& +\sum\limits_{j_{2}=1}^{2^2}\int_{-\tau^{-}-\frac{j_{2}}{2^2}\delta-\delta}^{-\tau^{-}-\frac{j_{2}-1}{2^2}\delta-\delta}\int_{-\tau^{-}-\frac{j_{2}}{2^2}\delta-\delta}^{\eta}\int_{t+\theta}^{t}\dot{x}^T(s)W_{2j_{2}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta+\cdots \\&&+\sum\limits_{j_{r}=1}^{2^r}\int_{-\tau^{-}-\frac{j_{r}}{2^r}\delta-(r-1)\delta}^{-\tau^{-}-\frac{j_{r}-1}{2^r}\delta-(r-1)\delta} \int_{\eta}^{-\tau^{-}-\frac{j_{r}-1}{2^r}\delta-(r-1)\delta}\int_{t+\theta}^{t}\dot{x}^T(s)U_{rj_{r}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta\\&& +\sum\limits_{j_{r}=1}^{2^r}\int_{-\tau^{-}-\frac{j_{r}}{2^r}\delta-(r-1)\delta}^{-\tau^{-}-\frac{j_{r}-1}{2^r}\delta-(r-1)\delta}\int_{-\tau^{-}-\frac{j_{r}}{2^r}\delta-(r-1)\delta}^{\eta}\int_{t+\theta}^{t}\dot{x}^T(s)W_{rj_{r}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta, \end{eqnarray*} $

$ \begin{eqnarray*} V_{4}(x_{t})&=&\sum\limits_{j_{1}=1}^{2}\int_{t-\tau^{-}-\frac{j_{1}}{2}\delta+\rho_{1}(t)}^{t-\tau^{-}-\frac{j_{1}-1}{2}\delta}x^T(s)M_{1j_{1}}x(s){\rm d}s \\&& +\sum\limits_{j_{2}=1}^{2^2}\int_{t-\tau^{-}-\frac{j_{2}}{2^2}\delta-\delta+\rho_{2}(t)}^{t-\tau^{-}-\frac{j_{2}-1}{2^2}\delta-\delta}x^T(s)M_{2j_{2}}x(s){\rm d}s +\cdots \\ && +\sum\limits_{j_{r}=1}^{2^r}\int_{t-\tau^{-}-\frac{j_{r}}{2^r}\delta-(r-1)\delta+\rho_{r}(t)}^{t-\tau^{-}-\frac{j_{r}-1}{2^r}\delta-(r-1)\delta}x^T(s)M_{rj_{r}}x(s){\rm d}s, \end{eqnarray*} $

$ \begin{eqnarray*} V_{5}(x_{t})&=&\sum\limits_{j_{1}=1}^{2}\int_{-\tau^{-}-\frac{j_1}{2} \delta}^{-\tau^{-}-\frac{j_1}{2}\delta+\rho_{1}(t)}\int_{t+\theta }^{t-\tau^{-}-\frac{j_1}{2}\delta +\rho_{1}(t)}\dot{x}^T(s)F_{1j_{1}}\dot{x}(s){\rm d}s{\rm d}\theta \\ && +\sum\limits_{j_{1}=1}^{2}\int_{-\tau^{-}-\frac{j_1}{2}\delta+ \rho_{1}(t)}^{-\tau^{-}-\frac{j_1-1}{2}\delta} \int_{t+\theta}^{t-\tau^{-} -\frac{j_1-1}{2}\delta}\dot{x}^T(s)Q_{1j_{1}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&+\sum\limits_{j_{2}=1}^{2^2}\int_{-\tau^{-}-\frac{j_2}{2^2}\delta-\delta}^{-\tau^{-}-\frac{j_2}{2^2}\delta-\delta+\rho_{2}(t)}\int_{t+\theta}^{t-\tau^{-}-\frac{j_2}{2^2}\delta-\delta+\rho_{2}(t)} \dot{x}^T(s)F_{2j_{2}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&+\sum\limits_{j_{2}=1}^{2^2}\int_{-\tau^{-}-\frac{j_2}{2^2}\delta-\delta+\rho_{2}(t)}^{-\tau^{-}-\frac{j_2-1}{2^2}\delta-\delta}\int_{t+\theta}^{t-\tau^{-}-\frac{j_2-1}{2^2}\delta-\delta}\dot{x}^T(s)Q_{2j_{2}}\dot{x}(s){\rm d}s{\rm d}\theta +\cdots \\ && +\sum\limits_{j_{r}=1}^{2^r}\int_{-\tau^{-}-\frac{j_r}{2^r}\delta-(r-1)\delta}^{-\tau^{-}-\frac{j_r}{2^r}\delta-(r-1)\delta+\rho_{r}(t)}\int_{t+\theta}^{t-\tau^{-}-\frac{j_r}{2^r}\delta-(r-1)\delta+\rho_{r}(t)}\dot{x}^T(s)F_{rj_{r}}\dot{x}(s){\rm d}s{\rm d}\theta\\ && +\sum\limits_{j_{r}=1}^{2^r}\int_{-\tau^{-}-\frac{j_r}{2^r}\delta-(r-1)\delta+\rho_{r}(t)}^{-\tau^{-}-\frac{j_r-1}{2^r}\delta-(r-1)\delta}\int_{t+\theta}^{t-\tau^{-}-\frac{j_r-1}{2^r}\delta-(r-1)\delta}\dot{x}^T(s)Q_{rj_{r}}\dot{x}(s){\rm d}s{\rm d}\theta, \end{eqnarray*} $

$ V_{6}(x_{t})=2\sum\limits_{i=1}^{n}\int_{t-\tau(t)}^{t}t_{i}(\sigma_{i}^{+}x_{i}(s)-f_{i}(x_{i}(s))x_{i}(s){\rm d}s, $

$ V_{7}(x_{t})=d^{+}\int_{-d^{+}}^{0}\int_{t+\beta}^{t}f^T(x(s))Mf(x(s)){\rm d}s{\rm d}\beta. $

现在, 沿着 (2.1) 式的解决方法对$V(x_{t})$进行时间求导, 可以得到

$ \dot{V}(x_{t})=\sum\limits_{i=1}^{6}\dot{V}_{i}(x_{t}), $

$ \begin{eqnarray} \dot{V}_{1}(x_{t})&=&2[f(x(t))-\Sigma^{-}x(t)]^TK_1\dot{x}(t)+2[\Sigma^{+}x(t)-f(x(t))]^TK_2\dot{x}(t)\\ & =&\eta^T(t)(W_{11}^TK_{1}W_{13}+W_{12}^TK_{2}W_{13}+W_{13}^TK_{1}W_{11}+W_{13}^TK_{2}W_{12})\eta(t)\\ & =&\eta^{T}(t)(W_{1}+W_{1}^{T})\eta(t), \end{eqnarray} $

(3.3)

$ \begin{eqnarray} \dot{V}_{2}(x_{t})&=&\dot{x}^T(t) \bigg\{\frac{\delta}{2}\sum\limits_{j_{1}=1}^{2} V_{1j_{1}}+\frac{\delta}{2^2}\sum\limits_{j_{2}=1}^{2^2}V_{2j_{2}}+\cdots +\frac{\delta}{2^r}\sum\limits_{j_{r}=1}^{2^r}V_{rj_{r}}\bigg\}\dot{x}(t) \\ &&-\frac{2}{\delta}\sum\limits_{j_{1}=1}^{2}(x(t-\tau^{-} -\frac{j_{1}-1}{2}\delta)-x(t-\tau^{-}-\frac{j_1}{2}\delta))^T V_{1j_{1}}(x(t-\tau^{-}-\frac{j_{1}-1}{2}\delta) \\ &&-x(t-\tau^{-}-\frac{j_1}{2}\delta)) -\frac{2}{\delta^{2}}\sum\limits_{j_{2}=1}^{2^2} (x(t-\tau^{-}-\frac{j_{2}-1}{2^2}\delta-\delta) \\ &&-x(t-\tau^{-} -\frac{j_2}{2^2}\delta-\delta))^TV_{2j_{2}}(x(t-\tau^{-} -\frac{j_{2}-1}{2^2}\delta-\delta) \\ &&-x(t-\tau^{-}-\frac{j_2}{2^2}\delta-\delta)) -\cdots -\frac{2}{\delta^{r}}\sum\limits_{j_{r}=1}^{2^r}(x(t-\tau^{- }-\frac{j_{r}-1}{2^r}\delta -(r-1)\delta) \\ &&-x(t-\tau^{-}-\frac{j_r}{2^r}\delta-(r-1)\delta))^TV_{rj_{r}} (x(t-\tau^{-}-\frac{j_{r}-1}{2^r}\delta -(r-1)\delta) \\ &&-x(t-\tau^{-}-\frac{j_r}{2^r}\delta-(r-1)\delta)) \\ & =&\dot{x}^T(t){\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}}\frac{\delta}{2^{i}} V_{ij_{i}}}\dot{x}(t)- \eta^T(t)\{\sum\limits_{i=1}^{r}\sum\limits_{j_{1}=1}^{2^i}\frac{2}{\delta^{i}} (e_{2^i+j_{i}-1}-e_{2^i+j_{i}})^{T}, \end{eqnarray} $

(3.4)

$ \begin{eqnarray*} &&V_{ij_{i}}(e_{2^i+j_{i}-1}-e_{2^i+j_{i}})\}\eta(t)\\ &=&\eta^T(t)\bigg\{\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}} \bigg[\frac{\delta}{2^{i}}W_{13}^{T}V_{ij_{i}}W_{13}-\frac{2}{\delta^{i}} (e_{2^i+j_{i}-1}-e_{2^i+j_{i}})^{T} V_{ij_{i}}(e_{2^i+j_{i}-1}-e_{2^i+j_{i}})\bigg]\bigg\}\eta(t) \\ &=&\eta^T(t)W_{2}\eta(t), \end{eqnarray*} $

$ \begin{eqnarray} \dot{V}_{3}(x_{t})&=&\frac{\delta^{2}}{2}\dot{x}^T(t) \bigg[\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}}(\frac{1}{2^i})^{2}(U_{ij_{i}} +W_{ij_{i}})\bigg]\dot{x}(t)\\ && -\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}} \bigg[\int_{-\tau^{-}-\frac{j_{i}}{2^i}\delta-(i-1)\delta}^{-\tau^{-}-\frac{j_{i}-1}{2^i} \delta-(i-1)\delta}\int_{t+\theta}^{t-\tau^{-}-\frac{j_{i}-1}{2^i} \delta-(i-1)\delta}\dot{x}^T(s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&+\int_{-\tau^{-}-\frac{j_{i}}{2^i}\delta-(i-1)\delta}^{-\tau^{-} -\frac{j_{i}-1}{2^i}\delta-(i-1)\delta}\int_{t-\tau^{-} -\frac{j_{i}}{2^i}\delta-(i-1)\delta}^{t+\theta}\dot{x}^T(s)W_{ij_{i}} \dot{x}(s){\rm d}s{\rm d}\theta\bigg], \end{eqnarray} $

(3.5)

我们令

$ \alpha_{i}=\rho_{i}(t) , \quad \beta_{i}=\frac{\delta}{2^{i}}-\rho_{i}(t), \quad i=1, 2, \cdots, r, $

$ \begin{eqnarray*} &&-\int_{-\tau^{-}-\frac{j_{i}}{2^i}\delta-(i-1)\delta}^{-\tau^{-} -\frac{j_{i}-1}{2^i}\delta-(i-1)\delta}\int_{t+\theta}^{t-\tau^{-} -\frac{j_{i}-1}{2^i}\delta-(i-1)\delta}\dot{x}^T(s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&-\int_{-\tau^{-}-\frac{j_{i}}{2^i}\delta-(i-1)\delta}^{-\tau^{-} -\frac{j_{i}-1}{2^i}\delta-(i-1)\delta}\int_{t-\tau^{-}-\frac{j_{i}}{2^i} \delta-(i-1)\delta}^{t+\theta}\dot{x}^T(s)W_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta\\ &\leq&- \int_{-\tau^{-}-\frac{j_i}{2^i}-(i-1)\delta}^{-\tau^{-}-\frac{j_i}{2^i} \delta-(i-1) \delta+\rho_{i}(t)} \int_{t+\theta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)} \dot{x}^T(s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&- \int_{-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{-\tau^{-} -\frac{j_{i}-1}{2^i}\delta-(i-1)\delta} \int_{t+\theta}^{t-\tau^{-}-\frac{j_{i}-1}{2^i}\delta-(i-1)\delta}\dot{x}^T (s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&-\frac{\alpha_{i}}{\beta_{i}}\int_{t-\tau^{-}-\frac{j_i}{2^i} \delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i} \delta-(i-1)\delta}\dot{x}^T(s){\rm d}sU_{ij_i} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^ {t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}(s){\rm d}s \\ && -\int_{-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{-\tau^{-} -\frac{j_i-1}{2^i}\delta-(i-1)\delta}\int_{t-\tau^{-}-\frac{j_i}{2^i} \delta-(i-1)\delta+\rho_{i}(t)}^{t+\theta} \dot{x}^T(s)W_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&-\int_{-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{-\tau^{- }-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\int_{t-\tau^{-} -\frac{j_i}{2^i}\delta-(i-1)\delta}^{t+\theta}\dot{x}^T(s)W_{ij_ {i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&-\frac{\beta_i}{\alpha_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta- (i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta +\rho_{i}(t)}\dot{x}^T(s){\rm d}sW_{ij_i} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-} -\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}(s){\rm d}s, \\ \\ && -\int_{-\tau^{-}-\frac{j_i}{2^i}-(i-1)\delta}^{-\tau^{-} -\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\int_{t+\theta}^ {t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x} ^T(s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&-\int_{-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{-\tau^{-} -\frac{j_{i}-1}{2^i}\delta-(i-1)\delta}\int_{t+\theta}^{t-\tau^{-} -\frac{j_{i}-1}{2^i}\delta-(i-1)\delta}\dot{x}^T(s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&-\int_{-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{-\tau^{-} -\frac{j_i-1}{2^i}\delta-(i-1)\delta}\int_{t-\tau^{-}-\frac{j_i} {2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t+\theta}\dot{x}^T(s) W_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &&-\int_{-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{-\tau^{-} -\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\int_{t-\tau^{-} -\frac{j_i}{2^i}\delta-(i-1)\delta}^{t+\theta}\dot{x}^T(s) W_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta \\ &\leq&-2\eta^{T}(t)\{(e_{2^{r+1}+2^i+j_i-2}-e_{2^{r+2}+2^i+j_i-4} )^TU_{ij_{i}}(e_{2^{r+1}+2^i+j_i-2}-e_{2^{r+2}+2^i+j_i-4})\\ &&+(e_{2^i+j_i-1}-e_{2^{r+2}+2^{r+1}+2^i+j_i-6})^TU_{ij_{i}} (e_{2^i+j_i-1}-e_{2^{r+2}+2^{r+1}+2^i+j_i-6}) \\ &&+(e_{2^{r+2}+2^{r+1}+2^i+j_i-6}-e_{2^{r+1}+2^i+j_i-2})^TW_{ij_i} (e_{2^{r+2}+2^{r+1}+2^i+j_i-6}-e_{2^{r+1}+2^i+j_i-2})\\ && +(e_{2^{r+2}+2^i+j_i-4}-e_{2^i+j_i})^TW_{ij_i}(e_{2^{r+2}+2^i+j_i-4} -e_{2^i+j_i})\}\eta(t), \end{eqnarray*} $

因此我们可以得到

$ \begin{eqnarray} \dot{V}_{3}(x_{t})&\leq&\eta^T(t)W_3\eta(t)-\sum\limits_{i=1}^{r} \sum\limits_{j_{i}=1}^{2^{i}} \bigg\{-\frac{\alpha_{i}}{\beta_{i}} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}^T(s){\rm d}sU_{ij_i}\\ && \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta} \dot{x}(s){\rm d}s-\frac{\beta_i}{\alpha_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}^T(s){\rm d}sW_{ij_i}\\ && \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}(s){\rm d}s\bigg\}, \end{eqnarray} $

(3.6)

$ \begin{eqnarray} \dot{V}_{4}(x_{t})&=&\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}} \bigg\{x^T(t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta)M_{ij_i} x(t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta)\\ &&-\dot{\rho_{i}}(t)x^T(t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t))M_{ij_1}x(t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)) \bigg\} \\ & \leq&\eta^T(t)\bigg\{\sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}}(e_{2^i+j_i-1}^TM_{ij_i}e_{2^i+j_i-1}-\frac{\mu}{2^{i}r}e_{2^{r+1}+2^i+j_i-2}^TM_{ij_i}e_{2^{r+1}+2^i+j_i-2}) \bigg\}\eta(t) \\ & =&\eta^T(t)W_4\eta(t), \end{eqnarray} $

(3.7)

$ \begin{eqnarray} \dot{V}_{5}(x_{t})&\leq& \eta^T(t)W_5\eta(t)-\frac{1}{\alpha_{i}} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}^T(s){\rm d}sF_{ij_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)} \\ &&\dot{x}(s){\rm d}s -\frac{1}{\beta_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}^T(s){\rm d}sQ_{ij_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}(s){\rm d}s. \\ &&\end{eqnarray} $

(3.8)

根据 (3.6), (3.8) 式和引理2.2, 如果$\left[\begin{array}{cc} Q_{ij_i} ~& Q_{i}^{\ast} \\ \ast~ & Q_{ij_i} \end{array} \right]>0 $, 有

$ \begin{eqnarray} &&-\frac{1}{\alpha_{i}} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}^T(s){\rm d}sF_{ij_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}(s){\rm d}s \\ && -\frac{1}{\beta_i} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}^T(s){\rm d}sQ_{ij_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}(s){\rm d}s \\ && -\frac{\beta_i}{\alpha_i} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}^T(s){\rm d}sW_{ij_i} \int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta}^{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}\dot{x}(s){\rm d}s \\ && -\frac{\alpha_{i}}{\beta_{i}}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}^T(s){\rm d}sU_{ij_i}\int_{t-\tau^{-}-\frac{j_i}{2^i}\delta-(i-1)\delta+\rho_{i}(t)}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta}\dot{x}(s){\rm d}s \\ & \leq&-\eta^T(t)\left[ \begin{array}{c} e_{2^{r+1}+2^i+j_i-2}-e_{2^i+j_i} \\ e_{2^i+j_i-1}-e_{2^{r+1}+2^i+j_i-2} \\ \end{array} \right]^T\left[ \begin{array}{cc} Q_{ij_i} & Q_{i}^{\ast} \\ \ast & Q_{ij_i} \\ \end{array} \right]\left[ \begin{array}{c} e_{2^{r+1}+2^i+j_i-2}-e_{2^i+j_i} \\ e_{2^i+j_i-1}-e_{2^{r+1}+2^i+j_i-2} \\ \end{array} \right]\eta(t), \\ && \end{eqnarray} $

(3.9)

从 (3.5), (3.6), (3.8) 和 (3.9) 式, 我们可以得到

$ \begin{eqnarray} \dot{V}_{3}(x_{t})+\dot{V}_{5}(x_{t})\leq\eta^T(t)(W_3+W_5-W^{\ast})\eta(t), \end{eqnarray} $

(3.10)

$ \begin{eqnarray} \dot{V}_{6}(x_{t})&=&2x^T(t)\Sigma^{+}x(t)-2f^T(x(t))Tx(t)-2(1-\mu) x^T(t-\tau(t))\Sigma^{+}Tx(t-\tau(t)) \\ &&+2(1-\mu)f^{T}(x(t-\tau(t)))Tx(t-\tau(t)) \\ &=&\eta^T(t)W_6\eta(t)+\eta^T(t)W_{6}^{T}(t)\eta(t), \end{eqnarray} $

(3.11)

$ \begin{eqnarray} \dot{V}_{7}(x_{t})&\leq&{(d^{+})}^2f(x(t))Mf(x(t))-\int_{t-d(t)}^{t}f^{T}(x(s)){\rm d}sM\int_{t-d(t)}^{t}f(x(s)){\rm d}s \\ & =&\eta^{T}(t)W_{7}\eta(t), \end{eqnarray} $

(3.12)

根据假设2.2, $[f_{i}(x_{i}(t))-\sigma_{i}^{+}x_{i}(t)][\sigma_{i}^{-}x_{i}(t)-f_{i}(x_{i}(t))]\geq0$, $i=1, 2, \cdots, n$, 我们很容易得到:对任意对角矩阵$T_{1}=diag(t_{11}, \cdots, t_{1n})\geq0$, $T_{2}=diag(t_{21}, \cdots, t_{2n})\geq0$, 有

$ \begin{eqnarray} &&-2f^{T}(x(t))T_1f(x(t))+2x^{T}(t)T_{1}(\Sigma^{+}+\Sigma^{-})f(x(t))-2x^T(t)\Sigma^{+}T_{1}\Sigma^{-}x(t) \\ & =&\eta^T(t)[-2e_{2^{r+3}-5}^TT_{1}e_{2^{r+3}-5}+e_{1}^TT_{1}(\Sigma^{+}+\Sigma^{-})e_{2^{r+3}-5}+e_{2^{r+3}-5}^T(\Sigma^{+}+\Sigma^{-})T_{1}e_{1} \\ && -2e_{1}^T\Sigma^{+}T_{1}\Sigma^{-}e_{1}]\eta(t) \\ & =&\eta^T(t)W_8\eta(t) \geq 0, \end{eqnarray} $

(3.13)

$ \begin{eqnarray} &&-2f^{T}(x(t-\tau(t)))T_2f(x(t-\tau(t)))+2x^{T}(t-\tau(t))T_{1}(\Sigma^{+}+\Sigma^{-})f(x(t-\tau(t))) \\ &&-2x^T(t-\tau(t))\Sigma^{+}T_{1}\Sigma^{-}x(t-\tau(t)) \\ & =&\eta^T(t)[-2e_{2^{r+3}-4}^TT_{1}e_{2^{r+3}-4}+e_{2^{r+3}-3}^TT_{1}(\Sigma^{+}+\Sigma^{-})e_{2^{r+3}-4} \\ &&+e_{2^{r+3}-4}^T(\Sigma^{+}+\Sigma^{-})T_{1}e_{2^{r+3}-3} -2e_{2^{r+3}-3}^T\Sigma^{+}T_{1}\Sigma^{-}e_{2^{r+3}-3}]\eta(t) \\ & =&\eta^T(t)W_9\eta(t) \geq0, \end{eqnarray} $

(3.14)

通过使用牛顿莱布尼兹公式和任意矩阵$\Lambda_{ij_{i}}$, $j_{i}=1, 2, \cdots, 2^i$, $i=1, 2, \cdots r$, 有

$ \begin{eqnarray} \Theta_{ij_{i}}&=&2\eta^T(t)\Lambda_{ij_{i}} \bigg[x(t-\tau^{-}-\frac{j_{i}-1}{2^i}-(i-1)\delta)-x(t-\tau^{-}-\frac{j_i}{2^i}-(i-1)\delta) \\ &&- \int_{t-\tau^{-}-\frac{j_i}{2^i}-(i-1)\delta}^{t-\tau^{-} -\frac{j_{i}-1}{2^i}-(i-1)\delta}\dot{x}(s){\rm d}s\bigg]=0, \end{eqnarray} $

(3.15)

我们可以得到

$ \begin{eqnarray*} &&2\eta^T(t)\Lambda_{ij_{i}}\int_{t-\tau^{-}-\frac{j_i}{2^i}-(i-1)\delta}^{ t-\tau^{-}-\frac{j_{i}-1}{2^i}-(i-1)\delta}\dot{x}(s){\rm d}s \\ &\leq&\eta^T(t)\Lambda_{ij_{i}}R_{i}^{-1}\Lambda_{ij_{i}}^T\eta(t)+ \int_{t-\tau^{-}-\frac{j_i}{2^i}-(i-1)\delta}^{t-\tau^{-}-\frac{j _{i}-1}{2^i}-(i-1)\delta}\dot{x}^T(s){\rm d}s R_{i}\int_{t-\tau^{-}-\frac{j_i}{2^i}-(i-1)\delta}^{t-\tau^{-} -\frac{j_{i}-1}{2^i}-(i-1)\delta}\dot{x}(s){\rm d}s\\ & =&\eta^T(t)[\Lambda_{ij_{i}}R_{i}^{-1}\Lambda_{ij_{i}}^T+(e_{2^i +j_{i}-1}-e_{2^i+j_{i}})^TR_{i}(e_{2^i+j_{i}-1}-e_{2^i+j_{i}})]\eta(t)\\ & =&\eta^T(t)\Lambda_{ij_{i}}R_{i}^{-1}\Lambda_{ij_{i}}^T\eta(t) +\eta^T(t)W_{10}\eta(t), \end{eqnarray*} $

由 (3.3) 到 (3.15) 式可以得到

$ \begin{eqnarray} \dot{V}(x_{t})&\leq&(\eta^T(t)W_{1}+W_{1}^T+\sum\limits_{i=2}^{5}W_{i} +\sum\limits_{i=7}^{10}W_{i}+W_{6}+W_{6}^T-W^{\ast} \\ &&+\sum\limits_{i=1}^{r} \sum\limits_{j_{i}=1}^{2^{i}}(\Lambda_{ij_{i}}e_{2^i+j_{i}-1} +e_{2^i+j_{i}-1}^T\Lambda_{ij_{i}}^T) \\ &&- \sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}}(\Lambda_{ij_{i}}e_{2^i+j_{i}} +e_{2^i+j_{i}}^T\Lambda_{ij_{i}}^T))\eta(t)+\eta^T(t) \Lambda_{ij_{i}}^TR_{i}^{-1}\Lambda_{ij_{i}}\eta(t) \\ & =&\eta^T(t)\Xi\eta(t)+\eta^T(t)\Lambda_{ij_{i}}^TR_{i}^{-1}\Lambda_{ij_{i}}\eta(t), \end{eqnarray} $

(3.16)

应用萧氏转换等价于 (3.2) 式, 可得

$ \begin{eqnarray} \dot{V}(x_{t})\leq\eta^T(t)\Xi\eta(t)+\eta^T(t)\Lambda_{ij_{i}}^TR_{i}^{-1}\Lambda_{ij_{i}}\eta(t)<0, \end{eqnarray} $

(3.17)

至此证得神经网络 (2.1) 是全局渐近稳定的.

注3.1  本文使用新不等时滞分割法对神经网络全局渐近稳定性问题进行了分析, 时间延迟区间$[\tau^{-}, \tau^{+}]$可被分为$r$段, 我们将每个子区间$[\tau^{-}+(k-1)\delta, \tau^{-}+k\delta]$分为$2^{r+1-k}$段.分区方法不同于以前的方法, 在本文中首次被提出.

注3.2  在文献[25]中, 时滞区间$[\tau_{2}^{-}, \tau_{2}^{+}]$的子区间$[0, \tau_{2}^{-}]$被分解为n个相等的段, 没有移除子区间时变延迟$\tau_{2}(t)$, 为了得到准确的结果, 本文分割时滞区间$[\tau_{2}^{-}, \tau_{2}^{+}]$并引入时间变量$\rho_{k}(t)$ ($\rho_{k}(t)=\frac{\tau(t)-\tau^{-}}{r \times 2^{r+1-k}}$).由于引入了时间变量$\rho_{k}(t)$, 我们可以构建Lyapunov-Krasovskii $V_{4}$和$V_{5}$, 如此类形式

$ \sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^i}\int_{-\tau^{-}-\frac{j_i}{2^i} \delta-(i-1)\delta+\rho_{i}(t)}^{-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta} \int_{t+\theta}^{t-\tau^{-}-\frac{j_i-1}{2^i}\delta-(i-1)\delta} \dot{x}^T(s)Q_{ij_{i}} \dot{x}(s){\rm d}s{\rm d}\theta. $

因为不等时滞分割法, 所引入的时间变量$\rho_{k}(t)$是不同的且在每个子区间中都是变化的.

注3.3  在本文中, 通过构造一个新的Lyapunov-Krasovskii函数, 其中一些项包含三重积分, 比如

$ \sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}}\int_{-\tau^{-}-\frac{j_{i}}{2^{i}}\delta-(i-1)\delta}^{-\tau^{-}-\frac{j_{i}-1}{2^{i}}\delta-(i-1)\delta}\int_{\eta}^{-\tau^{-}-\frac{j_{i}-1}{2^{i}}\delta-(i-1)\delta}\int_{t+\theta}^{t}\dot{x}^T(s)U_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta, $

$ \sum\limits_{i=1}^{r}\sum\limits_{j_{i}=1}^{2^{i}}\int_{-\tau^{-}-\frac{j_{i}}{2^{i}}\delta-(i-1)\delta}^{-\tau^{-}-\frac{j_{i}-1}{2^{i}}\delta-(i-1)\delta}\int_{-\tau^{-}-\frac{j_{i}}{2^{i}}\delta-(i-1)\delta}^{\eta}\int_{t+\theta}^{t}\dot{x}^T(s)W_{ij_{i}}\dot{x}(s){\rm d}s{\rm d}\theta {\rm d}\eta, $

这在改进低保守性的结果中起到重要作用.

注3.4  时滞分解的思想是受到文献[18]的启发, 但本文采用的是新不等时滞分割法, 这是一个特别的方法, 因为在每个子区间中拥有不同的变化趋势.在之前的方法中, 我们通常假设每个子区间拥有相同的状态, 但实际上状态会随着实间的改变而改变.为了获得普遍性的结论, 新不等时滞分割法在本文中被引入.

注3.5  延迟区间$[\tau^{-}, \tau^{+}]$被分解为$r$个子区间, 每个子区间的长度不相等, 在每个子区间中应用牛顿莱布尼兹公式, 并且选择不同的无权矩阵, 这将导致较低保守性的结果.

在此部分, 将给出两个数值实例来说明所提方法的有效性.

例1  对于系统 (2.1), 给出以下参数:

$ \begin{array}{*{20}{c}} {A = \left[{\begin{array}{*{20}{c}} {2}&0\\ {0}&2 \end{array}} \right], B = \left[{\begin{array}{*{20}{c}} {1}&1\\ {-1}&{-1} \end{array}} \right], C = \left[{\begin{array}{*{20}{c}} {0.88}&1\\ {1}&2 \end{array}} \right], }\\ {D = 0, {\Sigma ^ -} = diag(0, 0), {\Sigma ^ + } = diag(0.4, 0.8), {\tau ^ -} = 0, } \end{array} $

激活函数假设为$f_{i}(x_i)=0.5(|x_i+1|-|x_i-1|)$, $i=1, 2$.

对于不同的未知$\mu$的上界$ \tau^{+}$可通过本文中定理3.1得到, 将文献[6-8, 13, 16-20]的结果列在表 1中, 此例明显显示了对现有结果的改进.

注4.1  从表 1中可知, 相对于文献[6-8, 13, 16-20], 我们的结果有更低的保守性.进一步, 如果$r$值较大, 那么时间延迟的上界也将变大.因此, 通过使用新不等时滞分割法可得到更好的结果.

如果令$\tau^{+}=2.9184$, 初始状态$(-0.2, 0.2)^T$, 全局渐近稳定的结果可通过图 1被证实. 图 1显示了在所给参数的条件下, 系统 (2.1) 是全局渐近稳定的.

例2  对于系统 (2.1), 给出以下参数:

$ \begin{array}{*{20}{c}} {A = \left[{\begin{array}{*{20}{c}} {1.2769}&0&0&0\\ 0&{0.6231}&0&0\\ 0&0&{0.9230}&0\\ 0&0&0&{0.4480} \end{array}} \right], B = \left[{\begin{array}{*{20}{c}} {-0.0373}&{0.0482}&{-0.3351}&{0.2336}\\ {-1.6033}&{0.5988}&{ - 0.3224}&{1.2352}\\ {0.3394}&{ - 0.0860}&{ - 0.3824}&{ - 0.5785}\\ { - 0.1311}&{0.3253}&{ - 0.9534}&{ - 0.5015} \end{array}} \right], }\\ {C = \left[{\begin{array}{*{20}{c}} {0.8674}&{-1.2405}&{-0.5325}&{0.0220}\\ {0.0474}&{-0.9164}&{0.0360}&{0.9816}\\ { - 1.8495}&{2.6117}&{ - 0.3788}&{0.8428}\\ { - 2.0413}&{0.5179}&{1.1734}&{ - 0.2775} \end{array}} \right], {\Sigma ^ + } = \left[{\begin{array}{*{20}{c}} {0.1137}&0&0&0\\ 0&{0.1279}&0&0\\ 0&0&{0.7994}&0\\ 0&0&0&{0.2368} \end{array}} \right], }\\ {D = 0, {\Sigma ^ -} = 0, {\tau ^ -} = 0, } \end{array} $

激活函数假设为$f_{i}(x_i)=a_{i}(|x_i+1|-|x_i-1|)$, $i=1, 2, 3, 4$.其中$a_1=0.05685$, $a_2=0.06395$, $a_3=0.3997$, $a_4=0.1184$对于不同的未知$\mu$的上界$\tau^{+}$可通过本文中定理3.1得到, 将文献[6-8, 13, 16-20]的结果列在表 2中, 通过表 2, 此例明显显示了对现有结果的改进.

如果令$\tau^{+}=3.9957$, 初始状态$(-0.4, 0.5, -0.5, 0.4)^T$, 全局渐近稳定的结果可通过图 2被证实. 图 2显示了在所给参数的条件下, 系统 (2.1) 是全局渐近稳定的.

在该文中, 新的不等时滞分割法被应用于分析具有离散和分布式延迟的神经网络系统的稳定性判据.关于通过构建新的Lyapunov-Krasovskii函数而提出的新的不等时滞分割法, 其中考虑了涉及三重积分的一些项.结合一些有效的数学技术和新的不等时滞分割法, 可以显著增强获得时间延迟最大上限的效率.最后, 提出了保守性更小的稳定性标准.并给出了数值仿真以证明所提技术与一些现有结果相比的有效性.