近几年来, 由于分数阶系统可应用于许多科学和工程领域, 因此分数阶微积分模型受到人们越来越多的关注^[1-10].文献[3-6, 9]表明记忆效果(分数微分或积分算子)用到一个神经网络系统是一个非常重要的改进.

另一方面, 微分方程的稳定性分析一直是最重要的动力学行为, 与古典的李雅普诺夫稳定相比较, 有限时间稳定更符合实际需要.然而, 关于时滞分数阶模糊神经网络模型还没有这方面的讨论.本文考虑如下带有时滞的分数阶模糊神经网络模型:

$ \begin{eqnarray} \label{eq:3.1} \begin{array}{ll} \displaystyle D^\alpha x_i(t)=&-c_i x_i(t)+ \sum\limits_{j=1}^{n}a_{ij}f_j(x_j(t))+\sum\limits_{j=1}^{n}b_{ij}\mu_{j}+I_i +\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x_j(t-\tau_j))\\ &\displaystyle+\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x_j(t-\tau_j))+\bigwedge\limits_{j=1}^{n}T_{ij}\mu_{j}+\bigvee\limits_{j=1}^{n}H_{ij}\mu_{j}, %\ \ \ \ \ \ \ \ t\in\mathbb{T} \end{array} \end{eqnarray} $

(1.1)

其中 $t\in\mathbb{T}=[0, T]$, $i\in \aleph=\{1, 2, \cdots, n\}$, $n$是神经元的个数, $x_i(t)$表示第 $i$个神经元在 $t$时刻的状态变量; $\alpha_{ij}$, $\beta_{ij}$分别表示模糊反馈最小和最大模块的链接权重; $T_{ij}$及 $H_{ij}$分别表示模糊前向最小和最大模块的联接权重; $a_{ij}$表示第 $j$个神经元与第 $i$个神经元的联接权重; $b_{ij}$表示自由向前模块; $\wedge$, $\vee$分别表示模糊与(取小)和模糊或(取大)算子; $\mu_i$, $I_i$分别表示第 $i$个神经元的输入和偏差; $c_i>0$表示网络不连通和无外部附加电压差时第 $i$个神经元恢复独立静息状态的速率; $f_j(\cdot)$为激活函数; $\tau_j\geq 0$表示沿轴突的第 $j$个神经元的传输延迟.

系统(1.1) 的初始条件为

$ \begin{eqnarray} \label{eq:3.2} \begin{array}{ll} \displaystyle x_{i}=\phi_{i}(t), \ \ \ t\in[-\tau, 0], \ \ \ \ \ i\in \aleph, \end{array} \end{eqnarray} $

(1.2)

其中

$ \tau = _{\mathit{i} \in \aleph }^{\max }\{ {\tau _i}\}, \;\;\;\phi (t) = {({\phi _1}(t), {\phi _2}(t), \cdots, {\phi _n}(t))^T} \in C([-\tau, 0], {R^n}), $

这里 $C([-\tau, 0], R^n)$表示所有从 $[-\tau, 0]$到 $R^n$的连续函数组成的全体.定义范数为 $ \displaystyle\||\phi|\|=\sup_{t\in [-\tau, 0]}\|\phi(t) \|, $其中 $\|\phi(t) \|=\sum\limits_{i=1}^{n}|\phi_{i}(t)|$.为了证明方便, 定义如下符号

$ \hat c = _{\mathit{j} \in \aleph }^{\min }\{ {c_j}\}, \;\;\;\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over c} } = _{\mathit{j} \in \aleph }^{\max }\{ {c_j}\}, \;\;\;{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }}_i} = _{\mathit{j} \in \aleph }^{\max }\{ |{a_{ij}}|\}, \;\;\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} } = \sum\limits_{i = 1}^n {{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} }}_i}} . $

定义2.1^{[1, 3]}  函数 $f$的 $\alpha$分数阶积分定义为

$ I^{\alpha} f(t)=\frac{1}{\Gamma(\alpha)}\int_{t_0}^{t} \frac{f(s)}{(t-s)^{1-\alpha}}ds, \ \ t>0, \ \alpha>0, $

其中 $t\geq t_0$且 $\alpha>0$, $\Gamma(\cdot)$为Gamma函数, $\Gamma(s)=\displaystyle\int_{0}^{\infty}t^{s-1}e^{-t}dt$.

定义2.2^{[1, 5]}  函数 $f$的 $\alpha$阶Caputo分数阶微分定义为

$ D^{\alpha} f(t)=\frac{1}{\Gamma(m-\alpha)}\frac{d^m}{dt^m}\int_{t_0}^{t} \frac{f(s)}{(t-s)^{\alpha+1-m}}ds, \ \ t>0, $

其中 $t\geq t_0$. $m$是一个正整数满足 $m-1<\alpha<m$.特殊地, 当 $0<\alpha<1$时,

$ D^{\alpha} f(t)=\frac{1}{\Gamma(1-\alpha)}\int_{t_0}^{t} \frac{f'(s)}{(t-s)^{\alpha}}ds, \ \ t>0. $

定义2.3^{[7, 8]}  对任意 $t\in [0, +\infty)$, 如果常量 $x^*=(x_1^*, x_2^*, \cdots, x_n^*)^{T} \in R^n$满足

$ \begin{array}{ll} 0=&\displaystyle-c_i x_i^*+ \sum\limits_{j=1}^{n}a_{ij}f_j(x_j^*)+\sum\limits_{j=1}^{n}b_{ij}\mu_{j}+I_i +\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x_j^*)\\ & \displaystyle+\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x_j^*)+\bigwedge\limits_{j=1}^{n}T_{ij}\mu_{j}+\bigvee\limits_{j=1}^{n}H_{ij}\mu_{j}, \ \ \ i\in \aleph, %\ \ \ \ \ \ \ \ t\in\mathbb{T} \end{array} $

则称 $x^*$为系统(1.1) 的平衡点.

定义2.4^{[9, 10]}  若系统(1.1) 的平衡点 $x^*=(x_1^*, x_2^*, \cdots, x_n^*)^{T}$关于 $\{t_0, \delta, \varepsilon, \mathbb{T}, \tau\}$是有限时间稳定, 则对任意常数 $\varepsilon>0$, 都能找到 $0<\delta<\varepsilon$, 使得当初始值(1.2) 满足 $ \||\phi-x^*|\|<\delta $时, 系统(1.1) 的任意解 $x(t)=(x_1(t), x_2(t), \cdots, x_n(t))^{T}$有 $ \displaystyle\|x(t)-x^*\|<\varepsilon, \ \ \ \forall t\in \mathbb{T}=[t_0, t_0+T]. $

引理2.1^{[2, 4]}  若 $x(t)\in C^m\left([0, \infty), R^n\right)$, 且 $m-1<\alpha<m\in Z^{+}$, 则有

(1) $I^{\alpha}I^{\beta}x(t)=I^{\alpha+\beta}x(t), \ \ \ \alpha, \beta\geq 0, $

(2) $D^{\alpha}I^{\alpha}x(t)=x(t), \ \ \ \alpha\geq 0, $

(3) $I^{\alpha}D^{\alpha}x(t)=x(t)-\sum\limits_{j=0}^{k-1}\frac{t^j}{i!}x^{(j)}(0), \ \ \ \alpha\geq 0.$

引理2.2^[11]  对任意的 $\alpha_{ij}, \beta_{ij}\in R$及 $i, j\in \aleph$, 如下结论成立:

$ \begin{array}{ll} \displaystyle\mid \bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x_j)-\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(y_j)\mid\leq \sum\limits_{j=1}^{n}\mid \alpha_{ij}\mid\mid f_j(x_j)-f_j(y_j)\mid, \\ \displaystyle\mid \bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x_j)-\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(y_j)\mid\leq \sum\limits_{j=1}^{n}\mid \beta_{ij}\mid\mid f_j(x_j)-f_j(y_j)\mid. \end{array} $

引理2.3^[12]  若 $\kappa_1, \kappa_2, \cdots, \kappa_d$是非负实数, $d\in Z^+$, 因此对任意的 $\lambda>1$, 有

$ \left(\sum\limits_{i=1}^{d}\kappa_i\right)^\lambda\leq d^{\lambda-1}\sum\limits_{i=1}^{d}\kappa_i^\lambda. $

为了证明本文的主要结果, 对分数阶神经网络模型(1.1) 给出如下假设:

(H1) 激活函数 $f_j(\cdot)$满足Lipschitz条件, 即存在 $L_j>0$, 使得

$ \displaystyle\mid f_j(u)-f_j(v) \mid\leq L_j\mid u-v\mid, ~~~~~~u, v\in R, \ j\in \aleph. $

(H2) 对于 $c_i, \ a_{ij}, \ \alpha_{ij}, \ \beta_{ij}$和 $L_j$, 满足如下不等式 $ \displaystyle(\check{a}+\check{\alpha}+\check{\beta})\check{L}<\hat{c}. $

为了讨论分数阶神经网络系统(1.1) 解的存在性与唯一性, 先给出如下定理.

定理3.1  如果假设(H1) 和(H2) 成立, 则系统(1.1) 存在唯一的平衡点 $x^*$.

证  首先, 构造一个映射 $\Theta: R^n\rightarrow R^n$满足

$ \begin{eqnarray} \label{eq:3.3} \begin{array}{ll} \displaystyle\Theta_i (u_i)&\displaystyle=\sum\limits_{j=1}^{n}a_{ij}f_j(\frac{u_j}{c_j})+\sum\limits_{j=1}^{n}b_{ij}\mu_{j}+I_i +\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(\frac{u_j}{c_j})+\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(\frac{u_j}{c_j})\\ &\displaystyle\ \ \ +\bigwedge\limits_{j=1}^{n}T_{ij}\mu_{j}+\bigvee\limits_{j=1}^{n}H_{ij}\mu_{j}, \ \ \ i\in \aleph, \\ \end{array} \end{eqnarray} $

(3.1)

其中 $\Theta(u)=(\Theta_1(u_1), \Theta_2(u_2), \cdots, \Theta_n(u_n))^{T}$.对于任意两个不同的向量 $u=(u_1, u_2, \cdots, u_n)^{T}$和 $v=(v_1, v_2, \cdots, v_n)^{T}$, 有

$ \begin{eqnarray*} \label{eq:3.4} \begin{array}{ll} \displaystyle\|\Theta(u)-\Theta(v)\|&\displaystyle=\sum\limits^{n}_{i=1}|\Theta_i (u_i)-\Theta_i (v_i)|\\ &\displaystyle=\sum\limits_{i=1}^{n}\left|\sum\limits_{j=1}^{n}a_{ij}\left(f_j(\frac{u_j}{c_j})-f_j(\frac{v_j}{c_j})\right)\right. +\left(\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(\frac{u_j}{c_j})-\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(\frac{v_j}{c_j})\right)\\ &\displaystyle\ \ \ \left.+\left(\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(\frac{u_j}{c_j})-\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(\frac{u_j}{c_j})\right)\right|\\ &\displaystyle\leq\sum\limits_{i=1}^{n}\left[\sum\limits_{j=1}^{n}|a_{ij}|\left|f_j(\frac{u_j}{c_j})-f_j(\frac{v_j}{c_j})\right|\right. +\left|\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(\frac{u_j}{c_j})-\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(\frac{v_j}{c_j})\right|\\ &\displaystyle\ \ \ \left.+\left|\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(\frac{u_j}{c_j})-\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(\frac{u_j}{c_j})\right|\right]. \end{array} \end{eqnarray*} $

根据引理2.2和假设(H1), 可得

$ \begin{eqnarray*} \label{eq:3.5} \begin{array}{ll} \|\Theta(u)-\Theta(v)\| &\leq\sum\limits_{i=1}^{n}\left[\sum\limits_{j=1}^{n}|a_{ij}|L_j\frac{\left|u_j-v_j\right|}{c_j}\right. +\sum\limits_{j=1}^{n}|\alpha_{ij}|\left|f_j(\frac{u_j}{c_j})-f_j(\frac{v_j}{c_j})\right|\\ &\ \ \ \left.+\sum\limits_{j=1}^{n}|\beta_{ij}|\left|f_j(\frac{u_j}{c_j})-f_j(\frac{v_j}{c_j})\right|\right]\\ &\leq\sum\limits_{i=1}^{n}\left[\sum\limits_{j=1}^{n}|a_{ij}|L_j\frac{\left|u_j-v_j\right|}{c_j} +\sum\limits_{j=1}^{n}|\alpha_{ij}|L_j\frac{\left|u_j-v_j\right|}{c_j} +\sum\limits_{j=1}^{n}|\beta_{ij}|L_j\frac{\left|u_j-v_j\right|}{c_j}\right]\\ &\displaystyle\leq\frac{(\check{a}+\check{\alpha}+\check{\beta})\check{L}}{\hat{c}}\sum\limits_{j=1}^{n}\left|u_j-v_j\right|. \end{array} \end{eqnarray*} $

从假设条件(H2), 有

$ \begin{eqnarray} \label{eq:3.6} \begin{array}{ll} \|\Theta(u)-\Theta(v)\| \leq\left\|u-v\right\|. \end{array} \end{eqnarray} $

(3.2)

由(3.2) 式可知 $\Theta: R^n\rightarrow R^n$是压缩映射.故存在唯一不动点 $u^*\in R^n$使得 $\Theta(u^*)=u^*$.即

$ \begin{eqnarray} \label{eq:3.7} \begin{array}{ll} u_i^*&=\sum\limits_{j=1}^{n}a_{ij}f_j(\frac{u^*_j}{c_j})+\sum\limits_{j=1}^{n}b_{ij}\mu_{j}+I_i +\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(\frac{u^*_j}{c_j})+\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(\frac{u^*_j}{c_j})\\ &\ \ \ +\bigwedge\limits_{j=1}^{n}T_{ij}\mu_{j}+\bigvee\limits_{j=1}^{n}H_{ij}\mu_{j}, \ \ \ i\in \aleph. \end{array} \end{eqnarray} $

(3.3)

设 $c_i x_i^*=u^*_i$, $i\in \aleph$, 有

$ \begin{eqnarray} \label{eq:3.8} \begin{array}{ll} 0=&-c_ix^*_i+\sum\limits_{j=1}^{n}a_{ij}f_j(x^*_j)+\sum\limits_{j=1}^{n}b_{ij}\mu_{j}+I_i +\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x^*_j)+\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x^*_j)\\ & +\bigwedge\limits_{j=1}^{n}T_{ij}\mu_{j}+\bigvee\limits_{j=1}^{n}H_{ij}\mu_{j}, \ \ \ i\in\aleph. \end{array} \end{eqnarray} $

(3.4)

因为 $u^*$是唯一的不动点, 因此分数阶系统(1.1) 有唯一的平衡点 $x^*$.

定理3.2  如果定理3.1中的假设成立, 且系统(1.1) 的解 $x(t) \in C([0, T], R^n)$满足初始条件, 则系统(1.1) 存在唯一解 $x(t)$.

证  证明与定理3.1类似, 略.

本节应用定理3.1和定理3.2讨论分数阶神经网络系统(1.1) 解的稳定性.

定理4.1  当 $0.5\leq\alpha<1$时, 如果假设(H1) 和(H2) 成立, 并且满足不等式

$ \begin{eqnarray} \label{eq:3.77} \begin{array}{ll} \displaystyle\sqrt{\frac{6+(3N+MN+2M)e^{(N+2)t}}{N+2}}<\frac{\varepsilon}{\delta}, \ \ \ 0<\varepsilon, \ 0<\delta<\varepsilon, \ t\in \mathbb{T}, \end{array} \end{eqnarray} $

(4.1)

其中

$ M=\frac{3\Gamma(2\alpha-1)\check{L}^2\left(\check{\alpha}+ \check{\beta}\right)^2 (1-e^{-2\tau})}{4^\alpha\Gamma^2(\alpha)}, \ N=\!\frac{6\Gamma(2\alpha\!-1)}{4^\alpha\Gamma^2(\alpha)}\!\left[\left(\check{c}\! +\check{L}\check{a}\right)^2+ \check{L}^2\left(\check{\alpha}+ \check{\beta}\right)^2 \!\right], $

则(1.1) 式的唯一平衡点 $x^*=(x_1^*, x_2^*, \cdots, x_n^*)$关于 $\{t_0=0, \delta, \varepsilon, \mathbb{T}=[0, T], \tau\}$是有限时间稳定.

证  设 $x(t)=(x_1(t), x_2(t), \cdots, x_n(t))^{T}$是系统(1.1) 的任意解, 因此有

$ \begin{eqnarray*} \label{eq:3.9} \begin{array}{ll} D^\alpha (x_i(t)-x_i^*)=&-c_i \left(x_i(t)-x_i^*\right)+ \sum\limits_{j=1}^{n}a_{ij}\left(f_j(x_j(t))-f_j(x^*_j)\right)\\ &+\left(\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x_j(t-\tau_j)) -\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x^*_j(t))\right)\\ &+\left(\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x_j(t-\tau_j))-\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x^*_j)\right). %\ \ \ \ \ \ \ \ t\in\mathbb{T} \end{array} \end{eqnarray*} $

通过引理2.1, 可得出如下方程

$ \begin{eqnarray*} \label{eq:3.10} \begin{array}{ll} x_i(t)-x_i^*&\displaystyle=D^{-\alpha}\left[-c_i \left(x_i(t)-x_i^*\right)+ \sum\limits_{j=1}^{n}a_{ij}\left(f_j(x_j(t))-f_j(x^*_j)\right)\right.\\ &\displaystyle\ \ \ +\left(\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x_j(t-\tau_j)) -\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x^*_j)\right)\\ &\displaystyle\left.\ \ \ +\left(\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x_j(t-\tau_j)) -\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x^*_j)\right)\right]\\ &\displaystyle=\phi_{i}(0)-x_i^*+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\left[-c_i \left(x_i(s)-x_i^*\right)\right.\\ &\displaystyle\ \ \ + \sum\limits_{j=1}^{n}a_{ij}\left(f_j(x_j(s))-f_j(x^*_j)\right)\\ &\displaystyle\ \ \ +\left(\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x_j(s-\tau_j)) -\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x^*_j)\right)\\ &\displaystyle\left.\ \ \ +\left(\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x_j(s-\tau_j)) -\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x^*_j)\right)\right]ds. \end{array} \end{eqnarray*} $

根据引理2.2, 再由假设(H1) 和范数 $\| \cdot\|$的性质, 显然有

$ \begin{align} & \ \ \ \parallel x\left( t \right)-{{x}^{*}}\parallel =\sum\limits_{i=1}^{n}{xi\left( t \right)}-x_{i}^{*}| \\ & \le \parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}\sum\limits_{i=1}^{n}{\int_{0}^{t}{{{(t-s)}^{\alpha -1}}}} \\ & \ \ \ \ \ \ \left[ {{c}_{i}}\left| {{x}_{i}}(s)-x_{i}^{*} \right|+\sum\limits_{j=1}^{n}{|}{{a}_{ij}}|\left| {{f}_{j}}({{x}_{j}}(s))-{{f}_{j}}(x_{j}^{*}) \right| \right. \\ & \ \ \ \ \ \ +\left| \underset{j=1}{\overset{n}{\mathop{\wedge }}}\,{{\alpha }_{ij}}{{f}_{j}}({{x}_{j}}(s-{{\tau }_{j}}))-\underset{j=1}{\overset{n}{\mathop{\wedge }}}\,{{\alpha }_{ij}}{{f}_{j}}(x_{j}^{*}) \right| \\ & \left. \ \ \ \ \ \ +\left| \underset{j=1}{\overset{n}{\mathop{\wedge }}}\,{{\beta }_{ij}}{{f}_{j}}({{x}_{j}}(s-{{\tau }_{j}}))-\underset{j=1}{\overset{n}{\mathop{\vee }}}\,{{\beta }_{ij}}{{f}_{j}}(x_{j}^{*}) \right| \right]ds \\ & \le \parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}\sum\limits_{i=1}^{n}{\int_{0}^{t}{{{(t-s)}^{\alpha -1}}}}\left[ {{c}_{i}}\left| {{x}_{i}}(s)-x_{i}^{*} \right| \right. \\ & \ \ +\sum\limits_{j=1}^{n}{|}{{a}_{ij}}|{{L}_{j}}\left| {{x}_{j}}(s)-x_{j}^{*} \right|+\sum\limits_{j=1}^{n}{|}{{\alpha }_{ij}}|{{L}_{j}}\left| {{x}_{j}}(s-{{\tau }_{j}})-x_{j}^{*} \right| \\ & \ \ +\left. \sum\limits_{j=1}^{n}{|}{{\beta }_{ij}}|{{L}_{j}}\left| {{x}_{j}}(s-{{\tau }_{j}})-x_{j}^{*} \right| \right]ds \\ & \le \parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}\int_{0}^{t}{{{(t-s)}^{\alpha -1}}} \\ & \ \ \ \ \ \ \ \left[ \left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}}+\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right)\left\| x(s)-{{x}^{*}} \right\|+(\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }}+\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }})\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\| \right]ds \\ & =\parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}\int_{0}^{t}{}{{(t-s)}^{\alpha -1}}{{e}^{s}}\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}}+\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right){{e}^{-s}}\left\| x(s)-{{x}^{*}} \right\|ds \\ & \ \ +\frac{1}{\Gamma (\alpha )}\int_{0}^{\mathit{t}}{{{(t-s)}^{\alpha -1}}}{{e}^{s}}\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }}+\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}{{e}^{-s}}\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\|ds. \\ \end{align} $

(4.2)

对(4.2) 式应用Cauchy-Schwartz不等式, 可得

$ \begin{align} & \ \ \ \parallel x(t)-{{x}^{*}}\parallel \\ & \le \parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}{{\left( \int_{0}^{t}{{{(t-s)}^{2\alpha -2}}}{{e}^{2s}}ds \right)}^{1/2}} \\ & \ \ \ \ \ \ \ {{\left( \int_{0}^{t}{{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right)}^{2}}}{{e}^{-2s}}{{\left\| x(s)-{{x}^{*}} \right\|}^{2}}ds \right)}^{1/2}} \\ & \ \ \ \ \ \ \ \ \ +\frac{1}{\Gamma (\alpha )}{{\left( \int_{0}^{\mathit{t}}{{{(t-s)}^{2\alpha -2}}}{{e}^{2s}}ds \right)}^{1/2}} \\ & \ \ \ \ {{\left( \int_{0}^{\mathit{t}}{{{{\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}}}^{\rm{2}}}}{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)}^{2}}{{e}^{-2s}}{{\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\|}^{2}}ds \right)}^{1/2}} \\ & =\parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}{{\left( \int_{0}^{t}{{{(t-s)}^{2\alpha -2}}}{{e}^{2s}}ds \right)}^{1/2}} \\ & \ \ \ \ \ \ \ \ \left[ {{\left( \int_{0}^{t}{{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right)}^{2}}}{{e}^{-2s}}{{\left\| x(s)-{{x}^{*}} \right\|}^{2}}ds \right)}^{1/2}} \right. \\ & \ \left. \ \ \ \ +{{\left( \int_{0}^{\mathit{t}}{{{{\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}}}^{\rm{2}}}}{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)}^{2}}{{e}^{-2s}}{{\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\|}^{2}}ds \right)}^{1/2}} \right]. \\ \end{align} $

(4.3)

另一方面, 有

$ \begin{eqnarray} \label{eq:3.13} \displaystyle\int_{0}^{t}(t-s)^{2\alpha-2}e^{2s}ds&=&\displaystyle\int_{0}^{t}z^{2\alpha-2}e^{2(t-z)}dz \displaystyle=e^{2t}\int_{0}^{t}z^{2\alpha-2}e^{-2z}dz\nonumber\\ &\displaystyle=&\frac{e^{2t}}{2^{2\alpha-1}}\int_{0}^{2t}u^{2\alpha-2}e^{-u}du \displaystyle<\frac{2e^{2t}}{4^\alpha}\Gamma(2\alpha-1). \end{eqnarray} $

(4.4)

把(4.4) 式带入(4.3) 式, 可得

$ \begin{align} & \ \ \ \parallel x(t)-{{x}^{*}}\parallel \\ & \le \parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}{{\left( \frac{2{{e}^{2t}}}{{{4}^{\alpha }}}\Gamma (2\alpha -1) \right)}^{1/2}} \\ & \ \ \ \ \ \ \ \ \ \left[ {{\left( \int_{0}^{t}{{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right)}^{2}}}{{e}^{-2s}}{{\left\| x(s)-{{x}^{*}} \right\|}^{2}}ds \right)}^{1/2}} \right. \\ & \ \ \ \ \ +\left. {{\left( \int_{0}^{\mathit{t}}{{{{\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}}}^{2}}}{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)}^{2}}{{e}^{-2s}}{{\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\|}^{2}}ds \right)}^{1/2}} \right]. \\ \end{align} $

(4.5)

把引理2.3中 $\lambda=2$和 $d=3$应用到(4.5) 式, 可得

$ \begin{array}{l} \parallel x(t) - {x^*}{\parallel ^2} \le 3\parallel \phi (0) - {x^*}{\parallel ^2} + \frac{{6{e^{2t}}\Gamma (2\alpha - 1)}}{{{4^\alpha }{\Gamma ^2}(\alpha )}}\left( {\int\limits_0^t {{{\left( {\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over c} } + \mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over L} }} \right)}^2}} {e^{ - 2s}}{{\left\| {x(s) - {x^*}} \right\|}^2}ds} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \left. {\int_0^\mathit{t} {{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over L} }}^{\rm{2}}}} {{\left( {\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \alpha } } + \mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \beta } }} \right)}^2}{e^{ - 2s}}{{\left\| {x(s - {\tau _j}) - {x^*}} \right\|}^2}ds} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le 3\parallel \phi (0) - {x^*}{\parallel ^2} + \frac{{6{e^{2t}}\Gamma (2\alpha - 1)}}{{{4^\alpha }{\Gamma ^2}(\alpha )}}\left( {\int_0^t {{{\left( {\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over c} } + \mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over L} }} \right)}^2}} {e^{ - 2s}}{{\left\| {x(s) - {x^*}} \right\|}^2}ds} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \left. {\int_{ - \mathit{\tau j}}^\mathit{t} {{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over L} }}^2}{{\left( {\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \alpha } } + \mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \beta } }} \right)}^2}{e^{ - 2{\tau _j}}}{e^{ - 2s}}{{\left\| {x(s) - {x^*}} \right\|}^2}ds} } \right)\\ \;\;\;\;\;\; \le 3\parallel \phi (0) - {x^*}{\parallel ^2} + \frac{{6{e^{2t}}\Gamma (2\alpha - 1)}}{{{4^\alpha }{\Gamma ^2}(\alpha )}}{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over L} }}^2}{\left( {\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \alpha } } + \mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \beta } }} \right)^2}{e^{ - 2{\tau _j}}}\int_{ - {\tau _j}}^0 {{e^{ - 2s}}} {\left\| {x(s) - {x^*}} \right\|^2}ds\\ \;\;\;\;\;\;\;\;\;\;\;\; + \frac{{6{e^{2t}}\Gamma (2\alpha - 1)}}{{{4^\alpha }{\Gamma ^2}(\alpha )}}\left( {\int_0^\mathit{t} {\left[{{{\left( {\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over c} } + \mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over a} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over L} }} \right)}^2} + {{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over L} }}^2}{{\left( {\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \alpha } } + \mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \beta } }} \right)}^2}{e^{-2{\tau _j}}}} \right]{e^{ - 2s}}{{\left\| {x(s) - {x^*}} \right\|}^2}ds} } \right). \end{array} $

因为当 $t\in[-\tau, 0]$时, 有 $x(t)=\phi(t)$, 在结合范数 $\||\phi-x^*|\|=\sup\limits_{t\in [-\tau, 0]}\|\phi(t)-x^* \|$, 有

$ \begin{align} & \parallel x(t)-{{x}^{*}}{{\parallel }^{2}} \\ & \ \ \ \ \ \ \ \ \ \ \le \left[ 3+\frac{3{{e}^{2t}}\Gamma (2\alpha -1){{{\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}}}^{2}}{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)}^{2}}(1-{{e}^{-2{{\tau }_{j}}}})}{{{4}^{\alpha }}{{\Gamma }^{2}}(\alpha )} \right]\parallel |\phi -{{x}^{*}}|{{\parallel }^{2}} \\ & \ \ \ \ \ \ +\frac{6{{e}^{2t}}\Gamma (2\alpha -1)}{{{4}^{\alpha }}{{\Gamma }^{2}}(\alpha )}\left[ {{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right)}^{2}}+{{{\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}}}^{2}}{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)}^{2}}{{e}^{-2{{\tau }_{j}}}} \right] \\ & \ \ \ \ \ \ \ \ \ \ \ \ \int_{0}^{\mathit{t}}{{{e}^{-2s}}}{{\left\| x(s)-{{x}^{*}} \right\|}^{2}}ds. \\ \end{align} $

(4.6)

根据(4.6) 式, 可以有

$ \begin{eqnarray*} \label{eq:3.17} \begin{array}{ll} \displaystyle\|x(t)-x^*\|^2e^{-2t} &\displaystyle\!\leq\left[3e^{-2t}\!+\frac{3\Gamma(2\alpha-1)\check{L}^2\left(\check{\alpha}+ \check{\beta}\right)^2 (1-e^{-2\tau})}{4^\alpha\Gamma^2(\alpha)}\right]\!\||\phi\!-x^*|\|^2\!\\ &\displaystyle\ \ \ +\frac{6\Gamma(2\alpha-1)}{4^\alpha\Gamma^2(\alpha)}\left[\left(\check{c}+\check{a}\check{L} \right)^2 +\check{L}^2\left(\check{\alpha}+ \check{\beta}\right)^2 \right]\int_{0}^{t}e^{-2s}\left\|x(s)-x^*\right\|^2ds\\ &\displaystyle\leq\left(3e^{-2t}+M\right)\||\phi-x^*|\|^2+N\int_{0}^{t}e^{-2s}\left\|x(s)-x^*\right\|^2ds\\. \end{array} \end{eqnarray*} $

应用Gronwall不等式, 可得

$ \begin{eqnarray*} \label{eq:3.18} &&\|x(t)-x^*\|^2e^{-2t}\\ &\displaystyle\leq&\left(3e^{-2t}+M\right)\||\phi-x^*|\|^2 +\int_{0}^{t}N\left(3e^{-2t}+M\right)\||\phi-x^*|\|^2 \exp{\left( \int_{s}^{t}Ndu \right)}ds\nonumber\\ %&\displaystyle=\left(3e^{-2t}+M\right)\||\phi-x^*|\|^2+\int_{0}^{t}N\left(3e^{-2t}+M\right)\||\phi-x^*|\|^2 \exp{\left(N(t-s)\right)}ds\\ &\displaystyle=&\left(3e^{-2t}+M+\frac{3Ne^{Nt}-3Ne^{-2t}}{N+2}+Me^{Nt}-M \right)\||\phi-x^*|\|^2\nonumber\\ &\displaystyle=&\frac{6e^{-2t}+(3N+MN+2M)e^{Nt}}{N+2}\||\phi-x^*|\|^2. \end{eqnarray*} $

因此有

$ \begin{eqnarray*} \label{eq:3.19} \begin{array}{ll} \displaystyle\|x(t)-x^*\|\leq \sqrt{\frac{6+(3N+MN+2M)e^{(N+2)t}}{N+2}}\||\phi-x^*|\|. \end{array} \end{eqnarray*} $

由此可知当 $\||\phi-x^*|\|<\delta$时, 并且(4.1) 式成立, 则 $ \|x(t)-x^*\|<\varepsilon. $根据定义2.4, 可知系统(1.1) 中当 $0.5\leq \alpha <1$时平衡点 $x^*=(x_1^*, x_2^*, \cdots, x_n^*)^{T}$关于 $\{t_0=0, \delta, \varepsilon, \mathbb{T}=[0, T], \tau\}$是有限时间稳定.

定理4.2  当 $0<\alpha<0.5$时, 如果假设(H1) 和(H2) 成立, 并且满足不等式

$ \begin{eqnarray} \label{eq:3.22} \displaystyle\sqrt[q]{\frac{q\cdot3^{q-1}+(3^{q-1}\tilde{N}+q\tilde{M}+\tilde{MN})e^{(\tilde{N}+q)t}}{q+\tilde{N}}}<\frac{\varepsilon}{\delta}, \ \ \ 0<\varepsilon, \ 0<\delta<\varepsilon\ t\in \mathbb{T}, \end{eqnarray} $

(4.7)

其中

$ \begin{eqnarray*} \tilde{M}&=&3^{q-1}\left(\frac{\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p} \frac{\check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q (1-e^{-q\tau})}{q}, \\ \displaystyle\tilde{N}&=&\displaystyle3^{q-1}\left(\frac{\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p} \left[\left(\check{c}+\check{a}\check{L} \right)^q+ \check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q\right], \end{eqnarray*} $

$p=1+\alpha, \ q=1+1/\alpha, $则系统(1.1) 的唯一平衡点 $x^*=(x_1^*, x_2^*, \cdots, x_n^*)^{T}$关于 $\{t_0=0, \delta, \varepsilon, \mathbb{T}=[0, T], \tau\}$是有限时间稳定.

证  类似于定理4.1, 对系统(1.1) 有如下估计

$ \begin{align} & \parallel x(t)-{{x}^{*}}\parallel \le \parallel \phi (0)-{{x}^{*}}\parallel \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{\Gamma (\alpha )}\int_{0}^{\mathit{t}}{{{(t-s)}^{\alpha -1}}}{{e}^{s}}\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right){{e}^{-s}}\left\| x(s)-{{x}^{*}} \right\|ds \\ & \ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{\Gamma (\alpha )}\int_{0}^{\mathit{t}}{{{(t-s)}^{\alpha -1}}}{{e}^{s}}\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right){{e}^{-s}}\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\|ds. \\ \end{align} $

(4.8)

设 $p=1+\alpha$, $q=1+1/\alpha$, 显然, $p, q>1$且 $1/p+1/q=1$.利用Höder不等式, 可得

$ \begin{align} & \ \ \ \ \parallel x(t)-{{x}^{*}}\parallel \\ & \le \parallel \phi (0)-{{x}^{*}}\parallel +\frac{1}{\Gamma (\alpha )}{{\left( \int_{0}^{t}{{{(t-s)}^{p\alpha -p}}}{{e}^{ps}}ds \right)}^{1/p}} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \left[ {{\left( \int_{0}^{t}{{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right)}^{q}}}{{e}^{-qs}}{{\left\| x(s)-{{x}^{*}} \right\|}^{q}}ds \right)}^{1/q}} \right. \\ & \ \ \ \ \ \ +\left. {{\left( \int_{0}^{t}{{{{\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}}}^{\mathit{q}}}{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)}^{q}}{{e}^{-qs}}}{{\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\|}^{q}}ds \right)}^{1/q}} \right]. \\ \end{align} $

(4.9)

另一方面, 有

$ \begin{align} & \int_{0}^{\mathit{t}}{{{(t-s)}^{p\alpha -p}}}{{e}^{ps}}ds \\ & \ \ \ \ \ \ \ \ \ \ \ ={{e}^{pt}}\int_{0}^{\mathit{pt}}{{{u}^{p\alpha -p}}}{{e}^{-pu}}du=\frac{{{e}^{pt}}}{{{p}^{p(\alpha -1)+1}}}\int_{0}^{\mathit{pt}}{{{z}^{p\alpha -p}}}{{e}^{-z}}dz \\ & \ \ \ \ \ \ \ \ \ \ \ <\frac{{{e}^{pt}}}{{{p}^{p(\alpha -1)+1}}}\Gamma (p(\alpha -1)+1). \\ \end{align} $

(4.10)

将(4.10) 式代入(4.9) 式, 可得

$ \begin{align} & \|x(t)-{{x}^{*}}\| \\ & \le \|\phi (0)-{{x}^{*}}\|+{{\left( \frac{{{e}^{pt}}\Gamma (p(\alpha -1)+1)}{{{\Gamma }^{p}}(\alpha ){{p}^{p(\alpha -1)+1}}} \right)}^{1/p}} \\ & \ \ \ \ \ \ \ \left[ {{\left( \int_{0}^{\mathit{t}}{{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{c}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{a}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}} \right)}^{q}}}{{e}^{-qs}}{{\left\| x(s)-{{x}^{*}} \right\|}^{q}}ds \right)}^{1/q}} \right. \\ & \ \ \ \ \ +\left. {{\left( \int_{0}^{\mathit{t}}{{{{\mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{L}}}}^{q}}}{{\left( \mathit{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha }+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\beta }} \right)}^{q}}{{e}^{-qs}}{{\left\| x(s-{{\tau }_{j}})-{{x}^{*}} \right\|}^{q}}ds \right)}^{1/q}} \right]. \\ \end{align} $

(4.11)

把引理2.3中 $\lambda=q$和 $d=3$应用到(4.11) 式, 可知

$ \begin{eqnarray*} \label{eq:3.27} \begin{array}{ll} &\displaystyle\|x(t)-x^*\|^q\\ &\displaystyle\leq3^{q-1}\|\phi(0)-x^*\|^q+3^{q-1}\left(\frac{e^{pt}\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p}\\ & \displaystyle\ \ \ \times\left(\int_{0}^{t}\left(\check{c}+\check{a}\check{L} \right)^qe^{-qs}\left\|x(s)-x^*\right\|^qds\right. +\left.\int_{0}^{t}\check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q e^{-qs}\left\|x(s-\tau_j)-x^*\right\|^qds\right)\\ &\displaystyle\leq3^{q-1}\|\phi(0)-x^*\|^q+3^{q-1}\left(\frac{e^{pt}\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p}\\ & \displaystyle\ \ \ \times\left(\int_{0}^{t}\left(\check{c}+\check{a}\check{L} \right)^qe^{-qs}\left\|x(s)-x^*\right\|^qds\right. +\left.\int_{-\tau_j}^{t}\check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q e^{-q\tau_j}e^{-qs}\left\|x(s)-x^*\right\|^qds\right)\\ &\displaystyle\leq\left[3^{q-1}+3^{q-1}\left(\frac{e^{pt}\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p} \frac{\check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q (1-e^{-q\tau})}{q}\right]\||\phi-x^*|\|^q\\ &\displaystyle\ \ \ +3^{q-1}\left(\frac{e^{pt}\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p} \left[\left(\check{c}+\check{a}\check{L} \right)^q+\check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q \right]\\ &\int_{0}^{t}e^{-qs}\left\|x(s)-x^*\right\|^qds, \end{array} \end{eqnarray*} $

由此可得

$ \begin{eqnarray*} \label{eq:3.28} \begin{array}{ll} &\displaystyle\|x(t)-x^*\|^qe^{-qt}\\ &\displaystyle\leq\left[3^{q-1}e^{-qt}+3^{q-1}\left(\frac{\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p} \frac{\check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q (1-e^{-q\tau})}{q}\right]\||\phi-x^*|\|^q\\ &\displaystyle\ \ \ +3^{q-1}\left(\frac{\Gamma(p(\alpha-1)+1)}{\Gamma^p(\alpha)p^{p(\alpha-1)+1}}\right)^{q/p} \left[\left(\check{c}+\check{a}\check{L} \right)^q +\check{L}^q\left(\check{\alpha}+ \check{\beta}\right)^q \right]\int_{0}^{t}e^{-qs}\left\|x(s)-x^*\right\|^qds\\ &\displaystyle\leq\left(3^{q-1}e^{-qt}+\tilde{M}\right)\||\phi-x^*|\|^q+\tilde{N}\int_{0}^{t}e^{-qs}\left\|x(s)-x^*\right\|^qds. \end{array} \end{eqnarray*} $

应用Gronwall不等式, 可得

$ \begin{eqnarray*} \label{eq:3.30} \begin{array}{ll} \|x(t)-x^*\|^qe^{-qt} &\displaystyle\leq\left(3^{q-1}e^{-qt}+\tilde{M}\right)\||\phi-x^*|\|^q+\int_{0}^{t}\tilde{N}\left(3^{q-1}e^{-qt}+\tilde{M}\right)\||\\ &\phi-x^*|\|^qe^{\tilde{N}(t-s)}ds\\ &\displaystyle\leq\frac{q\cdot3^{q-1}e^{-qt}+\left(3^{q-1}\tilde{N}+q\tilde{M}+\tilde{M}\tilde{N}\right)e^{\tilde{N}t}}{q+\tilde{N}}\||\phi-x^*|\|^q, \end{array} \end{eqnarray*} $

因此有

$ \begin{eqnarray} \label{eq:3.31} \begin{array}{ll} \displaystyle\|x(t)-x^*\|\leq \sqrt[q]{\frac{q\cdot3^{q-1}+(3^{q-1}\tilde{N}+q\tilde{M}+\tilde{MN})e^{(\tilde{N}+q)t}}{q+\tilde{N}}}\||\phi-x^*|\|. \end{array} \end{eqnarray} $

(4.12)

由此可知当 $\||\phi-x^*|\|<\delta$且(4.7) 式成立, 则 $ \|x(t)-x^*\|<\varepsilon, $根据定义2.4, 可得出系统(1.1) 中当 $0< \alpha <0.5$时平衡点 $x^*=(x_1^*, x_2^*, \cdots, x_n^*)^{T}$关于 $\{t_0=0, \delta, \varepsilon, \mathbb{T}=[0, T], \tau\}$是有限时间稳定.

因为大部分的分数阶微分方程不能求出解析解, 所以在研究分数阶微分方程时, 近似的数值方法是必要的.在文献[13]中, 作者提出了一个数值算法(预估方法)求解分数阶微分方程.该方法是Adams-Bashforth-Moulton的方法的推广, 考虑如下形式的分数阶微分方程

$ \begin{eqnarray} \label{eq:3.32} \left\{ \begin{array}{ll} \displaystyle D^{\alpha}x(t)=F(t, x(t)), \ \ \ t\in \mathbb{T}, \ 0<\alpha<1, \ \mathbb{T}=[0, T], \\ x(0)=x_0, \end{array} \right. \end{eqnarray} $

(5.1)

故方程(5.1) 可以等价表示为

$ \begin{eqnarray} \label{eq:3.34} \begin{array}{ll} \displaystyle x(t)=x_0 +\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}F(s, x(s))ds. \end{array} \end{eqnarray} $

(5.2)

在本文中

$ \begin{array}{ll} F(t, x(t))=&-c_i x_i(t)+ \sum\limits_{j=1}^{n}a_{ij}f_j(x_j(t))+\sum\limits_{j=1}^{n}b_{ij}\mu_{j}+I_i +\bigwedge\limits_{j=1}^{n}\alpha_{ij}f_j(x_j(t-\tau_j))\\ &+\bigvee\limits_{j=1}^{n}\beta_{ij}f_j(x_j(t-\tau_j))+\bigwedge\limits_{j=1}^{n}T_{ij}\mu_{j} +\bigvee\limits_{j=1}^{n}H_{ij}\mu_{j}. %\ \ \ \ \ \ \ \ t\in\mathbb{T} \end{array} $

设 $h=T/N$, $t_k=kh$, $k=0, 1, 2, \cdots, N\in Z^+$, 因此(5.2) 式可写成

$ \begin{eqnarray} \label{eq:3.35} \begin{array}{ll} \displaystyle x_h(t_{k+1})=x_0 +\frac{h^\alpha}{\Gamma(\alpha+2)}F(t_{k+1}, X_h(t_{k+1}))+ \frac{h^\alpha}{\Gamma(\alpha+2)}\sum_{p=0}^{k}\bar{a}_{p, k+1}F(t_p, x_h(t_p)), \end{array} \end{eqnarray} $

(5.3)

其中

$ \begin{eqnarray*} \label{eq:3.37} \displaystyle X_h(t_{k+1})&=&x_0+\frac{1}{\Gamma(\alpha)}\sum_{p=0}^{k}\bar{b}_{p, k+1}F(t_p, x_h(t_p)), \ \ \displaystyle \bar{b}_{p, k+1}=\frac{h^\alpha}{\alpha}((k+1-p)^\alpha-(k-p)^\alpha), \\ \label{eq:3.36} \bar{a}_{p, k+1}&=&\left\{ \begin{array}{ll} \displaystyle k^{\alpha+1}-(k-\alpha)(N+1)^\alpha, \ \ \ &p=0, \\ \displaystyle (k-p+2)^{\alpha+1}+(k-p)^{\alpha+1}-2(k-p+1)^{\alpha+1}, \ \ \ &1\leq p\leq k, \\ 1, &p=k+1. \end{array} \right. \end{eqnarray*} $

此数值方法的误差为 $\max\limits_{k=0, 1, \cdots, N}|x(t_k)-x_h(t_k)|=O(h^{1+\alpha})$.基于此方法, 给出分数阶模糊神经网络模型的数值解.

本节将根据第5节的数值仿真算法, 并通过一个数值算例来验证本文理论结果的正确性和有效性.

考虑如下带有变时滞的分数阶模糊神经网络模型:

$ \begin{eqnarray} \label{eq:3.39} \begin{array}{ll} D^\alpha x_i(t)=&-c_i x_i(t)+ \sum\limits_{j=1}^{2}a_{ij}f_j(x_j(t))+\sum\limits_{j=1}^{2}b_{ij}\mu_{j}+I_i +\bigwedge\limits_{j=1}^{2}\alpha_{ij}f_j(x_j(t-\tau_j))\\ &+\bigvee\limits_{j=1}^{2}\beta_{ij}f_j(x_j(t-\tau_j))+\bigwedge\limits_{j=1}^{2}T_{ij}\mu_{j} +\bigvee\limits_{j=1}^{2}H_{ij}\mu_{j}, %\ \ \ \ \ \ \ \ t\in\mathbb{T} \end{array} \end{eqnarray} $

(6.1)

其中 $\alpha=0.98$, 并且有 $c_1=0.55$, $c_2=0.45$, $a_{11}=0.4$, $a_{12}=-0.01$, $a_{21}=0.01$, $a_{22}=0.1$, $\alpha_{11}=-0.01$, $\alpha_{12}=-0.4$, $\alpha_{21}=-0.01$, $\alpha_{22}=-0.01$, $\beta_{11}=0.01$, $\beta_{12}=-0.01$, $\beta_{21}=-0.1$, $\beta_{22}=0.01$, $T_{11}=0.02$, $T_{12}=0.01$, $T_{21}=-0.01$, $T_{22}=0.05$, $H_{11}=0.06$, $H_{12}=0.01$, $H_{21}=-0.01$, $H_{22}=0.1$, $I_1=-0.8$, $I_2=0.3$, $\mu_{1}=\mu_{2}=0.1$, $\tau_j=0.5$, $f_2(x_2(t))=\frac{1}{2}(\mid x_2(t)+1\mid+\mid x_2(t)-1\mid)$, $f_1(x_1)=\tanh(x_1)$.

显然, 可以得出激活函数 $f_j(x_j(t))~(j=1, 2)$满足假设条件(H1) 并且可得 $L_j=1~(j=1, 2)$.选取 $\delta=0.036$, $\varepsilon=1$, 当 $\alpha=0.98$时, 可以得到 $ M=0.131887442012802, \ \ \ N=1.839513983297748. $从不等式

$ \displaystyle\sqrt{\frac{6+(3N+MN+2M)e^{(N+2)t}}{N+2}}<\frac{\varepsilon}{\delta} $

中可以估计出时间 $T=1.6137$, 在根据定理3.1和定理4.1可知, 系统(6.1) 有唯一的平衡点 $(x_1^*, x_2^*)=(-2.98348, 1.10785)$, 并且关于 $\{t_0=0, \delta=0.036, \varepsilon=1, \mathbb{T}=[0, 1.6137], \tau=0.5\}$是有限时间稳定的.根据第5节的数值仿真, 考虑如下情况.

情况1  系统(6.1) 的初始值为 $(x_1, x_2)=(-2.9816, 1.1090)$.

情况2  系统(6.1) 的初始值为 $(x_1^*, x_2^*)=(-2.98348, 1.10785)$.

情况3  系统(6.1) 的初始值为 $(x_1, x_2)=(-2.9854, 1.1067)$.

根据图 6.1知当时间步长 $h=0.01$时, 对于情况1-3可得出系统(6.1) 的唯一平衡点是 $(x_1^*, x_2^*)=(-2.98348, 1.10785)$, 并关于 $\{t_0=0, \delta=0.036, \varepsilon=1, \mathbb{T}=[0, 1.6137], \tau=0.5\}$是有限时间稳定的.说明了理论结果的正确性.