最近几十年,分数阶微积分被应用到许多领域,如:电路系统,流体力学问题,控制理论,人工智能系统等方面^{[1-7, 24-25]}.而分数阶微分系统的Mittag-Leffler稳定性问题作为分数阶微分系统研究的核心内容之一,也取得了丰富的成果. Mittag-Leffler稳定性是由李岩及其同事^[8-9]提出,他给出了分数阶非线性动力系统的Mittag-Leffler稳定性定义.随后,分数阶微分系统的Mittag-Leffler稳定性引起了国内外学者的广泛关注和深入研究,并取得了很多成果^[10-15].文献[10-11]中Liu给出了非线性分数阶中立退化系统和分数阶非线性系统的Mittag-Leffler稳定性的充分条件.文献[12]给出了多变量广义Mittag-Leffler稳定性的定义,并引入多变量分数阶Lyapunov直接法,同时提出了一种新的方法研究多变量分数阶非线性动力系统的广义Mittag-Leffler稳定性.文献[13]讨论了带反馈控制的网络微分方程的分数阶耦合系统的全局Mittag-Leffler稳定性.利用压缩映射原理, Lyapunov方法,图理论方法和不等式技术,给出了平衡点的存在性、唯一性和全局Mittag-Leffler稳定性的条件.文献[14-15]中分析了分数阶基因调控网络的Mittag-Leffler稳定性并且研究了分数阶神经网络的局部Mittag-Leffler稳定性和局部渐近$ \omega $周期性.

在对实际系统的建模中,由于约束条件的存在,时滞和脉冲是必须要考虑的因素.到目前为止,对于脉冲分数阶微分方程的Mittag-Leffler稳定性已经取得了相关的结果^[16-20].现在,对于含有脉冲的相关文献主要集中在神经网络方面的研究,如在文献[16-18]中,作者使用Lyapunov方法给出了具有单侧Lipschitz条件的分数阶脉冲神经网络和分数阶脉冲神经网络的全局Mittag-Leffler稳定性的条件及可变时间脉冲分数阶神经网络的Mittag-Leffler稳定性分析.在文献[19]中研究了具有时变时滞分数阶脉冲神经网络的全局Mittag-Leffler稳定性及其同步问题.在非线性分数阶微分系统方面,文献[20]分析了含有脉冲的非线性分数阶微分系统的Mittag-Leffler稳定性的条件.对于分数阶神经网络已经出现了相关的应用,如文献[21]给出了分数阶神经网络模型并分析了分数阶神经网络的混沌行为.文献[22]研究了分数阶神经网络模型的分岔和混沌现象,同时给出了系统的稳定性条件.文献[23]通过对分数阶细胞神经网络和整数阶细胞神经网络的比较分析,说明了分数阶神经网络微分系统更加精确地描述系统的动力学行为.目前对于含有脉冲和时滞分数阶微分系统的Mittag-Leffler稳定性研究较少.受到现有研究成果的启发,在文献[20]的基础上,本文研究同时含有脉冲和时滞因素的分数阶非线性系统的Mittag-Leffler稳定性问题.讨论了含有脉冲的分数阶非线性时滞微分系统的全局Mittag-Leffler稳定性,研究的系统含有脉冲和时滞信息,在处理方法上有一定的改进,对现有结果进行了推广.

记$ {{\Bbb R}} ^{n} $是$ n $维欧几里得空间,且$ ||x|| = \sum\limits_{i = 1}^{n}|x_{i}| $定义为$ x\in {{\Bbb R}} ^{n} $的范数.记$ {{\Bbb R}} _{+} = [0, \infty) $且$ t_{0}\in {{\Bbb R}} _{+} $.

定义2.1^[1]  设可积函数$ f(t)\in C[t_{0}, +\infty), {\rm Re} \alpha>0, $则当$ t>t_{0} $时,记

$ _{t_{0}}D_{t}^{-\alpha}f(t) = \frac{1}{\Gamma(\alpha)}\int_{t_{0}}^{t}(t-s)^{\alpha-1}f(s){\rm d}s, (n-1<\alpha\leq n) $

为函数$ f(t) $的$ \alpha $阶$ \rm Riemann-liouville $分数阶积分.

定义2.2^[1]  对任意$ t\geq t_{0} $, $ q $阶$ \rm Caputo $分数阶导数, $ 0<q<1 $,下限为$ t_{0} $,函数$ f(t)\in C^{1}([t_{0}, b], {{\Bbb R}} ), b>t_{0}, $被定义为

$ ^{C}_{t_{0}}D^{q}_{t}f(t) = \frac{1}{\Gamma(1-q)}\int_{t_{0}}^{t}\frac{f^{\prime}(s)}{(t-s)^{q}}{\rm d}s. $

定义2.3^[1]  含有一个参数和两个参数的$ \rm Mittag-Leffler $函数分别被定义为

$ E_{\alpha}(t) = \sum\limits^{\infty}\limits_{k = 0}\frac{t^{k}}{\Gamma(k\alpha+1)}, (\alpha>0), $

$ E_{\alpha, \beta}(t) = \sum\limits^{\infty}\limits_{k = 0}\frac{t^{k}}{\Gamma(k\alpha+\beta)}, (\alpha>0, \beta>0). $

我们考虑下面含有时滞的脉冲分数阶非线性系统

(2.1) $ \begin{equation} \left\{\begin{array}{ll} { } ^{C}_{t_{0}}D^q_{t}x_{i}(t) = -\sum\limits_{j = 1}^{n}a_{ij}x_{j}(t)+f_{i}(t, x_{i}(t-\tau)), & t\neq t_{k}, t>t_{0}, \\ \Delta x_{i}(t_{k}) = x_{i}(t^{+}_{k})-x_{i}(t_{k}) = P_{ik}(x_{i}(t_{k})), &k = 1, 2, \cdots, i = 1, 2, \cdots, n. \end{array}\right. \end{equation} $

这里$ ^{C}_{t_{0}}D^q_{t} $是$ q $阶Caputo分数阶导数, $ 0<q<1 $. $ a_{ij} $是常数, $ x_{i}(t), f_{i}(t, x_{i}(t)) $是状态量. $ t_{k}, k = 1, 2, \cdots, $是脉冲扰动时刻满足: $ t_{0}<t_{1}<t_{2}<\cdots <t_{k}<t_{k+1}<\cdots , $且$ \lim\limits_{k\rightarrow\infty}t_{k} = \infty $. $ \tau\geq0 $是时滞.函数$ P_{ik} $表示在脉冲时刻$ t_{k} $状态$ x_{i}(t) $的突变函数.

记$ J\subset {{\Bbb R}} $是一个区间.定义下列的函数类: $ PC[J, {{\Bbb R}} ^{n}] = \{\sigma:J\rightarrow {{\Bbb R}} ^{n}:\sigma(t) $在$ t_{k}\in J $点以外都是连续的, $ \sigma(t^{-}_{k}) $和$ \sigma(t^{+}_{k}) $存在,且$ \sigma(t^{-}_{k}) = \sigma(t_{k})\} $; $ PCB[J, {{\Bbb R}} ^{n}] = \{\sigma\in PC[J, {{\Bbb R}} ^{n}]:\sigma(t) $在$ J $上是有界的$ \} $.

设$ \varphi_{0}\in PCB[[-\tau, 0], {{\Bbb R}} ^{n}] $.记$ x(t) = x(t;t_{0}, \varphi_{0})\in {{\Bbb R}} ^{n} $是系统(2.1)的解满足初值条件

(2.2) $ \begin{equation} \left\{\begin{array}{ll} x(t;t_{0}, \varphi_{0}) = \varphi_{0}(t-t_{0}), & t_{0}-\tau\leq t\leq t_{0}, \\ x(t^{+}_{0};t_{0}, \varphi_{0}) = \varphi_{0}(0). \end{array}\right. \end{equation} $

并且$ x(t) = x(t;t_{0}, \varphi_{0}) = (x_{1}(t;t_{0}, \varphi_{0}), \cdots, x_{n}(t;t_{0}, \varphi_{0}))^{T} $是有第一类间断点$ t_{k} $的分段连续函数, $ k = 1, 2, \cdots $,这里$ x_{i}(t) $是左连续的,即

$ x_{i}(t^{-}_{k}) = x_{i}(t_{k}), k = 1, 2, \cdots, x_{i}(t^{+}_{k}) = x_{i}(t_{k})+P_{ik}(x_{i}(t_{k})), t_{k}>t_{0}. $

作下列的假设

H2.1 对所有$ u, v\in {{\Bbb R}} , u\neq v, $且$ f_{i}(t, 0) = 0, i = 1, 2, \cdots, n $. $ \exists $常数$ L_{i}>0, $使得$ |f_{i}(t, u)-f_{i}(t, v)|\leq L_{i}|u-v| $.

H2.2 函数$ P_{ik} $在$ {{\Bbb R}} $上是连续的, $ i = 1, 2, \cdots, n, \; k = 1, 2, \cdots $.

H2.3 $ t_{0}<t_{1}<t_{2}<\cdots<t_{k}<t_{k+1}<\cdots $且$ t_{k}\rightarrow\infty(k\rightarrow\infty) $.

定义2.4  系统(2.1)的零解是

$ \rm (a) $稳定的,如果$ \forall t_{0}\in {{\Bbb R}} _{+}, \forall\varepsilon>0, \exists\delta = \delta(t_{0}, \varepsilon)>0, $使得$ \forall \varphi_{0}\in PC[[-\tau, 0], {{\Bbb R}} ^{n}]:||\varphi_{0}||_{\tau}<\delta $时,对$ \forall t\geq t_{0}:||x(t;t_{0}, \varphi_{0})||<\varepsilon $;

$ \rm (b) $全局吸引的,如果$ \lim\limits_{t\rightarrow\infty}x(t;t_{0}, \varphi_{0}) = 0 $;

$ \rm (c) $全局渐近稳定,如果零解是稳定且全局吸引.

其中$ \varphi\in PC[[-\tau, 0], {{\Bbb R}} ^{n}] $,定义$ ||\varphi||_{\tau} = \sup\limits_{-\tau\leq s\leq0}||\varphi(s)|| $.在$ \tau = \infty $的情况下

$ ||\varphi||_{\tau} = ||\varphi||_{\infty} = \sup\limits_{s\in(-\infty, 0]}||\varphi(s)||. $

定义2.5  系统(2.1)的零解是全局$ \rm Mittag-Leffler $稳定的,如果对$ \varphi_{0}\in PC[[-\tau, 0], {{\Bbb R}} ^{n}], $$ \exists $常数$ c>0 $和$ d>0 $,使得

$ ||x(t;t_{0}, \varphi_{0})||\leq\{m(\varphi_{0})E_{q}(-c(t-t_{0})^{q})\}^{d}, t\geq t_{0}, $

这里$ E_{q} $对应$ \rm Mittag-Leffler $函数, $ m(0) = 0, m(\varphi)\geq0, $且$ m(\varphi) $关于$ \varphi\in PC[[-\tau, 0], {{\Bbb R}} ^{n}] $是$ \rm Lipschitzian $.

推论2.1^[9]  全局$ \rm Mittag-Leffler $稳定是全局渐近稳定的.

进一步我们可以考虑分段连续Lyapunov函数$ V:[t_{0}, \infty)\times {{\Bbb R}} ^{n}\rightarrow {{\Bbb R}} _{+}. $

定义2.6  函数$ V:[t_{0}, \infty]\times {{\Bbb R}} ^{n}\rightarrow {{\Bbb R}} _{+}, $属于$ V_{0} $类,如果下列的条件成立

$ \rm 1) $函数$ V $在$ \bigcup\limits^{\infty}\limits_{k = 1}G_{k} $上是连续的且$ V(t, 0) = 0, t\in[t_{0}, \infty) $.

$ \rm 2) $函数$ V $在每个集合$ G_{k} $上关于$ x $满足局部Lipschitz条件.

$ \rm 3) $对每个$ k = 1, 2, \cdots $且$ x\in {{\Bbb R}} ^{n} $存在有限极限值,

$ V(t^{-}_{k}, x) = \lim\limits_{t\rightarrow t_{k}, t<t_{k}}V(t, x), V(t^{+}_{k}, x) = \lim\limits_{t\rightarrow t_{k}, t>t_{k}}V(t, x). $

$ \rm 4) $对每个$ k = 1, 2, \cdots $且$ x\in {{\Bbb R}} ^{n} $下列的等式是成立的: $ V(t^{-}_{k}, x) = V(t_{k}, x). $

其中$ G_{k} = (t_{k-1}, t_{k})\times {{\Bbb R}} ^{n}, k = 1, 2, \cdots;G = \bigcup\limits^{\infty}\limits_{k = 1} G_{k}. $

定义2.7  给定一个函数$ V\in V_{0} $.对$ t\in[t_{k-1}, t_{k}), k = 1, 2, \cdots, $且$ \varphi\in PC[[-\tau, 0], {{\Bbb R}} ^{n}], $$ 0<q<1 $,在$ \rm Caputo $意义下,对于系统(2.1), $ q $阶右上导数定义为

$ ^{C}D^{q}_{+}V(t, \varphi(0)) = \lim\limits_{h\rightarrow0^{+}}\sup\frac{1}{h^{q}}[V(t, \varphi(0))-V(t-h, \varphi(0)-h^{q}F(t, \varphi))], $

这里$ F(t, \varphi) = (F_{1}, F_{2}, \cdots, F_{n})^{T} $,且$ F_{i}(t, \varphi) = -\sum\limits_{j = 1}^{n}a_{ij}\varphi_{j}(t)+f_{i}(t, \varphi_{i}(t-\tau)), i = 1, 2, \cdots, n. $

定义2.8^[19]  假设函数$ V\in V_{0} $满足$ V(t+s, \varphi(s))\leq V(t, \varphi(0)), s\in[-\tau, 0] $时,有

$ ^{C}D^{q}_{+}V(t, \varphi(0))\leq MV(t, \varphi(0)), M\in {{\Bbb R}} $

是一个常数.则称函数$ V $满足$ \rm Razumikhin条件$.

引理2.1^[11]  记$ x(t)\in {{\Bbb R}} ^{n} $是一个可微函数向量.如果一个连续函数$ V:[t_{0}, \infty]\times {{\Bbb R}} ^{n}\rightarrow {{\Bbb R}} $满足

$ ^{C}_{t_{0}}D^{\beta}_{t}V(t, x(t))\leq -\alpha V(t, x(t)), $

那么

$ V(t, x(t))\leq V(t_{0}, x(t_{0}))E_{\beta}(-\alpha(t-t_{0})^{\beta}), $

这里$ 0<\beta<1 $, $ \alpha $是一个正常数.

引理2.2^[1]  假设$ 0<\alpha<2, \beta $是一个任意复数且$ \mu $是一个任意实数使得: $ \frac{\pi\alpha}{2}<\mu<{\rm min}\{\pi, \pi\alpha\} $,那么对任意整数$ p\geq1 $,有下列的展开式

$ E_{\alpha, \beta}(z) = \frac{1}{\alpha}z^{\frac{1-\beta}{\alpha}}{\rm exp}({z^{\frac{1}{\alpha}}}) -\sum\limits_{k = 1}^{p}\frac{z^{-k}}{\Gamma(\beta-\alpha k)}+O(|z|^{-1-p}), \; |\arg(z)|\leq\mu, \; |z|\longrightarrow\infty; $

$ E_{\alpha, \beta}(z) = -\sum\limits_{k = 1}^{p}\frac{z^{-k}}{\Gamma(\beta-\alpha k)}+O(|z|^{-1-p}), \; \mu\leq|\arg(z)|\leq\pi, \; |z|\rightarrow\infty. $

定理3.1  如果条件H2.1–H2.3成立,系统(2.1)满足

$ \min\limits_{1\leq i\leq n}(a_{ii}-\sum\limits^{n}\limits_{j = 1, j\neq i}|a_{ji}|)>\max\limits_{1\leq i\leq n}L_{i}>0, $

且函数$ P_{ik} $满足: $ P_{ik}(x_{i}(t_{k})) = -\sigma_{ik}x_{i}(t_{k}), 0<\sigma_{ik}<2, $$ i = 1, 2, \cdots, n, k = 1, 2, \cdots. $那么系统(2.1)的零解是全局$ \rm Mittag-Leffler $稳定的.

证  构造一个Lyapunov函数$ V(t, x(t)) = \sum\limits^{n}\limits_{i = 1}|x_{i}(t)|. $则当$ t\geq t_{0} $且$ t = t_{k} $时,由定理的条件,我们得到

(3.1) $\begin{eqnarray} \label{eq:3.1.3} V(t^{+}_{k}, x(t^{+}_{k}))& = &\sum\limits^{n}\limits_{i = 1}|x_{i}(t^{+}_{k})| = \sum\limits^{n}\limits_{i = 1}|x_{i}(t_{k})+P_{ik}(x_{i}(t_{k}))| \\& = &\sum\limits^{n}\limits_{i = 1}|x_{i}(t_{k})-\sigma_{ik}(x_{i}(t_{k}))| = \sum\limits^{n}\limits_{i = 1}|(1-\sigma_{ik})x_{i}(t_{k})| \\& = &\sum\limits^{n}\limits_{i = 1}|1-\sigma_{ik}||x_{i}(t_{k})| \\&<&\sum\limits^{n}\limits_{i = 1}|x_{i}(t_{k})| = V(t_{k}, x(t_{k})), k = 1, 2, \cdots. \end{eqnarray}$

设$ t\geq t_{0} $,且$ t\in[t_{k-1}, t_{k}). $

如果$ x_{i}(t) = 0, i = 1, 2, \cdots, n $,那么$ ^{C}D^{q}_{+}V(t, x(t)) = 0. $

如果$ x_{i}(t)>0, i = 1, 2, \cdots, n $,那么

$ \begin{eqnarray*} ^{C}_{t_{0}}D^{q}_{t}|x_{i}(t)| = \frac{1}{\Gamma(1-q)}\int^{t}_{t_{0}}\frac{|x_{i}(s)|^{\prime}}{(t-s)^{q}}{\rm d}s = \frac{1}{\Gamma(1-q)}\int^{t}_{t_{0}}\frac{x_{i}^{\prime}(s)}{(t-s)^{q}}{\rm d}s = ^{C}_{t_{0}}D^{q}_{t}x_{i}(t). \end{eqnarray*} $

如果$ x_{i}(t)<0, i = 1, 2, \cdots, n $,那么

$ \begin{eqnarray*} ^{C}_{t_{0}}D^{q}_{t}|x_{i}(t)| = \frac{1}{\Gamma(1-q)}\int^{t}_{t_{0}}\frac{|x_{i}(s)|^{\prime}}{(t-s)^{q}}{\rm d}s = -\frac{1}{\Gamma(1-q)}\int^{t}_{t_{0}}\frac{x_{i}^{\prime}(s)}{(t-s)^{q}}{\rm d}s = -^{C}_{t_{0}}D^{q}_{t}x_{i}(t). \end{eqnarray*} $

因此, $ ^{C}_{t_{0}}D^{q}_{t}|x_{i}(t)| = sgn(x_{i}(t))^{C}_{t_{0}}D^{q}_{t}x_{i}(t). $

那么当$ t\geq t_{0} $,且$ t\in[t_{k-1}, t_{k}) $,沿着系统(2.1)解的右上导数$ ^{C}D^{q}_{+}V(t, x(t)) $,我们得到

$ \begin{eqnarray*} ^{C}D^{q}_{+}V(t, x(t)) & = & {^{C}D^{q}_{+}\sum\limits^{n}\limits_{i = 1}|x_{i}(t)|} = \sum\limits^{n}\limits_{i = 1}(^{C}D^{q}_{+}|x_{i}(t)|) = \sum\limits^{n}\limits_{i = 1}[sgnx_{i}(t)\cdot^{C}D^{q}_{+}x_{i}(t)] \\& = &\sum\limits^{n}\limits_{i = 1}sgnx_{i}(t)[-\sum\limits_{j = 1}^{n}a_{ij}x_{j}(t)+f_{i}(t, x_{i}(t-\tau))] \\& = &-\sum\limits_{i = 1}^{n}sgnx_{i}(t)[\sum\limits_{j = 1}^{n}a_{ij}x_{j}(t)]+\sum\limits_{i = 1}^{n}sgnx_{i}(t)\cdot f_{i}(t, x_{i}(t-\tau)) \\& = &-sgnx_{1}(t)[a_{11}x_{1}(t)+a_{12}x_{2}(t)+\cdots+a_{1n}x_{n}(t)] \\&&-sgnx_{2}(t)[a_{21}x_{1}(t)+a_{22}x_{2}(t)+\cdots +a_{2n}x_{n}(t)] \\&& -\cdots-sgnx_{n}(t)[a_{n1}x_{1}(t)+a_{n2}x_{2}(t)+\cdots+a_{nn}x_{n}(t)] \\ && +\sum\limits_{i = 1}^{n}sgnx_{i}(t)\cdot f_{i}(t, x_{i}(t-\tau)) \\& = &-a_{11}|x_{1}(t)|-a_{22}|x_{2}(t)|-\cdots-a_{nn}|x_{n}(t)| -sgnx_{1}(t)[a_{12}x_{2}(t)\\ &&+\cdots+a_{1n}x_{n}(t)] -sgnx_{2}(t)[a_{21}x_{1}(t) +a_{23}x_{3}(t)+\cdots+a_{2n}x_{n}(t)]\\ &&-\cdots-sgnx_{n}(t)[a_{n1}x_{1}(t)+a_{n2}x_{2}(t) +\cdots+a_{nn-1}x_{n-1}(t)]\\ && +\sum\limits_{i = 1}^{n}sgnx_{i}(t)\cdot f_{i}(t, x_{i}(t-\tau)) \\ & \leq&-\sum\limits_{i = 1}^{n}a_{ii}|x_{i}(t)|+(|a_{21}|+|a_{31}|+\cdots+|a_{n1}|)|x_{1}(t)| \\ &&+(|a_{12}|+|a_{32}|+\cdots+|a_{n2}|)|x_{2}(t)| \\&&+\cdots+(|a_{1n}|+|a_{2n}|+\cdots+|a_{n-1, n}|)|x_{n}(t)|+\sum\limits_{i = 1}^{n}L_{i}|x_{i}(t-\tau)| \\& = &-\sum\limits_{i = 1}^{n}(a_{ii}-\sum\limits^{n}\limits_{j = 1, j\neq i}|a_{ji}|)|x_{i}(t)|+\sum\limits_{i = 1}^{n}L_{i}|x_{i}(t-\tau)| \\ &\leq&-\min\limits_{1\leq i\leq n}(a_{ii}-\sum\limits^{n}\limits_{j = 1, j\neq i}|a_{ji}|)\sum\limits_{i = 1}^{n}|x_{i}(t)|+\max\limits_{1\leq i\leq n}(L_{i})\sum\limits_{i = 1}^{n}|x_{i}(t-\tau)| \\ &\leq&-k_{1}V(t, x(t))+k_{2}\sup\limits_{t-\tau\leq s\leq t}V(s, x(s)). \end{eqnarray*} $

这里$ k_{1} = \min\limits_{1\leq i\leq n}(a_{ii}-\sum\limits^{n}\limits_{j = 1, j\neq i}|a_{ji}|)>0, k_{2} = \max\limits_{1\leq i\leq n}L_{i}>0. $

由以上的分析和系统(2.1)的任意解$ x(t) $满足Razumikhin条件,即有

$ \begin{eqnarray*} V(s, x(s))\leq V(t, x(t)), t-\tau\leq s\leq t. \end{eqnarray*} $

则有$ ^{C}D^{q}_{+}V(t, x(t))\leq-(k_{1}-k_{2})V(t, x(t)) $.

由定理的条件,存在一个实数$ \alpha>0 $,使得：$ k_{1}-k_{2}\geq\alpha $.有

(3.2) $ \begin{eqnarray} ^{C}D^{q}_{+}V(t, x(t))\leq-\alpha V(t, x(t)), t\neq t_{k}, t>t_{0}, \end{eqnarray} $

那么由(3.1)式, (3.2)式和引理2.1得到

$ \begin{eqnarray*} V(t, x(t))\leq V(t^{+}_{0}, x(t^{+}_{0}))E_{q}(-\alpha(t-t_{0})^{q}), t\in[t_{0}, \infty). \end{eqnarray*} $

因此

$ \begin{eqnarray*} ||x(t)||& = &\sum\limits^{n}\limits_{i = 1}|x_{i}(t)| \leq\sum\limits^{n}\limits_{i = 1}|x_{i}(t^{+}_{0})|E_{q}(-\alpha(t-t_{0})^{q}) \\& = &||x(t^{+}_{0})||E_{q}(-\alpha(t-t_{0})^{q}) \\& = &||\varphi_{0}(0)||E_{q}(-\alpha(t-t_{0})^{q}) \\&\leq&||\varphi_{0}||_{\tau}E_{q}(-\alpha(t-t_{0})^{q}), t>t_{0}. \end{eqnarray*} $

记$ m = ||\varphi_{0}||_{\tau} $,则$ ||x(t)||\leq mE_{q}(-\alpha(t-t_{0})^{q}), t>t_{0}. $这里$ m\geq0 $,且$ m = 0 $成立只有$ \varphi_{0}(s) = 0, s\in[-\tau, 0]. $

从而系统(2.1)的零解是全局$ \rm Mittag-Leffler $稳定的.证明完成.

定理3.2  如果系统(2.1)满足定理3.1的条件,那么系统(2.1)的零解是全局$ \rm Mittag-Leffler $稳定的,所以这个系统是全局渐近稳定的.

证  由于$ E_{\alpha, \beta}(z) = \sum\limits_{k = 0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)}, \alpha>0, \beta>0, z\in C. $则在$ \alpha>0, \beta>0 $的情况下,级数$ \sum\limits_{k = 0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)} $对于任意的$ z\in C $是收敛的.所以级数$ \sum\limits_{k = 0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)} $是有界的,从而$ E_{q}(-\alpha(t-t_{0})^{q}) $是有界的,即$ ||x(t)||\leq C_{1}. $

又知$ 0<q<1, \alpha>0, t>t_{0} $,则$ \arg(-\alpha(t-t_{0})^{q}) = \pi, $由引理2.2有

$ E_{q}(-\alpha(t-t_{0})^{q}) = -\sum\limits_{k = 1}^{p}\frac{[-\alpha(t-t_{0})^{q}]^{-k}}{\Gamma(1-qk)} +O(|-\alpha(t-t_{0})^{q}|^{-1-p}), $

可知$ \lim\limits_{t\rightarrow +\infty}E_{q}(-\alpha(t-t_{0})^{q}) = 0 $,所以$ \lim\limits_{t\rightarrow +\infty}||x(t)|| = 0 . $

所以这个系统的零解是全局渐近稳定的.证明完成.

考虑下面含有时滞的脉冲分数阶非线性微分系统

(4.1) $ \begin{eqnarray} ^{C}_{0}D^q_{t}x_{i}(t) = -\sum\limits_{j = 1}^{n}a_{ij}x_{j}(t)+f_{i}(t, x_{i}(t-\tau)), t\neq t_{k}, t>0, \end{eqnarray} $

这里$ 0<q<1, n = 2, f_{i}(t, x_{i}(t)) = \frac{1}{2}(|x_{i}+1|-|x_{i}-1|), i = 1, 2, \tau = 1. $

$ (a_{ij})_{2\times2} = \left(\begin{array}{cc} a_{11}{\quad}&a_{12}\\ a_{21}{\quad}&a_{22}\\ \end{array}\right) = \left(\begin{array}{cc} 1.2{\quad}&0.1\\ -0.1{\quad}&1.2\\ \end{array}\right), $

且

$ \left\{\begin{array}{ll} { } x_{1}(t^{+}_{k}) = \frac{x_{1}(t_{k})}{4}, {\quad}&k = 1, 2, \cdots, \\ { } x_{2}(t^{+}_{k}) = \frac{x_{2}(t_{k})}{3}, &k = 1, 2, \cdots. \end{array}\right. $

脉冲时刻使得: $ 0<t_{1}<t_{2}<\cdots $,且$ \lim\limits_{k\rightarrow\infty}t_{k} = \infty. $易证$ L_{1} = L_{2} = 1, k_{1} = 1.1, k_{2} = 1 $满足定理3.1的条件.又有$ 0<\sigma_{1k} = \frac{x_{1}(t_{k})-x_{1}(t^{+}_{k})}{x_{1}(t_{k})} = \frac{3}{4}<2, 0<\sigma_{2k} = \frac{x_{2}(t_{k})-x_{2}(t^{+}_{k})}{x_{2}(t_{k})} = \frac{2}{3}<2. $由定理3.1知:系统(4.1)的零解是全局$ \rm Mittag-Leffler $稳定的.