#### 期刊菜单

Stability Analysis of Cohen-Grossberg Neural Networks with Proportional Delays

Abstract: This paper investigates global exponential stability of Cohen-Grossberg neural networks with proportional delays. Firstly, the Cohen-Grossberg neural networks model with proportional delay is equivalent to the Cohen-Grossberg neural networks model with constant delay through appropriate transformation. Sufficient conditions for global exponential stability are established by applying M-matrix theory and inequality techniques. The validity of the obtained conclusions is verified by numerical simulation.

1. 引言

Cohen-Grossberg神经网络 [1] (Cohen-Grossberg neural networks, CGNNs)是一种特殊的递归神经网络。CGNNs比较普遍，涵盖了细胞神经网络 [2] 和Hopfield神经网络 [3] 。递归神经网络在图像处理 [4] [5] [6] 、优化 [7] [8] ，模式识别 [9] [10] ，联想记忆和并行计算等领域得到了广泛的应用。

2. 数学模型及预备知识

$\left\{\begin{array}{l}{\stackrel{˙}{u}}_{i}\left(t\right)=-{a}_{i}\left({u}_{i}\left(t\right)\right)\left[{b}_{i}\left({u}_{i}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{c}_{ij}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{d}_{ij}{g}_{j}\left({u}_{j}\left({q}_{j}t\right)\right)+{I}_{i}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 1\\ {u}_{i}\left(s\right)={\phi }_{i}\left(s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}s\in \left[\rho ,1\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n\end{array}$ (2.1)

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{i}\left(t\right)=-{e}^{t}{a}_{i}\left({x}_{i}\left(t\right)\right)\left[{b}_{i}\left({x}_{i}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{c}_{ij}{f}_{j}\left({x}_{j}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{d}_{ij}{g}_{j}\left({x}_{j}\left(t-{\tau }_{j}\right)\right)+{I}_{i}\right]\\ {x}_{i}\left(s\right)={\psi }_{i}\left(s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}s\in \left[-p,0\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n\end{array}$ (2.2)

${\alpha }_{i}^{-}\le {a}_{i}\left(u\right)\le {\alpha }_{i}^{+},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u\in R$

$\frac{{b}_{i}\left(u\right)-{b}_{i}\left(v\right)}{u-v}\ge {B}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u,v\in R\text{ }\text{ }且\text{ }\text{ }u\ne v$

$|{f}_{j}\left(u\right)-{f}_{j}\left(v\right)|\le {F}_{j}|u-v|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u,v\in R\text{ }\text{ }且\text{ }\text{ }u\ne v$

$|{g}_{j}\left(u\right)-{g}_{j}\left(v\right)|\le {G}_{j}|u-v|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u,v\in R\text{ }\text{ }且\text{ }\text{ }u\ne v$

$B=diag\left({B}_{1},{B}_{2},\cdots ,{B}_{n}\right),F=diag\left({F}_{1},{F}_{2},\cdots ,{F}_{n}\right),G=diag\left({G}_{1},{G}_{2},\cdots ,{G}_{n}\right)$

1) 矩阵A是非奇异M-矩阵。

2) 存在非零向量 $\xi ={\left({\xi }_{1},{\xi }_{2},\cdots ,{\xi }_{n}\right)}^{\text{T}}$ 使得 $A\xi >0$

$|{x}_{i}\left(t\right)-{x}_{i}^{*}|\le K‖\psi -{x}^{*}‖{\text{e}}^{-\lambda t},t\ge 0,$

$‖\psi -{x}^{*}‖=\underset{s\in \left[-\tau ,0\right]}{\mathrm{sup}}|{\psi }_{i}\left(s\right)-{x}_{i}^{*}|$

3. 主要结论

$\left\{\begin{array}{l}{\stackrel{˙}{y}}_{i}\left(t\right)=-{\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right)\left[{\stackrel{¯}{b}}_{i}\left({y}_{i}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{c}_{ij}{\stackrel{¯}{f}}_{j}\left({y}_{j}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{d}_{ij}{\stackrel{¯}{g}}_{j}\left({y}_{j}\left(t-{\tau }_{j}\right)\right)+{I}_{i}\right]\\ {y}_{i}\left(s\right)={\varphi }_{i}\left(s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}s\in \left[-p,0\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n\end{array}$ (3.1)

${\stackrel{¯}{a}}_{i}\left({y}_{i}\right)={a}_{i}\left({y}_{i}-{x}_{i}^{*}\right)-{a}_{i}\left({x}_{i}^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{b}}_{i}\left({y}_{i}\right)={b}_{i}\left({y}_{i}-{x}_{i}^{*}\right)-{b}_{i}\left({x}_{i}^{*}\right)$

${\stackrel{¯}{f}}_{j}\left({y}_{j}\right)={f}_{j}\left({y}_{j}-{x}_{j}^{*}\right)-{f}_{j}\left({x}_{j}^{*}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{g}}_{j}\left({y}_{j}\right)={g}_{j}\left({y}_{j}-{x}_{j}^{*}\right)-{g}_{j}\left({x}_{j}^{*}\right)$

${\varphi }_{i}\left(s\right)={\psi }_{i}\left(s\right)-{x}_{i}^{*},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i,j=1,2,\cdots ,n$

$-{B}_{i}{\xi }_{i}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\xi }_{j}|{c}_{ij}|{F}_{j}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\xi }_{j}|{d}_{ij}|{G}_{j}<0$ (3.2)

${H}_{i}\left(x\right)=-\left({B}_{i}-{x}_{i}\right){\xi }_{i}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\xi }_{j}|{c}_{ij}|{F}_{j}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\text{e}}^{{x}_{j}\tau }{\xi }_{j}|{d}_{ij}|{G}_{j}$ (3.3)

${H}_{i}\left(0\right)=-{B}_{i}{\xi }_{i}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\xi }_{j}|{c}_{ij}|{F}_{j}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\xi }_{j}|{d}_{ij}|{G}_{j}<0$ (3.4)

${H}_{i}\left(\sigma \right)=-\left({B}_{i}-{\sigma }_{i}\right){\xi }_{i}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\xi }_{j}|{c}_{ij}|{F}_{j}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\text{e}}^{{\sigma }_{j}\tau }{\xi }_{j}|{d}_{ij}|{G}_{j}=0$ (3.5)

${H}_{i}\left({\sigma }^{-}\right)=-\left({B}_{i}-{\sigma }_{i}^{-}\right){\xi }_{i}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\xi }_{j}|{c}_{ij}|{F}_{j}+\underset{j=1}{\overset{N}{\sum }}\text{ }\text{ }{\text{e}}^{{\sigma }_{j}^{-}\tau }{\xi }_{j}|{d}_{ij}|{G}_{j}<0$ (3.6)

${V}_{i}\left(t\right)={\text{e}}^{\lambda t}|{y}_{i}\left(t\right)|,i=1,2,\cdots ,n$ (3.7)

${V}_{i}\left(t\right)$ 求右上导：

$\begin{array}{c}{D}^{+}{V}_{i}\left(t\right)={\text{e}}^{\lambda t}\mathrm{sgn}\left({y}_{i}\left(t\right)\right)\left\{-{\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right)\left[{\stackrel{¯}{b}}_{i}\left({y}_{i}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{c}_{ij}{\stackrel{¯}{f}}_{j}\left({y}_{j}\left(t\right)\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{d}_{ij}{\stackrel{¯}{g}}_{j}\left({y}_{j}\left(t-{\tau }_{j}\right)\right)\right]\right\}+\lambda {\text{e}}^{\lambda t}|{y}_{i}\left(t\right)|\\ \le -{\text{e}}^{\lambda t}{\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right)\left[|{\stackrel{¯}{b}}_{i}\left({y}_{i}\left(t\right)\right)|-|\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{c}_{ij}{\stackrel{¯}{f}}_{j}\left({y}_{j}\left(t\right)\right)|-|\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{d}_{ij}{\stackrel{¯}{g}}_{j}\left({y}_{j}\left(t-{\tau }_{j}\right)\right)|\right]+\lambda {\text{e}}^{\lambda t}|{y}_{i}\left(t\right)|\end{array}$

$\begin{array}{l}\le -{\text{e}}^{\lambda t}{\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right)\left[{B}_{i}|\left({y}_{i}\left(t\right)\right)|-\underset{j=1}{\overset{n}{\sum }}|{c}_{ij}|{F}_{j}|{y}_{i}\left(t\right)|-\underset{j=1}{\overset{n}{\sum }}|{d}_{ij}|{G}_{j}|{y}_{j}\left(t-{\tau }_{j}\right)|\right]+\lambda {\text{e}}^{\lambda t}|{y}_{i}\left(t\right)|\\ \le -{\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right)\left[{B}_{i}{\text{e}}^{\lambda t}|\left({y}_{i}\left(t\right)\right)|-\underset{j=1}{\overset{n}{\sum }}|{c}_{ij}|{F}_{j}{\text{e}}^{\lambda t}|{y}_{i}\left(t\right)|-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{\text{e}}^{\lambda {\tau }_{j}}|{d}_{ij}|{G}_{j}{\text{e}}^{\lambda \left(t-{\tau }_{j}\right)}|{y}_{j}\left(t-{\tau }_{j}\right)|\right]+\lambda {\text{e}}^{\lambda t}|{y}_{i}\left(t\right)|\end{array}$

$\begin{array}{l}\le -{\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right)\left[{B}_{i}{V}_{i}\left(t\right)-\underset{j=1}{\overset{n}{\sum }}|{c}_{ij}|{F}_{j}{V}_{i}\left(t\right)-\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{\text{e}}^{\lambda {\tau }_{j}}|{d}_{ij}|{G}_{j}{V}_{i}\left(t-{\tau }_{j}\right)\right]+\lambda {\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right){V}_{i}\left(t\right)\\ ={\text{e}}^{t}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left(t\right)\right)\left[-\left({B}_{i}-\lambda \right){V}_{i}\left(t\right)+\underset{j=1}{\overset{n}{\sum }}|{c}_{ij}|{F}_{j}{V}_{i}\left(t\right)+\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{\text{e}}^{\lambda {\tau }_{j}}|{d}_{ij}|{G}_{j}{V}_{i}\left(t-{\tau }_{j}\right)\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n,\text{\hspace{0.17em}}t\ge 0\end{array}$ (3.8)

$q=\frac{‖\psi -{x}^{*}‖}{{\mathrm{min}}_{1\le i\le n}\left\{{\xi }_{i}\right\}}$ ，则

${V}_{i}\left(s\right)={\text{e}}^{\lambda s}|{y}_{i}\left(s\right)|\le |{\varphi }_{i}\left(s\right)|=|{\psi }_{i}\left(s\right)-{x}_{i}^{*}|\le {\xi }_{i}q$ (3.9)

${V}_{i}\left(t\right)\le {\xi }_{i}q,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 0,\text{\hspace{0.17em}}i=1,2,\cdots ,n$ (3.10)

${V}_{i}\left({t}^{*}\right)\le {\xi }_{i}q,{D}^{*}\left({V}_{i}\left({t}^{*}\right)\right)\ge 0$ ，且当 $t\in \left[-\rho ,{t}^{*}\right]$ 时， ${V}_{i}\left(t\right)\le {\xi }_{i}q$ (3.11)

$\begin{array}{c}{D}^{+}{V}_{i}\left({t}^{*}\right)\le {\text{e}}^{{t}^{*}}{\stackrel{¯}{a}}_{i}\left({y}_{i}\left({t}^{*}\right)\right)\left[-\left({B}_{i}-\lambda \right){V}_{i}\left({t}^{*}\right)+\underset{j=1}{\overset{n}{\sum }}|{c}_{ij}|{F}_{j}{V}_{i}\left({t}^{*}\right)+\underset{j=1}{\overset{n}{\sum }}\text{ }\text{ }{\text{e}}^{\lambda {\tau }_{j}}|{d}_{ij}|{G}_{j}{V}_{i}\left({t}^{*}-{\tau }_{j}\right)\right]\\ <0\end{array}$ (3.12)

$|{y}_{i}\left(t\right)|\le {\text{e}}^{-\lambda t}{\xi }_{i}q\le {\text{e}}^{-\lambda t}\frac{{\sum }_{i=1}^{n}{\xi }_{i}}{\mathrm{min}\left\{{\xi }_{i}\right\}}‖\psi -{x}^{*}‖=K‖\psi -{x}^{*}‖{\text{e}}^{-\lambda t}$

$|{x}_{i}\left(t\right)-{x}^{*}|\le K‖\psi -{x}^{*}‖{\text{e}}^{-\lambda t}$ (3.13)

4. 数值实验

$\left\{\begin{array}{l}{\stackrel{˙}{u}}_{1}\left(t\right)=-{a}_{1}\left({u}_{1}\left(t\right)\right)\left[{b}_{1}\left({u}_{1}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{c}_{1j}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{d}_{1j}{g}_{j}\left({u}_{j}\left(0.6t\right)\right)\right]\\ {\stackrel{˙}{u}}_{2}\left(t\right)=-{a}_{2}\left({u}_{2}\left(t\right)\right)\left[{b}_{2}\left({u}_{2}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{c}_{2j}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{d}_{2j}{g}_{j}\left({u}_{j}\left(0.8t\right)\right)\right]\end{array}$ (4.1)

${a}_{1}\left(u\right)=1+0.5\mathrm{cos}\left(u\right),\text{\hspace{0.17em}}{a}_{2}\left(u\right)=2+0.5\mathrm{cos}\left(u\right);\text{\hspace{0.17em}}{b}_{1}\left(u\right)=5u,\text{\hspace{0.17em}}{b}_{2}\left(u\right)=12u$

${f}_{i}\left(u\right)=0.5\left(|u+1|-|u-1|\right),\text{\hspace{0.17em}}{g}_{i}\left(u\right)=\mathrm{tanh}\left(u\right),\text{\hspace{0.17em}}i=1,2$

$C={\left({c}_{ij}\right)}_{2×2}=\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right],\text{\hspace{0.17em}}D={\left({d}_{ij}\right)}_{2×2}=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$

${\alpha }_{1}^{-}=0.5,\text{\hspace{0.17em}}{\alpha }_{1}^{+}=1.5,\text{\hspace{0.17em}}{\alpha }_{2}^{-}=1.5,\text{\hspace{0.17em}}{\alpha }_{2}^{+}=2.5$

$B=diag\left(5,12\right),F=G=diag\left(1,1\right)$

$B-|C|F-|D|G=\left[\begin{array}{cc}3& -3\\ -3& 10\end{array}\right]$

Figure 1. State response of (4.1)

Figure 2. Phase trajectories of system (4.1)

$\left\{\begin{array}{l}{\stackrel{˙}{u}}_{1}\left(t\right)=-{a}_{1}\left({u}_{1}\left(t\right)\right)\left[{b}_{1}\left({u}_{1}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{c}_{1j}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{d}_{1j}{g}_{j}\left({u}_{j}\left(0.5t\right)\right)-1\right]\\ {\stackrel{˙}{u}}_{2}\left(t\right)=-{a}_{2}\left({u}_{2}\left(t\right)\right)\left[{b}_{2}\left({u}_{2}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{c}_{2j}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{2}{\sum }}\text{ }\text{ }{d}_{2j}{g}_{j}\left({u}_{j}\left(0.5t\right)\right)+1\right]\end{array}$ (4.2)

${a}_{1}\left(u\right)=1+0.5\mathrm{cos}\left(u\right),\text{\hspace{0.17em}}{a}_{2}\left(u\right)=2+0.5\mathrm{cos}\left(u\right);\text{\hspace{0.17em}}{b}_{1}\left(u\right)=5u,\text{\hspace{0.17em}}{b}_{2}\left(u\right)=12u$

${f}_{i}\left(u\right)=0.5\left(|u+1|-|u-1|\right),\text{\hspace{0.17em}}{g}_{i}\left(u\right)=\mathrm{tanh}\left(u\right),\text{\hspace{0.17em}}i=1,2$

$C={\left({c}_{ij}\right)}_{2×2}=\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right],\text{\hspace{0.17em}}D={\left({d}_{ij}\right)}_{2×2}=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$

${\alpha }_{1}^{-}=0.5,\text{\hspace{0.17em}}{\alpha }_{1}^{+}=1.5,\text{\hspace{0.17em}}{\alpha }_{2}^{-}=1.5,\text{\hspace{0.17em}}{\alpha }_{2}^{+}=2.5$

$B=diag\left(5,12\right),F=G=diag\left(1,1\right)$

$B-|C|F-|D|G=\left[\begin{array}{cc}3& -3\\ -3& 10\end{array}\right]$

Figure 3. State response of (4.2)

Figure 4. Phase trajectories of system (4.2)

$\left\{\begin{array}{l}{\stackrel{˙}{u}}_{1}\left(t\right)=-{a}_{1}\left({u}_{1}\left(t\right)\right)\left[{b}_{1}\left({u}_{1}\left(t\right)\right)-\underset{j=1}{\overset{3}{\sum }}\text{ }\text{ }{c}_{1j}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{3}{\sum }}\text{ }\text{ }{d}_{1j}{g}_{j}\left({u}_{j}\left(0.5t\right)\right)\right]\\ {\stackrel{˙}{u}}_{2}\left(t\right)=-{a}_{2}\left({u}_{2}\left(t\right)\right)\left[{b}_{2}\left({u}_{2}\left(t\right)\right)-\underset{j=1}{\overset{3}{\sum }}\text{ }\text{ }{c}_{1j}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{3}{\sum }}\text{ }\text{ }{d}_{1j}{g}_{j}\left({u}_{j}\left(0.6t\right)\right)\right]\\ {\stackrel{˙}{u}}_{3}\left(t\right)=-{a}_{3}\left({u}_{3}\left(t\right)\right)\left[{b}_{3}\left({u}_{3}\left(t\right)\right)-\underset{j=1}{\overset{3}{\sum }}\text{ }\text{ }{c}_{1j}{f}_{j}\left({u}_{j}\left(t\right)\right)-\underset{j=1}{\overset{3}{\sum }}\text{ }\text{ }{d}_{1j}{g}_{j}\left({u}_{j}\left(0.8t\right)\right)\right]\end{array}$ (4.3)

${a}_{1}\left(u\right)=1+0.5\mathrm{cos}\left(u\right),\text{\hspace{0.17em}}{a}_{2}\left(u\right)=1+0.5\mathrm{cos}\left(u\right),\text{\hspace{0.17em}}{a}_{3}\left(u\right)=2+0.5\mathrm{sin}\left( u \right)$

${b}_{1}\left(u\right)=6u,\text{\hspace{0.17em}}{b}_{2}\left(u\right)=8u,\text{\hspace{0.17em}}{b}_{3}\left(u\right)=10u$

${f}_{i}\left(u\right)=0.5\left(|u+1|-|u-1|\right),\text{\hspace{0.17em}}{g}_{i}\left(u\right)=\mathrm{tanh}\left(u\right),\text{\hspace{0.17em}}i=1,2,3$

$C={\left({c}_{ij}\right)}_{3×3}=\left[\begin{array}{ccc}1& -2& 2\\ 2& -1& -1\\ -1& 3& 2\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}D={\left({d}_{ij}\right)}_{3×3}=\left[\begin{array}{ccc}1& -1& 2\\ -1& -1& -2\\ 2& -1& -1\end{array}\right]$

${\alpha }_{1}^{-}={\alpha }_{2}^{-}=0.5,\text{\hspace{0.17em}}{\alpha }_{3}^{-}=1.5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{1}^{+}={\alpha }_{2}^{+}=1.5,\text{\hspace{0.17em}}{\alpha }_{3}^{+}=2.5$

$B=diag\left(5,8,10\right),F=G=diag\left(1,1,1\right)$

$B-|C|F-|D|G=\left[\begin{array}{ccc}4& -3& -4\\ -3& 7& -3\\ -3& -4& 7\end{array}\right]$

Figure 5. State response of (4.3)

Figure 6. Phase trajectories of system (4.3)

5. 总结

NOTES

*通讯作者。

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