## 3 Main Results  
### Stability Analysis  
In this section, a sufficient condition of asymptotically stable is presented for the proposed flexible robotic arm system with an event-triggered controller in (20). **Theorem 1**. : _For given scalars \(\tau_{1}\), \(\tau_{2}\), \(\sigma\in(0,1)\), the closed-loop system in (20) is asymptotically stable with a \(\mathcal{H}_{\infty}\) performance \(\gamma\) if a matrix \(N\) can be found, in addition to positive definite symmetric matrices \(P_{i}\), \(Q_{1}\), \(Q_{2}\), \(R_{j}(j=1,2,3)\), \(W_{i}(i=1,2,3,4)\) satisfying the following inequalities_\[\Omega_{i}=\left[ \begin{array}{ccc}\Omega_{1i}&\Omega_{2i}&\Omega_{3i}\\ *&\Omega_{4}&0\\ *&*&\Omega_{5}\end{array} \right]-\varsigma(\kappa(t))<0,\;i\in\mathcal{S}. \tag{23}\]  
_where_  
\[\Omega_{1i}= \begin{bmatrix}\Phi_{11i}&P_{1}B_{2}LK_{i}&R_{1}&R_{2} \\ *&\Phi_{22}&R_{3}-N^{T}&R_{3}-N\\ *&*&\Phi_{33}&N\\ *&*&*&\Phi_{44}\end{bmatrix} ,\] \[\Omega_{2i}= \begin{bmatrix}-P_{i}B_{2}LK_{i}&P_{i}B_{1i}&\hat{ \mathcal{C}}\end{bmatrix} ,\;\hat{B}_{2}= \begin{bmatrix}B_{2}^{T}&0_{n}&0_{n}&0_{ n}\end{bmatrix} ^{T},\;\hat{B}_{1i}= \begin{bmatrix}B_{1i}^{T}&0_{n}&0_{n} \end{bmatrix} ^{T},\] \[\Omega_{3i}= \begin{bmatrix}\tau_{1}\mathcal{A}_{i}^{T}(t)&\tau_{2 }\mathcal{A}_{i}^{T}(t)&\tau_{12}\mathcal{A}_{i}^{T}(t)\end{bmatrix} ,\;\hat{ \mathcal{C}}= \begin{bmatrix}C&0_{n}&0_{n}&0_{n}\end{bmatrix} ^{T},\;\tau_{12}= \tau_{2}-\tau_{1},\] \[\Omega_{4}=-\operatorname{diag}\{W,\gamma^{2}I,I\},\;\Omega_{5}= -\operatorname{diag}\{R_{1}^{-1},R_{2}^{-1},R_{3}^{-1}\},\;\hat{P}_{i}=\sum_{j =1}^{s}\pi_{ij}P_{j},\] \[\Phi_{11i}=A_{i}^{T}P_{i}+P_{i}A_{i}+Q_{1}+Q_{2}-R_{1}-R_{2}+\hat{ P}_{i},\;\Phi_{22}=N+N^{T}-2R_{3}+\sigma W,\] \[\[PERSON]_{33}=-R_{3}-R_{1}-Q_{1},\;\;\[PERSON]_{44}=-R_{3}-R_{2}-Q_{2},\] \[\mathcal{A}_{i}(t)= \begin{bmatrix}A_{r}(t)&B_{2}LK_{i}&0_{n}&0_{n} &-B_{2}LK_{i}&B_{1i}\end{bmatrix} .\]  
Proof. : Let us build a Lyapunov-Krasovskii Functional (LKF) as follows:  
\[V(x(t),r(t),\kappa(t))=V_{1}(x(t),r(t)=i)+V_{2}(x(t))+V_{3}(x(t))+\kappa(t) \tag{24}\]  
where  
\[V_{1}(x(t),i) =x^{T}((t)P_{i}x(t),\] \[V_{2}(x(t)) =\int_{t-\tau_{1}}^{t}x^{T}(s)Q_{1}x(s)ds+\int_{t-\tau_{2}}^{t}x^ {T}(s)Q_{2}x(s)ds\] \[V_{3}(x(t)) =\tau_{1}\int_{-\tau_{1}}^{0}\int_{t+\theta}^{t}\dot{x}^{T}(s)R_{ 1}\dot{x}(s)dsd\theta\] \[\quad+\tau_{2}\int_{-\tau_{2}}^{0}\int_{t+\theta}^{t}\dot{x}^{T} (s)R_{2}\dot{x}(s)dsd\theta\] \[\quad+\tau_{12}\int_{-\tau_{2}}^{-\tau_{1}}\int_{t+\theta}^{t} \dot{x}^{T}(s)R_{3}\dot{x}(s)dsd\theta\]  
Taking the derivative of \(t\in \begin{bmatrix}t_{k}+\tau_{t_{k}},t_{k+1}+\tau_{t_{k+1}}\end{bmatrix} \) along the system track, we get  
\[\mathcal{L}V_{1}(x(t),i) =\lim_{\triangle\to 0}\frac{\mathbb{E}\{V(x(t+\triangle),r(t+ \triangle)\}-V(x,i)}{\triangle}\] \[=\dot{x}^{T}(t)P_{i}x(t)+x^{T}(t)P_{i}\dot{x}(t)+x^{T}(t)P_{i}x(t) \tag{25}\]  
where \(\hat{P}_{i}=\sum_{j=1}^{s}\pi_{ij}P_{j}\), and  
\[\dot{V}_{2}(x(t)) =x^{T}(t)Q_{1}x(t)-x^{T}(t-\tau_{1})Q_{1}x(t-\tau_{1})\] \[+x^{T}(t)Q_{2}x(t)-x^{T}(t-\tau_{2})Q_{2}x(t-\tau_{2}) \tag{26}\] \[\dot{V}_{3}(x(t)) =\dot{x}^{T}(t)\dot{R}\dot{x}(t)-\tau_{1}\int_{t-\tau_{1}}^{t} \dot{x}^{T}(s)R_{1}\dot{x}(s)ds\] \[\quad-\tau_{2}\!\!\!\int_{t-\tau_{2}}^{t}\dot{x}^{T}(s)R_{2}\dot{ x}(s)ds-\tau_{2}\!\!\!\int_{t-\tau_{2}}^{t-\tau_{1}}\!\!\!\dot{x}^{T}(s)R_{3}\dot{x}(s)ds\] \[\dot{x}(t) =-\varsigma(\kappa(t))-e^{T}(t)We(t)+\sigma x^{T}(t-\tau(t))WX(t- \tau(t)) \tag{27}\]  
according to [PERSON] complement lemma, \(\xi^{T}(t)(Y_{1}+Y_{2}+Y_{3})\xi(t)+z^{T}(t)z(t)-\gamma^{2}w^{T}(t)w(t)\)\(-\)\(\xi(\kappa(t))\) can be equivalent to (23).