**4. Event-Triggered Control Scheme**

In this section, we introduce an event generator in the controller node by using the following judgment algorithm

$$\left[\mathfrak{e}\left(\left(\mathfrak{k}+\mathfrak{j}\right)\mathfrak{h}\right)-\mathfrak{e}\left(\mathfrak{k}\mathfrak{h}\right)\right]^{T}\Phi\left[\mathfrak{e}\left(\left(\mathfrak{k}+\mathfrak{j}\right)\mathfrak{h}\right)-\mathfrak{e}\left(\mathfrak{k}\mathfrak{h}\right)\right]\leq\Sigma\mathfrak{e}^{T}\left(\left(\mathfrak{k}+\mathfrak{j}\right)\mathfrak{h}\right)\Phi\mathfrak{e}\left(\left(\mathfrak{k}+\mathfrak{j}\right)\mathfrak{h}\right),\tag{31}$$

where Φ is a positive definite matrix to be determined, k, j ∈ Z<sup>+</sup> and kh denotes the release instant, e((k + j)h) = v((k + j) − w((k + j)h) is the error information at the instant (k + j)h, and *σ* ∈ [0, 1) is a given constant. Cases A and B relate to the following delayed differential equation

$$\mathcal{O}^{\mathfrak{a}}\mathfrak{e}(\mathfrak{t}) = -(\mathcal{A} + \Delta\mathcal{A}(\mathfrak{t}))\mathfrak{e}(\mathfrak{t}) + (\mathcal{A} + \Delta\mathcal{C}(\mathfrak{t}))\mathfrak{h}(\mathfrak{e}(\mathfrak{t})) + (\mathcal{A} + \Delta\mathcal{A}(\mathfrak{t}))\mathfrak{h}(\mathfrak{e}(\mathfrak{t} - \sigma(\mathfrak{t})) $$

$$+ (\omega\mathfrak{t} + \Delta\omega\mathfrak{e}(\mathfrak{t})) \int\_{\mathfrak{t}-\eta}^{\mathfrak{t}} \mathfrak{h}(\mathfrak{e}(\mathfrak{s})) \mathfrak{g}\mathfrak{s} + \mathcal{A}\mathfrak{e}\mathcal{K}\mathfrak{e}(\mathfrak{t}\mathfrak{t})\mathfrak{h}\mathfrak{e}\mathfrak{t} \in [\mathfrak{t}\mathfrak{h} + \tau\mathfrak{e}, t\_{\mathfrak{t}}, t\_{\mathfrak{t}} + \tau\_{\mathfrak{t}+\mathbbm{1}}). \tag{32}$$

Case A: if *t*kh + h + *τ*¯ ≥ tk+1h + *τ*k+1, we can define *τ*(*t*) as

$$
\tau(\mathfrak{t}) = \mathfrak{t} - t\mathfrak{s}\mathfrak{h}, \\
\mathfrak{t} \in \left[t\mathfrak{h} + \tau\mathfrak{s}, t\mathfrak{e} + \mathfrak{s}\mathfrak{e}, \mathfrak{z}\right].
$$

It can be seen that

$$
\tau\_{\mathfrak{k}} \le \tau(\mathfrak{k}) \le (\mathfrak{k}\_{\mathfrak{k}+\mathfrak{a}} - \mathfrak{k}\_{\mathfrak{k}})\mathfrak{k} + \mathfrak{k}\_{\mathfrak{k}+\mathfrak{a}} \le \mathfrak{k} + \mathfrak{r}.
$$

Case B: if *t*kh + h + *τ*¯ < tk+1h + *τ*k+1, since t<sup>k</sup> ≤ *τ*¯, we can easily demonstrate that a positive constant m exists such that tkh + mh + *τ*¯ < tk+1h + *τ*k+<sup>1</sup> ≤ tkh + (m + 1)h + *τ*¯. For the time intervals [tkh + *τ*k,tk+1h + *τ*k+1), we divide them as F<sup>0</sup> = [tkh + *τ*k, *t*kh + h + *τ*¯), F<sup>i</sup> = [tkh + ih + *τ*¯, tkh + *i*h + h + *τ*¯), and F<sup>m</sup> = [tkh + mh + *τ*¯, tk+1h + *τ*k+1), and we define *τ*(t) as

$$\tau(\mathfrak{t}) = \mathfrak{t} - \mathfrak{t}\_{\mathfrak{k}}(\mathfrak{t}) - \mathfrak{i}\mathfrak{h}, \mathfrak{i}\mathfrak{t} \in \mathcal{F}\_{\mathfrak{k}}, \mathfrak{i} = 0, 1, \dots, \mathfrak{m}.$$

It is easy to prove that 0 ≤ *τ*<sup>k</sup> ≤ *τ*((*t*)) ≤ h + *τ*¯ = *τM*,t ∈ [tkh + *τ*k,tk+1h + *τ*k+1). Finally, we define

$$\mathfrak{e}\_{\mathbb{P}}(t) = \mathfrak{e}(t\_{\mathbb{P}}\mathfrak{h}) - \mathfrak{e}(t\_{\mathbb{P}}\mathfrak{h} + \mathrm{i}\mathfrak{h}), \mathfrak{t} \in \mathcal{\mathbb{P}}\_{\mathbb{P}}, \mathfrak{i} = \mathfrak{0}, 1, \dots, m. \tag{33}$$

For case A, *m* = 0, we have ek(t) = 0 from (33). Based on the analysis above, the event generator (31) can be rewritten as

$$
\mathfrak{e}\_{\mathbb{F}}^T(\mathfrak{t})\Phi\mathfrak{e}\_{\mathbb{F}}((t)) \le \Sigma\mathfrak{e}^T(\mathfrak{t}-\mathfrak{r}(\mathfrak{t}))\Phi\mathfrak{e}^T(\mathfrak{t}-\mathfrak{r}(\mathfrak{t}),\mathfrak{t} \in [\mathfrak{t}\mathfrak{h}+\mathfrak{r},\mathfrak{t}\_{\mathfrak{t}+\mathfrak{r}}\mathfrak{h}+\mathfrak{r}\_{\mathfrak{t}+\mathfrak{l}}).
$$

Then, the system is reduced to

$$\begin{split} \partial^{\mathfrak{a}}\mathfrak{e}(\mathfrak{t}) &= -\mathcal{A}\mathfrak{e}(\mathfrak{t}) + \mathcal{C}\mathfrak{h}(\mathfrak{e}(\mathfrak{t})) + \mathcal{A}\mathfrak{h}(\mathfrak{e}(\mathfrak{t}-\mathfrak{e}(\mathfrak{t})) \\ &+ \mathcal{A} \int\_{\mathfrak{t}-\eta}^{\mathfrak{t}} \mathfrak{e}(\mathfrak{s}) \mathfrak{s}\mathfrak{s} + \mathcal{A}^{\mathcal{C}}\mathcal{K}\mathfrak{e}(\mathfrak{t}) + \mathcal{A}^{\mathcal{C}}\mathcal{K}\mathfrak{e}(\mathfrak{t}-\mathfrak{e}(\mathfrak{t})). \end{split} \tag{34}$$

**Theorem 3.** *For the given scalars δ*1, *δ*2, *δ*3, *δ*4, *δ*5, *μ*1, *and σ and the diagonal matrices* L1, L2*, and* L3*, if there exist symmetric positive definite matrices* R<sup>1</sup> > 0, R<sup>2</sup> > 0, *then a feasible solution exists for the following LMIs:*

$$
\xi \ll 0.\tag{35}
$$

**Proof.** Now, let us define the Lyapunov–Krasovskii functional as follows:

$$\mathcal{V}'(\mathfrak{t}) = \mathcal{V}\_1(\mathfrak{t}) + \mathcal{V}\_2(\mathfrak{t}),\tag{36}$$

where

$$\begin{aligned} \mathcal{V}\_1(t) &= \mathfrak{e}^{\mathcal{F}}(t)\mathcal{R}\_1\mathfrak{e}(t), \\ \mathcal{V}\_2(t) &= \mathcal{R}^{(-\mathfrak{a}+1)} \int\_{t-\mathfrak{e}(t)}^t \mathfrak{e}^{\mathcal{F}}(\mathfrak{s})\mathcal{R}\_2\mathfrak{e}(\mathfrak{s})ds. \end{aligned}$$

Using Lemma 2, we have

2e <sup>T</sup> (t)R1ΔR(t)e(t) <sup>≤</sup> <sup>2</sup><sup>e</sup> <sup>T</sup> (t)R1JdK (t)Lde(t), <sup>≤</sup> *<sup>δ</sup>*−<sup>1</sup> <sup>1</sup> e <sup>T</sup> (t)R1JdJ <sup>T</sup> <sup>d</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*1e <sup>T</sup> (t)L <sup>T</sup> r Lre(t), (37) 2e <sup>T</sup> (t)R1Δ<sup>C</sup> (t)h(e(t)) <sup>≤</sup> <sup>2</sup><sup>e</sup> <sup>T</sup> (t)R1JcK (t)Lch(e(t)), <sup>≤</sup> *<sup>δ</sup>*−<sup>1</sup> <sup>2</sup> e <sup>T</sup> (t)R1JcJ *<sup>T</sup>* <sup>c</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*2e <sup>T</sup> (t)*φ*<sup>T</sup> L <sup>T</sup> c Lc*φ*e(t), (38) 2e <sup>T</sup> (t)R1ΔB(t)h(*e*(*<sup>t</sup>* <sup>−</sup> *<sup>σ</sup>*(*t*))) <sup>≤</sup> <sup>2</sup><sup>e</sup> <sup>T</sup> (t)R1Jb<sup>K</sup> (t)Lbh(e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*))), <sup>≤</sup> *<sup>δ</sup>*−<sup>1</sup> <sup>3</sup> e <sup>T</sup> (t)R1JbJ <sup>T</sup> <sup>b</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*3e <sup>T</sup> (<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(t))*φ*<sup>T</sup> <sup>L</sup> <sup>T</sup> <sup>b</sup> Lb*φ*e(*t* − *σ*(*t*)). (39)

Then, with the support of Lemma 1 and the linearity nature of the Caputo fractionalorder derivative, the fractional derivative along the trajectories of the system state is acquired as follows

*<sup>D</sup>α*<sup>V</sup> (t) <sup>≤</sup> <sup>2</sup><sup>e</sup> <sup>T</sup> (t)R1D*α*e(t), ≤ 2e <sup>T</sup> (t)R<sup>1</sup> − (<sup>R</sup> + ΔR(t))e(t)+(<sup>C</sup> + Δ<sup>C</sup> (t))h(e(t)) + (<sup>B</sup> <sup>+</sup> <sup>Δ</sup>B(t))h(e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*)) + H K <sup>e</sup>(t) + H K <sup>e</sup>(<sup>t</sup> <sup>−</sup> *<sup>τ</sup>*(t)) , ≤ −2e <sup>T</sup> (t)R1Re(t) + *δ*−<sup>1</sup> <sup>1</sup> e <sup>T</sup> (t)R1JdJ <sup>T</sup> <sup>d</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*1e <sup>T</sup> (t)L <sup>T</sup> d Lde(t) + 2e <sup>T</sup> (t)R1<sup>C</sup> h(e(t)) + *δ*−<sup>1</sup> <sup>2</sup> e <sup>T</sup> (t)R1JcJ <sup>T</sup> <sup>c</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*2e <sup>T</sup> (t)*φ*<sup>T</sup> L <sup>T</sup> c Lc*φ*e(t) + 2e <sup>T</sup> (t)R1Bh(*e*(*<sup>t</sup>* <sup>−</sup> *<sup>σ</sup>*(*t*))) + *<sup>δ</sup>*−<sup>1</sup> <sup>3</sup> e <sup>T</sup> (t)R1JbJ <sup>T</sup> <sup>b</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*3e <sup>T</sup> (<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*))*φ*<sup>T</sup> <sup>L</sup> <sup>T</sup> <sup>b</sup> <sup>L</sup>b*φ*e(t − *σ*(*t*)) + e <sup>T</sup> (t)R2e(t) <sup>−</sup> <sup>e</sup> <sup>T</sup> (<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*))R2e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*))(<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*). (40)

From Assumption 1, we have

$$
\mathbb{E}\begin{bmatrix}
\mathsf{e}(\mathsf{t}) \\
\mathsf{h}(\mathsf{e}(\mathsf{t}))
\end{bmatrix}^{T}
\begin{bmatrix}
\ast & -\mathcal{L}\ell\_{1}
\end{bmatrix}
\begin{bmatrix}
\mathsf{e}(\mathsf{t}) \\
\mathsf{h}(\mathsf{e}(\mathsf{t}))
\end{bmatrix} \leq 0\tag{41}
$$

$$
\begin{bmatrix}
\mathfrak{e}(\mathfrak{t}-\sigma(\mathfrak{t})) \\
\mathfrak{h}(\mathfrak{e}(\mathfrak{t}-\sigma(\mathfrak{t})))
\end{bmatrix}^{T} \begin{bmatrix}
\ast & -\mathcal{Q}\_{2}
\end{bmatrix} \begin{bmatrix}
\mathfrak{e}(\mathfrak{t}-\sigma(\mathfrak{t})) \\
\mathfrak{h}(\mathfrak{e}(\mathfrak{t}-\sigma(\mathfrak{t})))
\end{bmatrix} \le 0 \tag{42}
$$

$$
\begin{bmatrix}
\mathfrak{e}(\mathfrak{t}-\mathfrak{r}(\mathfrak{t})) \\
\mathfrak{h}(\mathfrak{e}(\mathfrak{t}-\mathfrak{r}(\mathfrak{t})))
\end{bmatrix}^{T} \begin{bmatrix}
\ast & -\mathcal{A}\_{3} \\
\end{bmatrix} \begin{bmatrix}
\mathfrak{e}(\mathfrak{t}-\mathfrak{r}(\mathfrak{t})) \\
\mathfrak{h}(\mathfrak{e}(\mathfrak{t}-\mathfrak{r}(\mathfrak{t})))
\end{bmatrix} \le 0. \tag{43}
$$

From (37)–(43), we obtain

*<sup>D</sup>α*<sup>V</sup> (t) <sup>≤</sup> <sup>2</sup><sup>e</sup> <sup>T</sup> (t)R1D*α*e(t), ≤ 2e <sup>T</sup> (t)R<sup>1</sup> − (R + ΔR(t))e(t)+(C + ΔC (t))h(e(t)) + (<sup>B</sup> <sup>+</sup> <sup>Δ</sup>B(t))h(e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*)) + H K <sup>e</sup>(t) + H K <sup>e</sup>(<sup>t</sup> <sup>−</sup> *<sup>τ</sup>*(t)) , ≤ −2e <sup>T</sup> (t)R1Re(t) + *δ*−<sup>1</sup> <sup>1</sup> e <sup>T</sup> (t)R1JdJ <sup>T</sup> <sup>d</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*1e <sup>T</sup> (t)L <sup>T</sup> d Lde(t) + 2e <sup>T</sup> (t)R1C h(e(t)) + *δ*−<sup>1</sup> <sup>2</sup> e <sup>T</sup> (t)R1JcJ <sup>T</sup> <sup>c</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*2e <sup>T</sup> (t)*φ*<sup>T</sup> L <sup>T</sup> c Lc*φ*e(t) + 2e <sup>T</sup> (t)R1Bh(*e*(*<sup>t</sup>* <sup>−</sup> *<sup>σ</sup>*(*t*))) + *<sup>δ</sup>*−<sup>1</sup> <sup>3</sup> e <sup>T</sup> (t)R1JbJ <sup>T</sup> <sup>b</sup> R<sup>T</sup> <sup>1</sup> e(t) + *δ*3e <sup>T</sup> (<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*))*φ*<sup>T</sup> <sup>L</sup> <sup>T</sup> <sup>b</sup> Lb*φ*e(t − *σ*(*t*)) + *δ*−<sup>1</sup> <sup>4</sup> e <sup>T</sup> (*t*)R1C *<sup>T</sup>*C R*<sup>T</sup>* <sup>1</sup> e(t) + *δ*4e <sup>T</sup> (*t*)*φTφ*e(t) + *δ*−<sup>1</sup> <sup>5</sup> e <sup>T</sup> (*t*)R1B*T*BR*<sup>T</sup>* <sup>1</sup> e(t) + *δ*5e <sup>T</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>σ</sup>*(*t*))*φTφ*e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(t)) + e <sup>T</sup> (t)R2e(t) <sup>−</sup> <sup>e</sup> <sup>T</sup> (<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*))R2e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(*t*)) + e(t) <sup>h</sup>(e(t)) *T* <sup>−</sup>L1Γ<sup>2</sup> <sup>L</sup>1Γ<sup>1</sup> ∗ −L<sup>1</sup> e(t) <sup>h</sup>(e(t)) + <sup>e</sup>(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(t)) <sup>h</sup>(e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(t))) *T* <sup>−</sup>L2Γ<sup>2</sup> <sup>L</sup>2Γ<sup>1</sup> ∗ −L<sup>2</sup> <sup>e</sup>(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(t)) <sup>h</sup>(e(<sup>t</sup> <sup>−</sup> *<sup>σ</sup>*(t))) + <sup>e</sup>(<sup>t</sup> <sup>−</sup> *<sup>τ</sup>*(t)) <sup>h</sup>(e(<sup>t</sup> <sup>−</sup> *<sup>τ</sup>*(t))) *T* <sup>−</sup>L3Γ<sup>2</sup> <sup>L</sup>3Γ<sup>1</sup> ∗ −L<sup>3</sup> <sup>e</sup>(<sup>t</sup> <sup>−</sup> *<sup>τ</sup>*(t)) <sup>h</sup>(e(<sup>t</sup> <sup>−</sup> *<sup>τ</sup>*(t))) .

Then,

$$
\Lambda = \begin{bmatrix}
\Lambda\_{11} & \Lambda\_{12} & 0 & \Lambda\_{14} & \Lambda\_{15} & 0 & \Lambda\_{17} & \Lambda\_{18} & \Lambda\_{19} \\
\* & \Lambda\_{22} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\* & \* & \Lambda\_{33} & \Lambda\_{34} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & \* & \Lambda\_{44} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & \Lambda\_{55} & 0 & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \Lambda\_{66} & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \* & \Lambda\_{77} & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \* & \Lambda\_{88} & 0 \\
\* & \* & \* & \* & \* & \* & \* & \* & \Lambda\_{99}
\end{bmatrix} < 0,
\tag{44}
$$

where

$$\begin{split} \Lambda\_{11} &= -2\mathcal{R}\_{1}\mathcal{R} + \delta\_{1}\mathcal{L}\_{r}^{\mathcal{F}}\mathcal{L}\_{r} + \delta\_{2}\mathcal{\Phi}^{\top}\mathcal{L}\_{c}^{\mathcal{F}}\mathcal{L}\_{c}\Phi + \mathcal{R}\_{2} + 2\mathcal{R}\_{1}\mathcal{H}, \mathcal{K} \\ &- \mathcal{L}\_{1}\Gamma\_{2} - \Phi, \Lambda\_{12} = \mathcal{R}\_{1}\mathcal{H} + \mathcal{L}\_{1}\Gamma\_{1}, \Lambda\_{14} = \mathcal{R}\_{1}\mathcal{R}, \Lambda\_{15} = \mathcal{R}\_{1}\mathcal{H}^{\top}\mathcal{K}, \\ \Lambda\_{17} &= \mathcal{R}\_{1}\mathcal{J}\_{5}, \Lambda\_{18} = \mathcal{R}\_{1}\mathcal{J}\_{6}, \Lambda\_{19} = \mathcal{R}\_{1}\mathcal{J}\_{6}, \Lambda\_{22} = -\mathcal{L}\_{1}, \Lambda\_{33} = \delta\_{3}\Phi^{T}\mathcal{L}\_{b}^{T}\mathcal{L}\_{b}\Phi \\ &- \mathcal{R}\_{2}(1-\mu) - \mathcal{L}\_{2}\Gamma\_{2}, \Lambda\_{34} = \mathcal{L}\_{2}\Gamma\_{1}, \Lambda\_{44} = -\mathcal{L}\_{2}, \Lambda\_{55} = \Sigma\Phi - \mathcal{L}\_{3}\Gamma\_{2}, \\ \Lambda\_{56} &= \mathcal{L}\_{3}\Gamma\_{1}, \Lambda\_{66} = -\mathcal{L}\_{3}, \Lambda\_{77} = -\delta\_{1}\mathcal{J}, \Lambda\_{88} = -\delta\_{2}\mathcal{J}, \Lambda\_{99} = -\delta\_{3}\mathcal{J}. \end{split}$$

We pre- and post-multiply <sup>Λ</sup> with {R−<sup>1</sup> <sup>1</sup> , <sup>I</sup> , <sup>I</sup> , <sup>I</sup> , <sup>R</sup>−<sup>1</sup> <sup>1</sup> , I , I , I , I }

$$\mathbf{Y} = \begin{bmatrix} \mathbf{Y}\_{11} & \mathbf{Y}\_{12} & \mathbf{0} & \mathbf{Y}\_{14} & \mathbf{Y}\_{15} & \mathbf{0} & \mathbf{Y}\_{17} & \mathbf{Y}\_{18} & \mathbf{Y}\_{19} \\ \ast & \mathbf{Y}\_{22} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \ast & \ast & \mathbf{Y}\_{33} & \mathbf{Y}\_{34} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \ast & \ast & \ast & \mathbf{Y}\_{44} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \ast & \ast & \ast & \ast & \mathbf{Y}\_{55} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \ast & \ast & \ast & \ast & \ast & \mathbf{Y}\_{66} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \ast & \ast & \ast & \ast & \ast & \ast & \mathbf{Y}\_{77} & \mathbf{0} & \mathbf{0} \\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \mathbf{Y}\_{88} & \mathbf{0} \\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \mathbf{Y}\_{99} \end{bmatrix} < 0,\tag{45}$$

where

<sup>Υ</sup><sup>11</sup> <sup>=</sup> <sup>−</sup>2RX<sup>1</sup> <sup>+</sup> <sup>X</sup>1*δ*1<sup>L</sup> <sup>T</sup> *<sup>r</sup> <sup>L</sup>*rX<sup>1</sup> + <sup>X</sup>1*δ*2*φ*<sup>T</sup> <sup>L</sup> <sup>T</sup> *<sup>c</sup>* Lc*φ*X<sup>1</sup> + X1R2X<sup>1</sup> + 2HKX<sup>1</sup> − X1L1Γ2X<sup>1</sup> − X1*φ*X1, Υ<sup>12</sup> = C + X1L1Γ1, Υ<sup>14</sup> = B, <sup>Υ</sup><sup>15</sup> <sup>=</sup> HKX1, <sup>Υ</sup><sup>17</sup> <sup>=</sup> <sup>J</sup>r, <sup>Υ</sup><sup>18</sup> <sup>=</sup> <sup>J</sup>c, <sup>Υ</sup><sup>19</sup> <sup>=</sup> <sup>J</sup>b, <sup>Υ</sup><sup>22</sup> <sup>=</sup> <sup>−</sup>L1, <sup>Υ</sup><sup>33</sup> <sup>=</sup> *<sup>δ</sup>*3*φT*<sup>L</sup> *<sup>T</sup>* b Lb*φ* − R2(1 − *μ*) − L2Γ2, Υ<sup>34</sup> = L2Γ1, Υ<sup>44</sup> = −L2, Υ<sup>55</sup> = X1ΣΦX<sup>1</sup> − X1L3Γ2X1, Υ<sup>56</sup> = L3Γ1, Υ<sup>66</sup> = −L3, Υ<sup>77</sup> = −*δ*1I , Υ<sup>88</sup> = −*δ*2I , Υ<sup>99</sup> = −*δ*3I .

At the same time, the controller gain matrix K can be obtained as Y<sup>1</sup> = K X<sup>1</sup>

$$
\zeta = \begin{bmatrix}
\mathbb{Z}\_{11} & \mathbb{Z}\_{12} & 0 & \mathbb{Z}\_{14} & \mathbb{Z}\_{15} & 0 & \mathbb{Z}\_{17} & \mathbb{Z}\_{18} & \mathbb{Z}\_{19} \\
\* & \mathbb{Z}\_{22} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\* & \* & \mathbb{Z}\_{33} & \mathbb{Z}\_{34} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & \* & \mathbb{Z}\_{44} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & \mathbb{Z}\_{55} & 0 & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \mathbb{Z}\_{66} & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \mathbb{Z}\_{77} & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \* & \mathbb{Z}\_{89} & 0 \\
\* & \* & \* & \* & \* & \* & \* & \* & \mathbb{Z}\_{99}
\end{bmatrix} < 0,\tag{46}
$$

where

*<sup>ξ</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup>2RX<sup>1</sup> <sup>+</sup> <sup>X</sup>1*δ*1<sup>L</sup> <sup>T</sup> *<sup>r</sup> <sup>L</sup>*rX<sup>1</sup> + <sup>X</sup>1*δ*2*φ*<sup>T</sup> <sup>L</sup> <sup>T</sup> *<sup>c</sup>* Lc*φ*X<sup>1</sup> + X1R2X<sup>1</sup> + 2H Y − X1L1Γ2X<sup>1</sup> − X1*φ*X1, *ξ*<sup>12</sup> = C + X1L1Γ1, *ξ*<sup>14</sup> = B, *ξ*<sup>15</sup> = H Y , *ξ*<sup>17</sup> = Jr, *<sup>ξ</sup>*<sup>18</sup> <sup>=</sup> <sup>J</sup>c, *<sup>ξ</sup>*<sup>19</sup> <sup>=</sup> <sup>J</sup>b, *<sup>ξ</sup>*<sup>22</sup> <sup>=</sup> <sup>−</sup>L1, *<sup>ξ</sup>*<sup>33</sup> <sup>=</sup> *<sup>δ</sup>*3*φT*<sup>L</sup> *<sup>T</sup>* <sup>b</sup> Lb*φ* − R2(1 − *μ*) − L2Γ2, *ξ*<sup>34</sup> = L2Γ1, *ξ*<sup>44</sup> = −L2, *ξ*<sup>55</sup> = X1ΣΦX<sup>1</sup> − X1L3Γ2X1, *ξ*<sup>56</sup> = L3Γ1, *ξ*<sup>66</sup> = −L3, *ξ*<sup>77</sup> = −*δ*1I , *ξ*<sup>88</sup> = −*δ*2I , *ξ*<sup>99</sup> = −*δ*3I .

$$D^{\alpha} \mathcal{V}'(\mathfrak{t}) \le \mathfrak{q}^{T}(\mathfrak{t}) \mathfrak{q} \mathfrak{q}(\mathfrak{t}),\tag{47}$$

where

$$\begin{aligned} \boldsymbol{\varrho}(\mathbf{t}) &= \boldsymbol{\varrho} \boldsymbol{\varrho} [\boldsymbol{\mathfrak{e}}(\mathbf{t}), \ \mathfrak{h}(\mathbf{e}(\mathbf{t})), \ \mathfrak{e}(\mathbf{t} - \boldsymbol{\sigma}(\mathbf{t})), \ \mathfrak{h}(\mathbf{e}(\mathbf{t} - \boldsymbol{\sigma}(\mathbf{t})), \ \mathfrak{e}(\mathbf{t} - \boldsymbol{\tau}(\mathbf{t})), \\ &\quad \mathfrak{h}(\mathbf{e}(\mathbf{t} - \boldsymbol{\tau}(\mathbf{t})))]. \end{aligned}$$

By the Lypunov stability theory analysis, the event-triggered synchronization of the fractional-order uncertain neural networks' error system (34) is globally asymptotic stable if LMI (35) holds. This completes the proof.
