**3. Main Results**

Consider the following uncertain delayed neural network described by

$$\mathcal{O}^{\mathsf{d}}\mathfrak{w}\_{\mathsf{i}}(\mathsf{t}) = -(\mathsf{r}\_{\mathsf{i}} + \Delta\mathsf{r}\_{\mathsf{i}}(\mathsf{t}))\mathfrak{w}\_{\mathsf{i}}(\mathsf{t}) + \sum\_{j=1}^{\mathsf{n}} (\mathsf{c}\_{\mathsf{i}\mathsf{j}} + \Delta\mathsf{c}\_{\mathsf{i}\mathsf{j}}(\mathsf{t}))\mathfrak{h}\_{\mathsf{j}}(\mathfrak{w}\_{\mathsf{j}}(\mathsf{t})) $$

$$\begin{split} & \quad + \sum\_{j=1}^{\mathsf{n}} (\mathsf{b}\_{\mathsf{i}\mathsf{j}} + \Delta\mathsf{b}\_{\mathsf{i}\mathsf{j}}(\mathsf{t}))\mathfrak{h}\_{\mathsf{i}}(\mathfrak{w}\_{\mathsf{j}}(\mathsf{t} - \sigma\_{\mathsf{i}}(\mathsf{t}))) \\ & \quad + \sum\_{j=1}^{\mathsf{n}} (\mathsf{a}\_{\mathsf{i}\mathsf{j}} + \Delta\mathsf{a}\_{\mathsf{i}\mathsf{j}}(\mathsf{t})) \int\_{\mathsf{t}-\mathsf{n}}^{\mathsf{t}} \mathfrak{w}\_{\mathsf{j}}(\mathsf{s})\mathfrak{s}\mathfrak{s} + \mathfrak{p}\_{\mathsf{i}}(\mathsf{t}). \end{split} \tag{1}$$

Conveniently, we write the master system as

$$\begin{split} \mathcal{O}^{\mathsf{d}} \mathfrak{w}(\mathsf{t}) &= -\left( \mathscr{A} + \Delta \mathscr{A}(\mathsf{t}) \right) \mathfrak{w}(\mathsf{t}) + \left( \mathscr{C} + \Delta \mathscr{C}(\mathsf{t}) \right) \mathfrak{h}(\mathfrak{w}(\mathsf{t})) + \left( \mathscr{A} + \Delta \mathscr{B}(\mathsf{t}) \right) \mathfrak{h}(\mathfrak{w}(\mathsf{t} - \mathscr{C}(\mathsf{t})) ) \\ &+ \left( \mathscr{A} + \Delta \mathscr{A}(\mathsf{t}) \right) \int\_{\mathsf{t} - \overline{\eta}}^{\mathsf{t}} \left( \mathfrak{w}(\mathsf{s}) \right) \mathfrak{h} + \mathscr{B}^{\mathsf{d}}(\mathsf{t}), \end{split} \tag{2}$$

in which <sup>w</sup>(t)=(w1(t), <sup>w</sup>2(t),..., <sup>w</sup>n(t))<sup>T</sup> <sup>∈</sup> <sup>R</sup>n, is the state vector associated with n neurons, the diagonal matrix ri(t) = *diag*{r1(t),r2(t), ... ,rn(t)}, and C (t), B(t), and A (t) are the known constant matrices of appropriate dimensions; the symbol Δ denotes the uncertain term, and ΔC (t), ΔB(t), and ΔA (t) are known matrices that represent the time-varying parameter uncertainties. h(w(t)) is the neuron activation function.

Next, we consider the corresponding slave system as follows:

$$\begin{split} \partial^{\mathsf{u}} \mathfrak{v}\_{\mathsf{i}}(\mathsf{t}) &= - (\mathsf{r}\_{\mathsf{i}} + \Delta \mathsf{r}\_{\mathsf{i}}(\mathsf{t})) \mathfrak{v}\_{\mathsf{i}}(\mathsf{t}) + \sum\_{\mathsf{j}=1}^{\mathsf{n}} (\mathsf{c}\_{\mathsf{i}\mathsf{j}} + \Delta \mathsf{c}\_{\mathsf{i}\mathsf{j}}(\mathsf{t})) \mathfrak{h}\_{\mathsf{j}}(\mathsf{v}\_{\mathsf{j}}(\mathsf{t})) \\ &+ \sum\_{\mathsf{j}=1}^{\mathsf{n}} (\mathsf{b}\_{\mathsf{i}\mathsf{j}} + \Delta \mathsf{b}\_{\mathsf{i}\mathsf{j}}(\mathsf{t})) \mathfrak{h}\_{\mathsf{j}}(\mathsf{v}\_{\mathsf{j}}(\mathsf{t}-\mathsf{v}\_{\mathsf{i}}(\mathsf{t}))) \\ &+ \sum\_{\mathsf{j}=1}^{\mathsf{n}} (\mathsf{a}\_{\mathsf{i}\mathsf{j}} + \Delta \mathsf{a}\_{\mathsf{i}\mathsf{j}}(\mathsf{t})) \int\_{\mathsf{t}-\mathsf{v}}^{\mathsf{t}} \mathfrak{v}\_{\mathsf{j}}(\mathsf{s}) \mathfrak{g} \mathfrak{s} + \mathfrak{p}\_{\mathsf{i}}(\mathsf{t}) + \mathsf{b} \mathfrak{q}\_{\mathsf{i}}(\mathsf{t}). \end{split} \tag{3}$$

The compact form of (3) is

$$
\begin{split} \mathcal{O}^{\mathfrak{a}} \mathfrak{v}(\mathfrak{t}) &= -\left( \mathscr{A} + \Delta \mathscr{A}(\mathfrak{t}) \right) \mathfrak{v}(\mathfrak{t}) + \left( \mathscr{A}' + \Delta \mathscr{C}(\mathfrak{t}) \right) \mathfrak{h} \left( \mathfrak{v}(\mathfrak{t}) \right) + \left( \mathscr{A} + \Delta \mathscr{B}(\mathfrak{t}) \right) \mathfrak{h} \left( \mathfrak{v}(\mathfrak{t} - \mathscr{C}(\mathfrak{t})) \right) \\ &+ \left( \mathscr{A}' + \Delta \mathscr{A}'(\mathfrak{t}) \right) \int\_{\mathfrak{t} - \eta}^{\mathfrak{t}} \mathfrak{v}(\mathfrak{s}) \mathfrak{A} + \mathscr{B}'(\mathfrak{t}) + \mathscr{A} \mathscr{C} \mathscr{B}(\mathfrak{t}). \end{split} \tag{4}
$$

Now, we introduce the e(t) = v(t) − w(t):

$$
\begin{split}
\mathscr{G}^{a}\mathfrak{e}(\mathfrak{t}) &= -\left(\mathscr{A}^{\flat} + \Delta\mathscr{A}^{\flat}(\mathfrak{t})\right)\mathfrak{e}(\mathfrak{t}) + \left(\mathscr{C} + \Delta\mathscr{C}(\mathfrak{t})\right)\mathfrak{h}(\mathfrak{e}(\mathfrak{t})) + \left(\mathscr{A}^{\flat} + \Delta\mathscr{A}^{\flat}(\mathfrak{t})\right)\mathfrak{h}(\mathfrak{e}(\mathfrak{t} - \sigma(\mathfrak{t}))) \\ &+ \left(\mathscr{A}^{\flat} + \Delta\mathscr{A}^{\flat}(\mathfrak{t})\right)\int\_{\mathfrak{t}-\mathfrak{Y}}^{\mathsf{t}} \mathfrak{e}(\mathfrak{s})\mathfrak{g}\mathfrak{s} + \mathscr{A}^{\flat}\mathscr{B}(\mathfrak{t}).
\end{split}
\tag{5}
$$

The purpose of this paper is to design a controller Q(t) = K e(t), such that the slave system (3) synchronizes with the master system (1), and K is the controller gain to be determined.

Without distributed delays in the system (1), it is easy to obtain the error system

$$\begin{split} \partial^{a} \mathbf{e}(\mathbf{t}) &= -(\mathcal{A} + \Delta \mathcal{S}(\mathbf{t})) \mathbf{e}(\mathbf{t}) + (\mathcal{C} + \Delta \mathcal{C}(\mathbf{t})) \mathfrak{h}(\mathbf{e}(\mathbf{t})) + (\mathcal{A} + \Delta \mathcal{A}(\mathbf{t})) \mathfrak{h}(\mathbf{e}(\mathbf{t} - \mathbf{e}(\mathbf{t})) \\ &+ \mathcal{A}^{\boldsymbol{\theta}} \mathcal{X} \mathfrak{e}(\mathbf{t}). \end{split} \tag{6}$$

**Theorem 1.** *The FNNs (1) and (3) are globally asymptotically synchronized under the eventtriggered control scheme, for the given scalars δ*1, *δ*2, *δ*3, *δ*4, *δ*5, *and μ*1, *and if there exist symmetric positive definite matrices* R<sup>1</sup> > 0, R<sup>2</sup> > 0, *such that a feasible solution exists for the following LMIs,*

$$
\Omega = \begin{bmatrix}
\Omega\_{11} & \mathcal{R}\_{1}\mathcal{J}\_{\mathsf{r}} & \mathcal{R}\_{1}\mathcal{J}\_{\mathsf{r}} & \mathcal{R}\_{1}\mathcal{J}\_{\mathsf{r}} & \mathcal{R}\_{1}\mathcal{J}\_{\mathsf{r}} & \mathcal{R}\_{1}\mathcal{J}\_{\mathsf{r}} & \mathcal{R}\_{1}\mathcal{J}\_{\mathsf{r}} & 0 \\
\* & -\delta\_{1}\mathcal{J} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & -\delta\_{2}\mathcal{J} & 0 & 0 & 0 & 0 \\
\* & \* & \* & -\delta\_{3}\mathcal{J} & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & -\delta\_{4}\mathcal{J} & 0 \\
\* & \* & \* & \* & \* & \* & \Omega\_{6}\mathcal{J}
\end{bmatrix} < 0,\tag{7}
$$

*where*

$$\begin{split} \Omega\_{11} &= -2\mathcal{A}\_{1}\mathcal{A} + \delta\_{1}\mathcal{L}\_{r}^{\mathcal{P}}\mathcal{L}\_{\mathfrak{r}} + \delta\_{2}\Phi^{\mathcal{P}}\mathcal{L}\_{\mathfrak{c}}^{\mathcal{P}}\mathcal{L}\_{\mathfrak{c}}\Phi + \delta\_{4}\Phi^{\mathcal{P}}\Phi + \mathcal{A}\_{2} + \mathcal{A}\_{1}\mathcal{H}^{\mathfrak{c}}\mathcal{K}\_{\mathfrak{c}}, \\ \Omega\_{66} &= \delta\_{3}\Phi^{\mathcal{T}}\mathcal{L}\_{\mathfrak{b}}^{\mathcal{T}}\mathcal{L}\_{\mathfrak{b}}\Phi + \delta\_{5}\Phi^{\mathcal{T}}\Phi - \mathcal{A}\_{2}(1-\mu) \end{split}$$

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

$$\mathcal{V}(\mathfrak{t}) = \mathcal{V}\_1(\mathfrak{t}) + \mathcal{V}\_2(\mathfrak{t}),\tag{8}$$

where

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

By using Lemma 2, we have,

$$\begin{split} 2\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_{1}\Delta\mathcal{R}(\mathfrak{t})\mathfrak{e}(\mathfrak{t}) &\leq 2\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_{1}\mathcal{J}\_{\mathfrak{P}}\mathcal{K}(\mathfrak{t})\mathcal{L}\mathfrak{e}(\mathfrak{t}),\\ &\leq \delta\_{1}^{-1}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_{1}\mathcal{J}\_{\mathfrak{P}}\mathcal{J}\_{\mathfrak{P}}^{\mathcal{F}}\mathcal{R}\_{1}^{\mathcal{F}}\mathfrak{e}(\mathfrak{t})\\ &+ \delta\_{1}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{L}\mathcal{E}\_{\mathfrak{r}}^{\mathcal{F}}\mathcal{L}\_{\mathfrak{r}}\mathfrak{e}(\mathfrak{t}),\\ \varepsilon\_{1} &\leq \varepsilon. \end{split} \tag{9}$$

$$\begin{split} 2\mathbf{c}^{\mathcal{F}}(\mathfrak{t})\mathcal{A}\_{1}\mathsf{A}\mathfrak{t}^{\mathcal{C}}(\mathfrak{t})\mathfrak{h}(\mathfrak{c}(\mathfrak{t})) &\leq 2\mathbf{c}^{\mathcal{F}}(\mathfrak{t})\mathcal{A}\_{1}\mathcal{J}\_{\mathfrak{c}}\mathcal{K}(\mathfrak{t})\mathcal{L}\mathfrak{C}\mathfrak{h}(\mathfrak{c}(\mathfrak{t})),\\ &\leq \delta\_{2}^{-1}\mathbf{c}^{\mathcal{F}}(\mathfrak{t})\mathcal{A}\_{1}\mathcal{J}\_{\mathfrak{c}}\mathcal{J}\_{\mathfrak{c}}\mathcal{J}\_{\mathfrak{c}}^{T}\mathfrak{A}\_{1}^{\mathcal{F}}\mathfrak{c}(\mathfrak{t})\\ &+ \delta\_{2}\mathbf{c}^{\mathcal{F}}(\mathfrak{t})\mathfrak{d}^{\mathcal{F}}\mathcal{A}\_{\mathfrak{c}}^{\mathcal{F}}\mathcal{A}\_{\mathfrak{c}}\mathfrak{c}\mathfrak{C}\mathfrak{c}(\mathfrak{t}), \end{split} \tag{10}$$

$$\begin{split} 2\mathbf{e}^{\mathcal{F}}(\mathbf{t})\mathcal{R}\_{1}\Delta\mathcal{A}(\mathbf{t})\mathfrak{h}(\mathbf{c}(t-\sigma(\mathbf{t}))) &\leq 2\mathbf{e}^{\mathcal{F}}(\mathbf{t})\mathcal{R}\_{1}\mathcal{J}\_{\mathbf{b}}\mathcal{K}(\mathbf{t})\mathcal{L}\mathfrak{h}(\mathbf{c}(t-\sigma(\mathbf{t}))),\\ &\leq \delta\_{3}^{-1}\mathbf{e}^{\mathcal{F}}(\mathbf{t})\mathcal{R}\_{1}\mathcal{J}\_{\mathbf{b}}\mathcal{J}\_{\mathbf{b}}\mathcal{J}\_{\mathbf{b}}^{\mathcal{F}}\mathcal{R}\_{1}^{\mathcal{F}}\mathfrak{e}(\mathbf{t})\\ &+ \delta\_{3}\mathbf{e}^{\mathcal{F}}(\mathbf{t}-\sigma(\mathbf{t}))\mathfrak{q}^{\mathcal{F}}\mathcal{L}\_{\mathbf{b}}^{\mathcal{F}}\mathcal{L}\_{\mathbf{b}}\mathfrak{e}(\mathbf{t}-\sigma(\mathbf{t})),\\ 2\mathbf{e}^{\mathcal{F}}(\mathbf{t})\mathcal{R}\_{1}\mathfrak{e}(\mathbf{t})\mathfrak{e}(\mathbf{t})) &\leq \delta\_{4}^{-1}\mathbf{e}^{\mathcal{F}}(\mathbf{t})\mathcal{R}\_{1}\mathfrak{e}^{\mathcal{F}}\mathcal{R}\_{1}^{\mathcal{F}}\mathfrak{e}(\mathbf{t}) \end{split} \tag{11}$$

$$\begin{split} 2\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_{1}\mathfrak{C}\mathfrak{h}(\mathfrak{e}(\mathfrak{t})) &\leq \delta\_{\mathfrak{t}}^{-1}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_{1}\mathfrak{C}\mathfrak{e}^{\mathcal{F}}\mathfrak{H}\_{1}^{T}\mathfrak{e}(\mathfrak{t}) \\ &+ \delta\_{\mathfrak{t}}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathfrak{e}^{\mathcal{F}}\mathfrak{g}\mathfrak{e}(\mathfrak{t}), \\ 2\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_{1}\mathfrak{A}\mathfrak{H}(\mathfrak{e}(\mathfrak{t}-\sigma(\mathfrak{t}))) &\leq \delta\_{\mathfrak{t}}^{-1}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_{1}\mathfrak{A}\mathfrak{C}\mathfrak{g}^{T}\mathcal{R}\_{1}^{T}\mathfrak{e}(\mathfrak{t}) \\ &+ \delta\mathfrak{e}^{\mathcal{F}}(\mathfrak{t}-\sigma(\mathfrak{t}))\mathfrak{e}^{\mathcal{F}}\mathfrak{g}\mathfrak{e}(\mathfrak{t}-\sigma(\mathfrak{t})). \end{split} \tag{12}$$

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> − (R + ΔR(t))e(t)+(C + ΔC (t))h(e(t)) + (B + ΔB(t))h(e(t − *σ*(*t*)) + H K e(*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*))(<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*). (13)

From (9)–(13), the following can be obtained.

$$D^{\mathfrak{a}}\mathcal{V}'(\mathfrak{t}) \le \mathcal{Z}^T(\mathfrak{t})\Omega\mathcal{Z}(\mathfrak{t}),\tag{14}$$

where

$$\zeta(\mathfrak{t}) = \operatorname{col}[\mathfrak{e}(\mathfrak{t}), \ \mathfrak{e}(\mathfrak{t} - \sigma(\mathfrak{t}))].$$

From the aforementioned part, we know that matrix inequality (7) guarantees Ω < 0.

Thereby, the master system (1) is synchronized with the slave system (3). The proof of Theorem 1 is complete.

**Theorem 2.** *The FNNs (1) and (3) are globally asymptotically synchronized, for given scalars δ*1, *δ*2, *δ*3, *δ*4, *δ*5, *and σ, if there exist symmetric positive definite matrices* R<sup>1</sup> > 0, R<sup>2</sup> > 0, *such that the following LMIs hold:*

$$
\pi = \begin{bmatrix}
\pi\_{11} & \mathcal{J}\_{\mathbf{r}} & \mathcal{J}\_{\mathbf{t}} & \mathcal{J}\_{\mathbf{t}} & \mathcal{J}\_{\mathbf{t}} & \mathcal{J}\_{\mathbf{t}} & \mathcal{R} & 0 \\
\* & -\delta\_1 \mathcal{J} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & -\delta\_2 \mathcal{J} & 0 & 0 & 0 & 0 \\
\* & \* & \* & -\delta\_3 \mathcal{J} & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & -\delta\_4 \mathcal{J} & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \pi\_{6\delta} \\
\* & \* & \* & \* & \* & \* & \pi\_{6\delta}
\end{bmatrix} < 0,\tag{15}
$$

*where*

$$\begin{aligned} \pi\_{11} &= -2\mathcal{A}\_{1}\mathcal{X}\_{1} + \mathcal{X}\_{1}\delta\_{1}\mathcal{L}\_{\tau}^{\mathcal{F}}\mathcal{L}\_{\mathsf{f}}\mathcal{X}\_{1} + \mathcal{X}\_{1}\delta\_{2}\Phi^{\mathcal{F}}\mathcal{L}\_{\mathsf{c}}^{\mathcal{F}}\mathcal{L}\_{\mathsf{c}}\Phi\mathcal{X}\_{1} \\ &+ \mathcal{X}\_{1}\delta\_{4}\Phi^{\mathcal{F}}\Phi\mathcal{X}\_{1} + \mathcal{X}\_{1}\delta\_{2}\mathcal{X}\_{1} + \mathcal{X}^{\rho}\mathcal{Y}\_{1}, \\ \pi\_{66} &= \delta\_{3}\Phi^{\mathcal{T}}\mathcal{L}\_{\mathsf{b}}^{\mathcal{T}}\mathcal{L}\_{\mathsf{b}}\Phi + \delta\_{5}\Phi^{\mathcal{T}}\Phi - \mathcal{R}\_{2}(1-\mu), \end{aligned} \tag{16}$$

*and the other parameters are the same as in Theorem 1; among them, the gain matrix is defined with* R−<sup>1</sup> <sup>1</sup> = X1.

**Proof.** We pre- and post-multiply <sup>Ω</sup> by {R−<sup>1</sup> <sup>1</sup> , <sup>I</sup> , <sup>I</sup> , <sup>I</sup> , <sup>I</sup> , <sup>I</sup> , <sup>I</sup> } and <sup>R</sup>−<sup>1</sup> <sup>1</sup> = X<sup>1</sup>

$$
\Phi = \begin{bmatrix}
\Phi\_{11} & \mathcal{J}\_{\mathbf{r}} & \mathcal{J}\_{\mathbf{c}} & \mathcal{J}\_{\mathbf{c}} & \mathcal{J}\_{\mathbf{b}} & \mathcal{J}\_{\mathbf{c}} & \mathcal{R} & 0 \\
\* & -\delta\_{1}\mathcal{J} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & -\delta\_{2}\mathcal{J} & 0 & 0 & 0 & 0 \\
\* & \* & \* & -\delta\_{3}\mathcal{J} & 0 & 0 & 0 \\
\* & \* & \* & \* & -\delta\_{4}\mathcal{J} & 0 & 0 \\
\* & \* & \* & \* & \* & -\delta\_{5}\mathcal{J} & 0 \\
\* & \* & \* & \* & \* & \* & \Phi\_{66}
\end{bmatrix} < 0,\tag{17}
$$

where

$$\begin{split} \Phi\_{11} &= -2\mathcal{R}\_{1}\mathcal{X}\_{1} + \mathcal{X}\_{1}\delta\_{1}\mathcal{L}\_{r}^{\rho}\mathcal{X}\_{1}\mathcal{X}\_{1} + \mathcal{X}\_{1}\delta\_{2}\phi^{\mathcal{T}}\mathcal{L}\_{c}^{\rho}\mathcal{L}\_{\mathfrak{C}}\phi\mathcal{X}\_{1} \\ &+ \mathcal{X}\_{1}\delta\_{4}\phi^{\mathcal{T}}\phi\mathcal{X}\_{1} + \mathcal{X}\_{1}\delta\_{2}\mathcal{X}\_{1} + \mathcal{X}\_{\mathcal{C}}\mathcal{X}\_{\mathcal{C}}\mathcal{X}\_{1} \\ \Phi\_{66} &= \delta\_{3}\phi^{T}\mathcal{L}\_{\mathfrak{b}}^{\rho T}\mathcal{L}\_{\mathfrak{b}}\Phi + \delta\_{5}\phi^{T}\Phi - \mathcal{R}\_{2}(1-\mu) .\end{split}$$

At the same time, the controller gain matrix K can be obtained as Y<sup>1</sup> = K X1,

$$
\pi = \begin{bmatrix}
\pi\_{11} & \mathcal{J}\_{\mathsf{r}} & \mathcal{J}\_{\mathsf{t}} & \mathcal{J}\_{\mathsf{t}} & \mathcal{J}\_{\mathsf{b}} & \mathcal{J}\_{\mathsf{t}} & \mathcal{R} & 0 \\
\* & -\delta\_{1}\mathcal{J} & 0 & 0 & 0 & 0 & 0 \\
\* & \* & -\delta\_{2}\mathcal{J} & 0 & 0 & 0 & 0 \\
\* & \* & \* & -\delta\_{3}\mathcal{J} & 0 & 0 & 0 \\
\* & \* & \* & \* & \* & -\delta\_{4}\mathcal{J} & 0 & 0 \\
\* & \* & \* & \* & \* & \* & \pi\_{6}\mathcal{J}
\end{bmatrix} < 0. \tag{18}
$$

Hence, (15) guarantees that

$$
\pi < 0.\tag{19}
$$

Thereby, the master system (1) is synchronized with the slave system (3). The proof of Theorem 2 is complete.

**Remark 2.** *Specifically, when there are no uncertainties in the given system, the neural network (6) reduces to*

$$\begin{split} \partial^{\mathfrak{a}}\mathfrak{e}(\mathfrak{t}) &= -\mathcal{R}\mathfrak{e}(\mathfrak{t}) + \mathcal{C}\mathfrak{h}(\mathfrak{e}(\mathfrak{t})) + \mathcal{R}\mathfrak{h}(\mathfrak{e}(\mathfrak{t}-\mathfrak{e}(\mathfrak{t})) \\ &+ \varkappa \mathfrak{d} \int\_{\mathfrak{t}-\mathfrak{h}}^{\mathfrak{t}} \mathfrak{e}(\mathfrak{s}) \mathfrak{g}\mathfrak{s} + \mathcal{R}^{\mathfrak{e}} \mathcal{K}\mathfrak{e}(\mathfrak{t}). \end{split} \tag{20}$$

**Corollary 1.** *The scalars are δ*4, *δ*5, *η*, , *and σ*, *and if there exist symmetric positive definite matrices* R<sup>1</sup> > 0, R<sup>2</sup> > 0, *a feasible solution exists for the following LMIs:*

$$
\beta < 0. \tag{21}
$$

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

$$\mathcal{V}'(\mathfrak{t}) = \mathcal{V}\_1(\mathfrak{t}) + \mathcal{V}\_2(\mathfrak{t}),\tag{22}$$

where

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

By using Lemma 2, we have

$$\begin{split} 2\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{A}\_{1}\mathfrak{C}\mathfrak{h}(\mathfrak{e}(\mathfrak{t})) &\leq \delta\_{\mathfrak{t}}^{-1}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{A}\_{1}\mathfrak{C}\mathfrak{e}^{\mathcal{F}}\mathcal{A}\_{1}^{\mathcal{T}}\mathfrak{e}(\mathfrak{t}) \\ &+ \delta\_{\mathfrak{t}}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathfrak{q}^{\mathcal{F}}\mathfrak{g}\mathfrak{e}(\mathfrak{t}), \\ 2\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{A}\_{1}\mathfrak{C}\mathfrak{h}(\mathfrak{e}(\mathfrak{t}-\sigma(t))) &\leq \delta\_{\mathfrak{s}}^{-1}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{A}\_{1}\mathcal{A}\_{1}\mathfrak{C}\mathfrak{f}^{\mathcal{F}}\mathcal{A}\_{1}^{\mathcal{T}}\mathfrak{e}(\mathfrak{t}) \\ &+ \delta\_{\mathfrak{s}}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t}-\sigma(t))\mathfrak{g}^{\mathcal{F}}\mathfrak{e}(\mathfrak{t}-\sigma(t)). \end{split} \tag{24}$$

Further, the above term is computed in view of the procedure in [47], and by employing Lemma 2.1 in [47] and the Cauchy matrix inequality, we have

$$\begin{split} 2\epsilon^{\mathcal{F}}(\mathfrak{t})\,\bar{\mathcal{R}}\_{1}\omega^{\prime}(\mathfrak{t}) \int\_{\mathfrak{t}-\eta}^{\mathfrak{t}} \mathfrak{c}(\mathfrak{s}) \mathfrak{d}\mathfrak{s} &\leq \eta \epsilon^{\mathcal{F}}(\mathfrak{t})\,\bar{\mathcal{R}}\_{1}\omega^{\prime}\mathcal{R}\_{1}^{-1}\omega^{\prime T}\mathcal{R}\_{1}\mathfrak{c}(\mathfrak{t}) \\ &\quad + \frac{1}{\eta} \Big(\int\_{\mathfrak{t}-\eta}^{\mathfrak{t}} \mathfrak{c}(\mathfrak{s})\,\mathfrak{d}\mathfrak{s}\Big)^{T} \mathcal{R}\_{1} \Big(\int\_{\mathfrak{t}-\eta}^{\mathfrak{t}} \mathfrak{c}(\mathfrak{s})\Big) \mathfrak{d}\mathfrak{s} \Big) \\ &\leq \eta \epsilon^{\mathcal{F}}(\mathfrak{t})\,\bar{\mathcal{R}}\_{1}\omega^{\prime}\mathcal{R}\_{1}^{-1}\omega^{\prime T}\mathcal{R}\_{1}\mathfrak{c}(\mathfrak{t}) \\ &\quad + \frac{1}{\eta} \Big(\int\_{\mathfrak{t}-\eta}^{\mathfrak{t}} \mathfrak{c}^{\prime}(\mathfrak{s})\,\bar{\mathcal{R}}\_{1}\mathfrak{c}(\mathfrak{s})\Big)\mathfrak{d}\mathfrak{s} \Big) \\ &\leq \eta \epsilon^{\mathcal{F}}(\mathfrak{t})\,\bar{\mathcal{R}}\_{1}\omega^{\prime}\mathcal{R}\_{1}^{-1}\omega^{\prime T}\mathcal{R}\_{1}\mathfrak{c}(\mathfrak{t}) \\ &\quad + \frac{1}{\eta} \Big(\int\_{\mathfrak{t}-\eta}^{\mathfrak{t}} \mathfrak{c}^{\prime T}(\mathfrak{t}+\mathfrak{s})\,\bar{\mathcal{R}}\_{1}\mathfrak{c}(\mathfrak{t}+\mathfrak{s})\Big)\mathfrak{d}\mathfrak{s} \Big), \tag{25} \end{split}$$

since V (t + s,x(t + s)) ≤ V (t,x(t))

$$2\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_1\omega'(\mathfrak{t})\int\_{\mathfrak{t}-\eta}^{\mathfrak{t}}\mathfrak{e}(\mathfrak{s})\mathfrak{d}\mathfrak{s} \leq \eta\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_1\omega'\mathcal{R}\_1^{-1}\omega'^T\mathcal{R}\_1\mathfrak{e}(\mathfrak{t}) + \eta\mathfrak{e}\mathfrak{e}^{\mathcal{F}}(\mathfrak{t})\mathcal{R}\_1\mathfrak{e}(\mathfrak{t}).\tag{26}$$

Then, with the support of Lemma 1 and the linearity nature of the Caputo fractional-order 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> − Re(t) + C h(e(t)) + Bh(e(t − *σ*(*t*)) + 2e <sup>T</sup> (t)R1A (t) t <sup>t</sup>−*<sup>η</sup>* e(s))ds + K e(*t*) , ≤ −2e <sup>T</sup> (t)R1Re(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*)) + *η*e <sup>T</sup> (t)R1A R−<sup>1</sup> <sup>1</sup> <sup>A</sup> *<sup>T</sup>*R1e(t) + *η*<sup>e</sup> <sup>T</sup> (t)R1e(t) <sup>+</sup> <sup>e</sup>*T*(t)R2e(t) <sup>−</sup> <sup>e</sup>*T*(<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>*). (27)

From (23)–(27) and applying Lemma 4, we obtain

$$
\Theta = \begin{bmatrix}
\Theta\_{11} & \mathcal{R}\_1 \mathcal{I} & \mathcal{R}\_1 \mathcal{A} & \eta \mathcal{R}\_1 \mathcal{A} & 0 \\
\* & -\delta\_4 \mathcal{I} & 0 & 0 & 0 \\
\* & \* & -\delta\_5 \mathcal{I} & 0 & 0 \\
\* & \* & \* & \eta \mathcal{R}\_1 & 0 \\
\* & \* & \* & \* & \delta\_5 \Phi^T \Phi - \mathcal{A}\_2
\end{bmatrix} < 0,\tag{28}
$$

Θ<sup>11</sup> = −2R1R + *η*R<sup>1</sup> + R<sup>2</sup> + R1H K .

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

$$
\Sigma = \begin{bmatrix}
\Sigma\_{11} & \mathcal{H} & \mathcal{A} & \eta \mathcal{A} \mathcal{K}\_1 & 0 \\
\* & -\delta\_4 \mathcal{J} & 0 & 0 & 0 \\
\* & \* & -\delta\_5 \mathcal{J} & 0 & 0 \\
\* & \* & \* & -\eta \mathcal{K}\_1 & 0 \\
\* & \* & \* & \* & \delta \mathfrak{s} \mathfrak{q}^T \mathfrak{g} - \mathcal{A} \mathfrak{q}
\end{bmatrix}, \tag{29}
$$

$$
\text{where } \Sigma\_{11} = -2\mathcal{A}\mathcal{K}\_1 + \mathcal{K}\_1 \delta\_4 \mathfrak{q}^T \boldsymbol{\phi} \mathcal{K}\_1 + \mathcal{K}\_1 \eta \epsilon + \mathcal{K}\_1 \delta \mathfrak{q} \mathcal{K}\_1 + \mathcal{K}\_1 \mathcal{K}\_1 \mathcal{K}\_1
$$

$$
\boldsymbol{\xi} = \begin{bmatrix}
\xi\_{11} & \mathcal{E} & \mathcal{B} & \eta \mathcal{A} \mathcal{K}\_1 & 0 \\
\* & -\delta\_4 \mathcal{J} & 0 & 0 & 0 \\
\* & \* & -\delta\_5 \mathcal{J} & 0 & 0 \\
\* & \* & \* & -\eta \mathcal{K}\_1 & 0 \\
\* & \* & \* & \* & \delta\_5 \mathfrak{q}^T \boldsymbol{\phi} - \mathcal{A} \mathfrak{q}
\end{bmatrix}, \tag{30}
$$

where *<sup>ς</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup>2RX<sup>1</sup> <sup>+</sup> <sup>X</sup>1*δ*4*φTφ*X<sup>1</sup> <sup>+</sup> <sup>X</sup>1*η* <sup>+</sup> <sup>X</sup>1R2X<sup>1</sup> <sup>+</sup> H Y .

Thereby, the master system (1) is synchronized with the slave system (3).
