**5. Numerical Example**

**Example 1.** *Consider the following uncertain neural networks (5) with time-varying delays described by*

$$\begin{split} \mathcal{H}^{\mathfrak{g}} \mathfrak{e}(\mathfrak{t}) &= -(\mathcal{A} + \Delta \mathcal{A}(\mathfrak{t})) \mathfrak{e}(\mathfrak{t}) + (\mathcal{C} + \Delta \mathcal{C}(\mathfrak{t})) \mathfrak{h}(\mathfrak{e}(\mathfrak{t})) + (\mathcal{A} + \Delta \mathcal{A}(\mathfrak{t})) \mathfrak{h}(\mathfrak{e}(\mathfrak{t} - \sigma(\mathfrak{t})) \\ &+ \mathcal{H}^{\mathfrak{e}} \mathcal{B}(\mathfrak{t}), \end{split} \tag{48}$$

*with the following parameters*


55


*Moreover, the activation functions are* f(e(t)) = *tanh*(e(t)) *and* f(e(*t* − *σ*(*t*))) = *sinh*(e(t))*. Solving the LMI conditions provided in (7) based on the MATLAB toolbox returns the following feasible solutions:*

$$
\begin{aligned}
\mathscr{R}\_{1} &= \begin{bmatrix}
0.0284 & 0.0154 & 0.0180 & -0.0127 & -0.0074 \\
0.0154 & 0.0244 & 0.0120 & 0.0070 & -0.0260 \\
0.0180 & 0.0120 & 0.0209 & -0.0054 & -0.0102 \\
\end{bmatrix}, \\
\mathscr{R}\_{2} &= \begin{bmatrix}
36.6572 & 0.0000 & 0.0000 & -0.0000 & -0.0000 \\
0.0000 & 36.6572 & 0.0000 & 0.0000 & -0.0000 \\
0.0000 & 0.0000 & 36.6572 & -0.0000 & -0.0000 \\
\end{bmatrix}.
\end{aligned}
$$

*The gain matrix of the designed controller can be obtained as:*


*δ*<sup>1</sup> = 20.2099, *δ*<sup>2</sup> = 20.2097, *δ*<sup>3</sup> = 20.2099, *δ*<sup>4</sup> = 20.2099, *and δ*<sup>5</sup> = 20.2099*, which preserves system (48) as synchronous.*

**Example 2.** *Consider the following uncertain neural networks with time-varying delays described by*

$$\begin{split} \mathcal{O}^{\#} \mathfrak{e}(\mathfrak{t}) &= -(\mathcal{A} + \Delta \mathcal{A}(\mathfrak{t})) \mathfrak{e}(\mathfrak{t}) + (\mathcal{C} + \Delta \mathcal{C}(\mathfrak{t})) \mathfrak{h}(\mathfrak{e}(\mathfrak{t})) + (\mathcal{A} + \Delta \mathcal{A}(\mathfrak{t})) \mathfrak{h}(\mathfrak{e}(\mathfrak{t} - \sigma(\mathfrak{t})) \\ &+ \mathcal{A}^{\mathfrak{e}} \mathcal{B} \mathcal{A}^{\prime}(\mathfrak{t}) \end{split} \tag{49}$$


,

,

,


*Moreover, the activation functions are* f(e(t)) = *tanh*(e(t)) *and* f(e(*t* − *σ*(*t*))) = *sinh*(e(t))*. Solving the LMI conditions provided in (15) based on the MATLAB toolbox returns the following feasible solutions:*


.

*The gain matrix of the designed controller can be obtained as:*

$$
\mathcal{K} = \begin{bmatrix}
1.0518 & -4.4548 & 0.0090 & -1.0158 & -1.3140 \\
1.7014 & 0.0090 & -4.7390 & -0.8662 & -0.7067 \\
\end{bmatrix}.
$$

*δ*<sup>1</sup> = 21.1589, *δ*<sup>2</sup> = 21.1589, *δ*<sup>3</sup> = 21.1567, *δ*<sup>4</sup> = 21.1590, *andδ*<sup>5</sup> = 21.1583, *which preserves (49) as synchronous.*

**Example 3.** *Consider the following neural networks (20), with the following parameters*


*Moreover, the activation functions are* f(e(t)) = *tanh*(e(t)) *and* f(e(*t* − *σ*(*t*))) = *sinh*(e(t))*. Solving the LMI conditions provided in (21) based on the MATLAB toolbox returns the following feasible solutions:*


*The gain matrix of the designed controller and trigger parameters can be obtained as follows:*


*δ*<sup>4</sup> = 4.3607 *and δ*<sup>5</sup> = 4.5189. *Therefore, preserves system (20) is synchronous.*
