**4. Application**

$$\begin{aligned} \, \_0D\_t^q[\mathbf{x}(t) + p\mathbf{x}(t-\tau)] &= \, \_-a\mathbf{x}(t) + b\tanh\mathbf{x}(t-\sigma) \quad t > 0, \\ \mathbf{x}(t) &= \, \_0\mathbf{q}(t), \quad t \in [-\kappa, 0], \end{aligned} \tag{24}$$

for 0 < *q* ≤ 1, the state vector *x*(*t*) ∈ R, *a*, *b*, *p* are real constants with |*p*| < 1, *τ*, *σ* are positive real constants ∈ *<sup>C</sup>*([−*κ*, 0]; R) with *κ* = max{*<sup>τ</sup>*, *<sup>σ</sup>*}.

**Corollary 3.** *If there are positive real constants ki*(*i* = 1, 2, 3, 4, 5) *and real constants qj*(*j* = 1, 2, 3) *such that satisfy*

$$
\begin{bmatrix}
\ast & 2q\_2 + k\_2 + k\_3 + k\_4\tau + k\_5 & q\_2p + q\_3 & 0 & 0 \\
\ast & \ast & 2q\_3p - k\_2 & 0 & 0 \\
\ast & \ast & \ast & -k\_5 - \sigma & 0 \\
\ast & \ast & \ast & \ast & -k\_3 + \sigma
\end{bmatrix} < 0. \tag{25}
$$

*Then the Equation* (24) *is asymptotically stable.*

**Proof of Corollary 3.** For positive real constants *ki*(*i* = 1, 2, 3, 4, 5) and real constants *qj*(*j* = 1, 2, <sup>3</sup>). Consider the Lyapunov-Krasovskii functional

$$V(t) = \sum\_{i=1}^{2} V\_i(t),\tag{26}$$

for

$$\begin{aligned} V\_1(t) &= \begin{array}{rcl} k\_{1t\_0} D\_t^{q-1} \Psi^2(t) \\ V\_2(t) &=& k\_2 \int\_{t-\tau}^t \mathbf{x}^2(\mathbf{s}) ds + k\_3 \int\_{t-\sigma}^t \mathbf{x}^2(\mathbf{s}) ds \\ &+ k\_4 \int\_{t-\tau}^t (\tau - t + \mathbf{s}) \mathbf{x}^2(\mathbf{s}) ds + k\_5 \int\_{t-\sigma}^t \tanh \mathbf{x}^2(\mathbf{s}) ds. \end{array} \end{aligned}$$

According to Theorem 1, we present the asymptotic stability criterion (25) of system (3).

#### **5. Numerical Examples**

**Example 1.** *The fractional neutral system :*

$$\begin{bmatrix} \mathbf{1}\_{t\_0} D\_t^q [\mathbf{x}(t) + \mathbf{C} \mathbf{x}(t-0.5)] \end{bmatrix} = \begin{bmatrix} -A\mathbf{x}(t) + B\mathbf{x}(t-\sigma) + f(\mathbf{x}(t-\sigma)) \end{bmatrix} \tag{27}$$

*Solving the LMI* (10) *when A* = '1.45 0 0 1.45( , *B* = ' 0 0.4 0.4 0 ( , *C* = '−0.1 0 0 −0.1(*, we have a set of parameters that ensures asymptotic stability of system* (27) *which η* = 5 × <sup>10</sup>3*, δ* = 1 *and σ* = 0.5 *as follows: K*1 = 10<sup>8</sup> × '3.5993 0 0 3.5993(*, K*2 = 10<sup>7</sup> × '1.3106 0 0 1.3106(*, K*3 = 10<sup>8</sup> × '1.5730 0 0 1.5730(*, K*4 = 10<sup>6</sup> × '9.7620 0 0 9.7620(*, K*5 = 10<sup>8</sup> × '3.5456 0 0 3.5456(*, Q*1 = 10<sup>8</sup> × '2.9931 0 0 2.9931(*, Q*2 = 10<sup>8</sup> × '−3.0980 0 0 −3.0980(*, Q*3 = 10<sup>7</sup> × '−2.2267 0 0 −2.2267(*.*

*Moreover, the least upper bound of the parameter σ that ensures the asymptotic stability of system* (27) *is* 1.3227 *when η* = 5 × 10<sup>3</sup> *and δ* = 1*. Table 1 represents the least upper bound σ of this example for various values of η, δ.*

**Table 1.** The least upper bound of *σ* for Example 1.


**Example 2.** *The fractional neutral system :*

$$\left[\mathbf{x}\_{t0}D\_t^q\mathbf{\dot{x}}(t) + \mathbf{C}\mathbf{x}(t-\tau)\right] \quad = \quad -A\mathbf{x}(t) + B\mathbf{x}(t-1.2). \tag{28}$$

*Solving the LMI* (18) *when A* = '1.45 0 0 1.45( , *B* = ' 0 0.4 0.4 0 ( , *C* = '−0.1 0 0 −0.1(*, we have a set of parameters that ensures asymptotic stability of system* (28) *which τ* = 0.6 *as follows:*

$$\begin{aligned} K\_1 &= \begin{bmatrix} 44.0782 & 0 \\ 0 & 44.0782 \end{bmatrix}, \quad K\_2 = \begin{bmatrix} 32.9861 & 0 \\ 0 & 32.9861 \end{bmatrix}, \quad K\_3 = \begin{bmatrix} 32.6501 & 0 \\ 0 & 32.6501 \end{bmatrix}, \\\ K\_4 &= \begin{bmatrix} 31.8793 & 0 \\ 0 & 31.8793 \end{bmatrix}, \quad Q\_1 = \begin{bmatrix} 14.6090 & 0 \\ 0 & 14.6090 \end{bmatrix}, \quad Q\_2 = \begin{bmatrix} -56.7801 & 0 \\ 0 & -56.7801 \end{bmatrix}, \\\ Q\_3 &= \begin{bmatrix} -3.3600 & 0 \\ 0 & -3.3600 \end{bmatrix}. \end{aligned}$$

*Moreover, the least upper bound of the parameter τ that ensures the asymptotic stability of system* (28) *is* 3.7 × 1022*.*

**Example 3.** *The fractional neutral system :*

$$\left[\mathbf{A}\_{t0}D\_t^q\right]\mathbf{x}(t) + \mathbb{C}\mathbf{x}(t-\tau)\left[\mathbf{ \ = }-A\mathbf{x}(t) + B\mathbf{x}(t-\tau)\mathbf{.}\tag{29}$$

*Solving the LMI* (21) *when A* = '3 −1 0 1 ( , *B* = '0.2 0.1 0 0.1( , *C* = '0.1 0 0 0.2(*, we obtain the least upper*

*bound of the parameter τ that ensures the asymptotic stability is* 2.86 × 1024*. By the criterion in [35], the least upper bound of the parameter τ is* 2.99 × 1021*. This example represents our result is less conservative than these in [35].*

**Example 4.** *The differential equation, which is considered in [25,27,30–32]:*

$$\frac{d}{dt}[\mathbf{x}(t) + 0.35\mathbf{x}(t-0.5)] = -1.5\mathbf{x}(t) + b\tanh\mathbf{x}(t-0.5). \tag{30}$$

*By using linear matrix inequality* (25)*, the comparison for the least upper bound b that ensures asymptotic stability of Equation* (30) *are represented in Table 2.*


**Table 2.** The least upper bound of *b* for Example 4.

**Example 5.** *The differential equation in [27,30,31,38]:*

$$\frac{d}{dt}[\mathbf{x}(t) + 0.2\mathbf{x}(t-0.1)] = -0.6\mathbf{x}(t) + 0.3\tanh\mathbf{x}(t-\sigma). \tag{31}$$

*By using linear matrix inequality* (25)*, the comparison for the least upper bound delay σ that ensures asymptotic stability of Equation* (31) *are represented in Table 3.*


*:*

**Table 3.** The least upper bound of *σ* for Example 5.

**Example 6.** *The fractional neutral equation* 

$$\_{t\_0}D\_t^q[\mathbf{x}(t) + p\mathbf{x}(t-0.5)] = -a\mathbf{x}(t) + b\tanh\mathbf{x}(t-0.5).\tag{32}$$

*Solving the LMI* (25)*, we have a set of parameters that ensures asymptotic stability of Equation* (32) *which a* = 0.75, *b* = 0.3 *and p* = 0.4 *as follows:*

*k*1 = 3.1544, *k*2 = 1.0324, *k*3 = 1.0749, *k*4 = 0.7170, *k*5 = 0.7385, *q*1 = 0.7587, *q*2 = −1.9721, *q*3 = 0.4433*.*

*Furthermore, the least upper bound of b that ensures the asymptotic stability of Equation* (32) *is* 0.6873 *with a* = 0.75, *p* = 0.4*. Table 4 represents the least upper bound b of this example for various values of a*, *p.*


**Table 4.** The least upper bound of *b* for Example 6.

## **6. Conclusions**

The aim of this paper is a novel asymptotic stability analysis of differential and Riemann-Liouville fractional differential neutral systems with constant delays and nonlinear perturbation by applying zero equations, model transformation and other inequalities. The new asymptotic stability condition is given in the form of LMIs. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we propose the improved delay-dependent asymptotic stability criterion of differential and Riemann-Liouville fractional differential neutral systems with single constant delay and the new delay-dependent asymptotic stability criterion of differential and Riemann-Liouville fractional differential neutral equations with constant delays. Numerical examples illustrate the advantages and applicability of our results.

**Author Contributions:** All authors claim to have contributed significantly and equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by Science Achievement Scholarship of Thailand (SAST), Research and Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal year 2020 and National Research Council of Thailand and Khon Kaen University, Thailand (6200069).

**Acknowledgments:** The authors thank the reviewers for their valuable comments and suggestions, which led to the improvement of the content of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
