**2. Preliminaries**

In this section, we recall the basic definition and some properties concerning fractionalorder calculus. In addition, definition, remark, assumption and some lemmas are presented.

**Definition 1** ([43])**.** *The Caputo fractional derivative of order β for a function* f*(t) is defined as*

$$D^{\beta}f(t) = \frac{1}{\Gamma(m-\beta)} \int\_0^t \frac{f^m(\gamma)}{(t-\gamma)^{\beta-m+1}} d\gamma,$$

*where t* <sup>≥</sup> <sup>0</sup>*, and m* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>* <sup>&</sup>lt; *<sup>m</sup>* <sup>∈</sup> *<sup>Z</sup>*+*. In particular, when <sup>β</sup>* <sup>∈</sup> (0, 1)*,*

$$D^{\mathfrak{G}}f(t) = \frac{1}{\Gamma(1-\beta)} \int\_0^t \frac{f'(\gamma)}{(t-\gamma)^{\beta}} d\gamma.$$

**Lemma 1** ([44])**.** *Let a vector-valued function* (*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup> be differentiable. Then, for any <sup>t</sup>* <sup>&</sup>gt; <sup>0</sup>*, one has*

$$
\partial^{\mathfrak{a}}(\varrho^{T}(\mathfrak{t})\mathcal{P}\varrho(\mathfrak{t})) \le 2\varrho^{T}(\mathfrak{t})\mathcal{P}\partial^{\mathfrak{a}}\varrho(\mathfrak{t}), 0 < \mathfrak{a} < 1.
$$

**Lemma 2** ([45])**.** *For the given positive scalar <sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup>*,* <sup>l</sup>,<sup>r</sup> <sup>∈</sup> <sup>R</sup>*<sup>m</sup> and matrix* <sup>D</sup>*,*

$$\mathbf{t}^T \boldsymbol{\mathcal{Q}} \mathbf{r} \le \frac{\lambda^{-1}}{2} \mathbf{t}^T \boldsymbol{\mathcal{Q}} \boldsymbol{\mathcal{Q}}^T \mathbf{l} + \frac{\lambda}{2} \mathbf{r}^T \mathbf{r}.$$

**Lemma 3** ([46])**.** *If* N > 0*, and the given matrices are* S , Q, N *, then*

$$
\begin{bmatrix}
\mathcal{Q} & \mathcal{J}^T \\
\mathcal{J} & -\mathcal{J}
\end{bmatrix} < 0,
$$

*if and only if*

$$\mathcal{Q} + \mathcal{P}^T \mathcal{J}^{-1} \mathcal{J}^{\vee} < 0.$$

**Lemma 4** ([47])**.** *For a vector function* <sup>Ξ</sup> : [t1,t2] <sup>→</sup> <sup>R</sup><sup>n</sup> *and any positive definite matrix* <sup>P</sup>*, we have*

$$\left(\int\_{t\_1}^{t\_2} \Xi(\mathfrak{s}) \mathfrak{d}\mathfrak{s}\right)^T \mathcal{P}\left(\int\_{t\_1}^{t\_2} \Xi(\mathfrak{s}) \mathfrak{d}\mathfrak{s}\right) \le \left(t\_2 - t\_1\right) \int\_{t\_1}^{t\_2} \Xi^{\mathcal{P}}\left(\mathfrak{s}\right) \mathcal{P}\Xi(\mathfrak{s}) \mathfrak{s} \mathbf{e}\_1$$

**Assumption 1.** *Let* gi(·) *be continuous and bounded;* X <sup>−</sup> <sup>s</sup> *and* X <sup>+</sup> s *are constants,*

$$\mathcal{R}\_{\mathfrak{s}}^{\cdot-} \le \frac{\mathfrak{g}\_{\mathfrak{s}}(\mathfrak{r}\_1) - \mathfrak{g}\_{\mathfrak{s}}(\mathfrak{r}\_2)}{\mathfrak{r}\_1 - \mathfrak{r}\_2} \le \mathcal{R}\_{\mathfrak{s}}^{\cdot+}, \mathfrak{s} = 1, 2, \dots, n\_{\mathfrak{s}}$$

*where* <sup>r</sup>1,r<sup>2</sup> <sup>∈</sup> <sup>R</sup> *and* <sup>r</sup><sup>1</sup> <sup>=</sup> <sup>r</sup>2*.*

**Remark 1.** *From the literature survey, it is clear that most of the results on fractional order neural networks (FONNs) are derived with fractional-order Lyapunov stability criteria having quadratic terms. However, in this paper, we introduce the integral term* <sup>D</sup>(−*α*+1) # <sup>t</sup> <sup>t</sup>−*<sup>η</sup>* <sup>e</sup><sup>T</sup> (s)R2e(s)ds *in the Lyapunov functional candidate, which is solved by utilizing the properties of Caputo fractional-order derivatives and integrals. The Lyapunov functional is novel, as it contains the quadratic term. By* *applying fractional-order derivatives in the error system of the FCNNs under suitable adaptive update laws, a new sufficient condition can be derived in terms of solvable LMIs.*
