Finite-Time Stabilization Criteria of Delayed Inertial Neural Networks with Settling-Time Estimation Protocol and Reliable Control Mechanism

Abstract

This work investigates the finite-time stability (FTS) issue for a class of inertial neural networks (INNs) with mixed-state time-varying delays, proposing a novel analytical approach. Firstly, we establish a novel FTS lemma, which is entirely different from the existing FTS theorems, and extend the current research results. Secondly, an improved discontinuous reliable control mechanism is developed, which is more valid and widens the application scope compared to previous results. Then, by using a novel non-reduced order approach (NROA) and the Lyapunov functional theory, novel sufficient criteria are established using FTS theorems to estimate the settling time with respect to a finite-time stabilization of INNs. Finally, the simulation results are given to validate the usefulness of the theoretical results.

1. Introduction

Inertial neural networks (INNs) have aroused the increasing interest of multitudinous researchers because they have extensive applications in various areas of science and engineerings, such as model identification, secure communication, and image encryption [1,2]. INNs differ from conventional NNs, and the dynamic properties of INNs have proved more complex [3]. In fact, INNs have actual backgrounds in biological and engineering [4,5,6]. To date, much of the work has been devoted to the dynamical behaviors of INNs, including their stability, synchronization, and dissipativity [7,8,9,10,11].

Naturally, time delays unavoidably exist in neural networks (NNs) because of the intrinsic communication time between the neurons and the limited switching velocity of amplifiers [12,13,14,15]. Meanwhile, NNs have a large number of parallel pathways, which may cause distributed time delays [16,17]. However, the existence of delays has been proven to lead to possible divergences, oscillations, and even instability [18,19,20,21,22]. The stability of delayed NNs is the first and most important condition for their application. Hence, it is important to research the stabilization of NNs with mixed-state time delays (MSTDs). In addition, actuator and controller failure occurs in many actual systems, which is another reason for their undesirable performance and even instability. Attempts to obtain the desired reliable controllers have been costly [23,24,25,26]. However, a reliable control for delayed INNs has been not fully studied to date.

Some new research results on the stability and stabilization analysis of delayed INNs or MSTDs have been obtained. The authors of [27] obtained the global exponential stability of generalized discrete-time delayed INNs. In [28], the authors studied the exponential stabilization of INNs with MSTDs. Based on the non-smooth analysis and adaptive control, the authors of [29] stabilized delayed INNs. However, most theoretical criteria regarding the stability and stabilization of INNs focus on the infinite-time exponential and asymptotical stability. Compared with infinite-time stability, finite-time stabilization has a better convergence rate and exhibits serval other desirable features [30,31,32,33,34,35,36,37,38,39]. As biologies and apparatuses have a limited lifepspan, we often desire to acquire a faster convergent speed in practice.

However, in several earlier works [40,41,42,43,44,45,46] that studied the dynamical properties of INNs, the criteria were basically obtained by utilizing an alterable substitution to transform INNs into a low-order form. Few articles [10,29,47] researched the infinite-time dynamical behaviors of INNs using non-reduced order methods. To our knowledge, the variable substitution method requires the introduction of an intermediate variable to indirectly study the dynamic behaviors of INNs. A natural issue arises: how can a finite-time stabilization analysis of INNs be directly achieved, without employing the reduced-order method? This is a fundamental and important issue to be addressed in the community.

In this paper, we mean to study the FTS issue of INNs with MSTDs using a novel analysis method and reliable control mechanism. The main contributions are highlighted below:

• Inspired by the existing FTS theorems [48,49,50], a new FTS Lemma is proposed, which is completely different from the classical FTS theorems and extends the existing research results (see Remark 2).
• By combining a novel NROD with FTS theorems, we create a new approach, which is entirely different from the existing variable transformation approach, and low-order INNs are used to develop novel FTS criteria to ensure the stabilization of the discussed INNs in finite time.
• In contrast to several earlier works, a wider range of settling-time estimation mechanisms is analyzed.

Notation: In this paper, the solutions of all systems are obtained in the sense of Filippov [51]. $\mathbb{R}$ denotes the space of real numbers. $C([t_0-d,t_0],\mathbb{R})$ is the set of all continuous functions from $[t_0-d,t_0]$ to $\mathbb{R}.$ $\overline{co}[\Omega ]$ denotes the closure of the convex hull generated by set $\Omega .$ ${\overline{d}}_{ij}=max\left\{\right|d_{ij}^{\prime }|,|d_{ij}^{″}\left|\right\},$ ${\overline{f}}_{ij}=max\left\{\right|f_{ij}^{\prime }|,|f_{ij}^{″}\left|\right\},$ ${\overline{c}}_{ij}=max\left\{\right|c_{ij}^{\prime }|,|c_{ij}^{″}\left|\right\},$ for all $i,j=1,2,\cdots ,n.$

2. Preliminaries

The INNs with MSTDs:

$$\begin{array}{cc}\hfill \frac{d^2{x}_i\left(t\right)}{d{t}^2}=& -a_i\frac{d{x}_i\left(t\right)}{dt}-b_i{x}_i\left(t\right)+\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)\hfill \\ \hfill & +\sum _{j=1}^n{e}_{ij}{\int }_{t-{\ell }_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)ds,\hfill \end{array}$$

(1)

$i=1,2,\cdots ,n,$ where $x_i\left(t\right)$ is the state of the ith neuron, and $\frac{d^2{x}_i\left(t\right)}{d{t}^2}$ is the inertial term of INNs (1). $f_j(·)$ is the activation function of jth neuron. ${\tau }_j\left(t\right)$ and ${\ell }_j\left(t\right)$ are the time-varying delays, which satisfy $0\le {\tau }_j\left(t\right)\le \tau ,$ $0\le {\ell }_j\left(t\right)\le \ell ,$ and $\tau$, ℓ, $a_i>0$ and $b_i>0$ are constants. The connection weights $c_{ij},$ $d_{ij}$ and $e_{ij}$ are constants.

The initial condition of the INNs (1) is as follows:

$$\begin{array}{c}\hfill x_i\left(s\right)={\varphi }_i\left(s\right),\frac{d{x}_i\left(s\right)}{ds}={\varphi }_i^{\prime }\left(s\right),t_0\ge 0,s\in [t_0-\tau ,t_0],\end{array}$$

where ${\varphi }_i\left(s\right),{\varphi }_i^{\prime }\left(s\right)\in C([t_0-\tau ,t_0],\mathbb{R}),$ $i=1,2,\cdots ,n.$

To achieve stabilization in finite time, by designing the controller $U_i\left(t\right)$ for the INNs (1), we obtain

$$\begin{array}{cc}\hfill \frac{d^2{x}_i\left(t\right)}{d{t}^2}=& -a_i\frac{d{x}_i\left(t\right)}{dt}-b_i{x}_i\left(t\right)+\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)\hfill \\ \hfill & +\sum _{j=1}^n{e}_{ij}{\int }_{t-{\ell }_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)ds+U_i\left(t\right).\hfill \end{array}$$

(2)

If actuator failures occur in actual engineering, we consider the following reliable control design:

$$\begin{array}{c}\hfill U_i^r\left(t\right)=r_i\left(t\right)U_i\left(t\right),\end{array}$$

where the time-varying fault functions $r_i\left(t\right)$ are unknown, which satisfies $0\le {\check{r}}_i\le r_i\left(t\right)\le {\widehat{r}}_i\le 1,$ $i=1,2,\cdots ,n.$ ${\check{r}}_i$ and ${\widehat{r}}_i$ are known constants.

Remark 1.

Due to the actuator aging, environmental influence, etc., actuator failures inevitably exist in dynamic systems. Therefore, reliable controllers have garnered unprecedented attention from researchers. For example, $H_{\infty }$ reliable controllers were designed for Markovian switching systems in [23]. Zeng et al. designed a reliable aperiodic event-triggered control for complex-valued memristive NNs [24]. In [25], the adaptive reliable control mechanism of dynamical systems with actuator fault was investigated. However, a reliable control strategy has not been considered for INNs to date. To the best of our knowledge, actuator failures inevitably exist in actual situations. Hence, research focusing on obtaining a reliable control strategy for INNs is essential.

The controlled system (1) under the reliable controller can be presented as

$$\begin{array}{c}\hfill \frac{d^2{x}_i\left(t\right)}{d{t}^2}=-a_i\frac{d{x}_i\left(t\right)}{dt}-b_i{x}_i\left(t\right)+F_i\left(t\right)+U_i^r\left(t\right),\end{array}$$

(3)

where

$$\begin{array}{c}\hfill F_i\left(t\right)=\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)+\sum _{j=1}^n{e}_{ij}{\int }_{t-{\ell }_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)ds.\end{array}$$

Assumption 1.

The activation functions $f_j:\mathbb{R}\to \mathbb{R}$ in INNs (1) satisfy the Lipschitz condition and are bound; that is, there are scalar $l_j>0$ and $M_j>0$ such that, for all $x,y\in \mathbb{R},$ $x\ne y$, one has $|f_j\left(x\right)-f_j\left(y\right)|\le l_j|x-y|$ and $|f_j\left(x\right)|\le M_j,j=1,2,\cdots ,n.$

Definition 1

([33]). The origin of the INNs (3) is FTS by a suitable controller. If, for any initial conditions, there exists a time $T>0,$ such that ${\displaystyle \underset{t\to t_0+T}{lim}x\left(t\right)=0,}$ and $x\left(t\right)\equiv 0$ for all $t\ge t_0+T,$ and T is called the settling time.

Lemma 1.

Assume that a continuous positive definite and differentiable Lyapunov function $V\left(t\right)$ satisfies

$$\begin{array}{c}\hfill V^{\prime }\left(t\right)\le -kV\left(t\right)-\theta ,\forall t\ge 0,V\left(t\right)\ge 0,\end{array}$$

(4)

where $k>0,$ $\theta >0$ are real numbers. Then, the origin of the system is FTS. Moreover, the settling time T satisfies

$$\begin{array}{c}\hfill T\le t_0+\frac{1}{k}ln\frac{kV\left(t_0\right)+\theta }{\theta }.\end{array}$$

Proof. 

Multiplying (4) with $e^{kt}$ provides

$$\begin{array}{c}\hfill \frac{d\left[V\left(t\right)e^{kt}\right]}{dt}\le -\theta e^{kt}.\end{array}$$

(5)

Integrating both sides of (5) over $[t_0,t]$ yields

$$\begin{array}{cc}\hfill V\left(t\right)e^{kt}\le & V\left(t_0\right)e^{k{t}_0}-\theta {\int }_{t_0}^t{e}^{ks}ds\hfill \\ \hfill =& V\left(t_0\right)e^{k{t}_0}+\frac{\theta }{k}(e^{k{t}_0}-e^{kt}).\hfill \end{array}$$

Namely,

$$\begin{array}{c}\hfill 0\le V\left(t\right)\le (V\left(t_0\right)+\frac{\theta }{k})e^{k(t_0-t)}-\frac{\theta }{k}.\end{array}$$

(6)

When $t\ge t_0+\frac{1}{k}ln\frac{kV\left(t_0\right)+\theta }{\theta },$ we have $V\left(t\right)=0,$ and the proof is completed. □

Lemma 2

([48]). Assume that a continuous positive definite and differentiable Lyapunov function $V\left(t\right)$ satisfies

$$\begin{array}{c}\hfill V^{\prime }\left(t\right)\le -\alpha V^{\eta }\left(t\right),\forall t\ge 0,V\left(t\right)\ge 0,\end{array}$$

where $\alpha >0,$ $\eta \in (0,1)$ are real numbers. Then, the origin of the system is FTS. Moreover, the settling time T satisfies

$$\begin{array}{c}\hfill T\le t_0+\frac{V^{1-\eta }\left(t_0\right)}{\alpha (1-\eta )}.\end{array}$$

Lemma 3

([49]). Assume that a continuous positive definite and differentiable Lyapunov function $V\left(t\right)$ satisfies

$$\begin{array}{c}\hfill V^{\prime }\left(t\right)\le -\alpha V^{\eta }\left(t\right)-kV\left(t\right),\forall t\ge 0,V\left(t\right)\ge 0,\end{array}$$

where $\alpha >0,$ $k>0,$ $\eta \in (0,1)$ are real numbers. Then, the origin of the system is FTS. Moreover, the settling time T satisfies

$$\begin{array}{c}\hfill T\le t_0+\frac{ln(1+\frac{k}{\alpha }{V}^{1-\eta }\left(t_0\right))}{k(1-\eta )}.\end{array}$$

Remark 2.

The novel FTS lemma is completely different from the classical FTS theorems and extends the existing research results. In [50], the Lyapunov function needs to satisfy $V^{\prime }\left(t\right)\le -\theta ,$ and the settling time $T=t_0+\frac{V\left(t_0\right)}{\theta }$. It is clear that $\frac{1}{k}ln\frac{kV\left(t_0\right)+\theta }{\theta }<\frac{V\left(t_0\right)}{\theta },$ i.e., Lemma 1 can provide a tighter estimate value of settling time than FTST in [50]. In contrast to Lemma 2 [48] and Lemma 3 [49], the derivative of the Lyapunov function in Lemma 1 does not need to have exponential terms, and the settling time contains fewer parameters, which makes it easier to achieve in practical applications. Therefore, the results obtained by utilizing the new Lemma 1 are more concise and easier to test than the existing results.

Lemma 4

([45]). Let $x_1,x_2,\cdots ,x_n\ge 0,0<p\le 1;$ then, the inequality holds as follows:

$$\begin{array}{c}\hfill \sum _{i=1}^n{x}_i^p\ge {\left(\sum _{i=1}^n{x}_i\right)}^p.\end{array}$$

3. Main Results

In this section, by combining FTS theorems with a novel NROD, the FTS issue for the discussed INNs is resolved.

The delay-dependent control mechanism is designed as follows:

$$\begin{array}{c}\hfill U_i^r\left(t\right)=-\mathrm{sign}\left(x_i^{\prime }\left(t\right)\right)r_i\left(t\right)[{\kappa }_i|x_i\left(t\right)|+{\iota }_i|x_i^{\prime }\left(t\right)|+{\lambda }_i+\frac{1}{{\check{r}}_i}\sum _{j=1}^n|d_{ij}|l_j|x_j(t-{\tau }_j\left(t\right))|],\end{array}$$

(7)

where ${\kappa }_i,{\iota }_i>0$ and ${\lambda }_i>0$ are constants.

Theorem 1.

Under Assumption 1 and the designed delay-dependent reliable feedback controller (7), if the related parameters satisfy $a_i+{\check{r}}_i{\iota }_i>1$, ${\displaystyle {\check{r}}_i{\kappa }_i\ge b_i+\sum _{j=1}^n\left|c_{ji}\right|l_i}$ and ${\displaystyle {\check{r}}_i{\lambda }_i>\sum _{j=1}^n\ell \left|e_{ij}\right|M_j}$ for all $i,$ then the INNs (3) with MSTDs can be stabilized in finite time. Moreover, the estimated value of the settling time is as follows:

$$\begin{array}{c}\hfill T\le t_0+\frac{1}{a}ln\frac{aV\left(t_0\right)+\Lambda }{\Lambda },\end{array}$$

(8)

where $a=min\{\widehat{a},\check{a}\}$, ${\displaystyle \widehat{a}=\underset{1\le i\le n}{min}\{a_i+{\check{r}}_i{\iota }_i-1\}>0,}$ ${\displaystyle \check{a}=\underset{1\le i\le n}{min}\{{\check{r}}_i{\kappa }_i-b_i-\sum _{j=1}^n|c_{ji}|l_i\}>0}$ and ${\displaystyle \Lambda =\sum _{i=1}^n\left({\check{r}}_i{\lambda }_i-\sum _{j=1}^n\ell e_{ij}{M}_j\right)>0.}$

Proof. 

The following Lyapunov functions can be chosen for system (3):

$$\begin{array}{c}\hfill V\left(t\right)=\sum _{i=1}^n\left(\right|x_i\left(t\right)|+|x_i^{\prime }\left(t\right)\left|\right).\end{array}$$

Computing the time-derivative of $V\left(t\right)$, we have

$$\begin{array}{cc}\hfill D^{+}V\left(t\right)=& \sum _{i=1}^n(sign\left(x_i\left(t\right)\right)x_i^{\prime }\left(t\right)+sign\left(x_i^{\prime }\left(t\right)\right)x_i^{″}\left(t\right))\hfill \\ \hfill =& \sum _{i=1}^{n}sign\left(x_i\left(t\right)\right)x_i^{\prime }\left(t\right)+\sum _{i=1}^{n}sign\left(x_i^{\prime }\left(t\right)\right)\left[-a_i{x}_i^{\prime }\left(t\right)-b_i{x}_i\left(t\right)+F_i\left(t\right)+U_i^r\left(t\right)\right]\hfill \\ \hfill =& \sum _{i=1}^{n}sign\left(x_i\left(t\right)\right)x_i^{\prime }\left(t\right)+\sum _{i=1}^{n}sign\left(x_i^{\prime }\left(t\right)\right)[-a_i{x}_i^{\prime }\left(t\right)-b_i{x}_i\left(t\right)+\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)\hfill \\ \hfill & +\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)+\sum _{j=1}^n{e}_{ij}{\int }_{t-{\ell }_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)ds]+\sum _{i=1}^{n}sign\left(x_i^{\prime }\left(t\right)\right)U_i^r\left(t\right).\hfill \end{array}$$

(9)

Based on Assumption 1, we obtain

$$\begin{array}{cc}\hfill D^{+}V\left(t\right)\le & \sum _{i=1}^n[-(a_i-1)|x_i^{\prime }\left(t\right)|+b_i|x_i\left(t\right)|+\sum _{j=1}^n|c_{ij}|l_j|x_j\left(t\right)|+\sum _{j=1}^n|d_{ij}|l_j|x_j(t-{\tau }_j\left(t\right))|\hfill \\ \hfill & +\sum _{j=1}^n\ell \left|e_{ij}\right|M_j]+\sum _{i=1}^{n}sign\left(x_i^{\prime }\left(t\right)\right)U_i^r\left(t\right)\hfill \\ \hfill \le & -\sum _{i=1}^n(a_i+{\check{r}}_i{\iota }_i-1)|x_i^{\prime }\left(t\right)|-\sum _{i=1}^n\left({\check{r}}_i{\kappa }_i-b_i-\sum _{j=1}^n\left|c_{ji}\right|l_i\right)\left|x_i\left(t\right)\right|\hfill \\ \hfill & -\sum _{i=1}^n\left({\check{r}}_i{\lambda }_i-\sum _{j=1}^n\ell \left|e_{ij}\right|M_j\right).\hfill \end{array}$$

(10)

Using the condition given by Theorem 1, from (10), we can obtain

$$\begin{array}{cc}\hfill D^{+}V\left(t\right)\le & -\widehat{a}\sum _{i=1}^n|x_i^{\prime }\left(t\right)|-\check{a}\sum _{i=1}^n\left|x_i\left(t\right)\right|-\Lambda \hfill \\ \hfill =& -aV\left(t\right)-\Lambda .\hfill \end{array}$$

(11)

where $a=min\{\widehat{a},\check{a}\}$, ${\displaystyle \widehat{a}=\underset{1\le i\le n}{min}\{a_i+{\check{r}}_i{\iota }_i-1\}>0,}$ ${\displaystyle \check{a}=\underset{1\le i\le n}{min}\{{\check{r}}_i{\kappa }_i-b_i-\sum _{j=1}^n|c_{ji}|l_i\}>0}$ and ${\displaystyle \Lambda =\sum _{i=1}^n\left({\check{r}}_i{\lambda }_i-\sum _{j=1}^n\ell e_{ij}{M}_j\right)>0.}$

Using Lemma 1, the finite-time stabilization for the INNs (3) can be realized with the controller (7). In addition, the estimated value of settling time is as follows:

$$\begin{array}{c}\hfill T\le t_0+\frac{1}{a}ln\frac{aV\left(t_0\right)+\Lambda }{\Lambda }.\end{array}$$

(12)

The proofs are completed. □

Remark 3.

Compared to the INNs with a discrete or constant delay and the NNs without an inertial term [7,8,9,10,27,30,31,32,33,40], the INNs proposed in this work are a more general case, which effectively widens their application scope. Therefore, the new theoretical criteria are acquired within a more general framework and are more valid than the previous works; that is, we analyze a wider range of setting-time estimation approaches.

Remark 4.

The FTS considered in this paper requires that the state trajectories of the system converge to the equilibrium point in finite time. However, annular FTS [36,52] means the state trajectory of the system does not exceed a given annular domain with upper and lower bounds within a finite time. Compared with the annular FTS, the FTS considered in this paper ensures that the controlled system has high precision and convergence. In addition, the FTS presents the settling-time estimation strategy, which can be used to estimate and adjust the convergence rate of the system.

Next, the following delay-dependent control is designed:

$$\begin{array}{cc}\hfill U_i^r\left(t\right)=& -\mathrm{sign}\left(x_i^{\prime }\left(t\right)\right)r_i\left(t\right)[{\kappa }_i|x_i\left(t\right)|++{\iota }_i|x_i^{\prime }\left(t\right)|+{\lambda }_i+\frac{1}{{\check{r}}_i}\sum _{j=1}^n|d_{ij}|l_j\left|x_j(t-{\tau }_j\left(t\right))\right|\hfill \\ \hfill & +\varrho |x_i{\left(t\right)|}^{\mu }+\rho {\left|x_i^{\prime }\left(t\right)\right|}^{\mu }],\hfill \end{array}$$

(13)

where $\rho ,\varrho >0$ and $0<\mu <1$ are constants.

Theorem 2.

If Assumption 1 holds, then FTS of the INNs (3) with MSTDs is achieved under the designed controller (13) if the related parameters satisfy $a_i+{\check{r}}_i{\iota }_i>1$, ${\displaystyle {\check{r}}_i{\kappa }_i\ge b_i+\sum _{j=1}^n\left|c_{ji}\right|l_i}$ and ${\displaystyle {\check{r}}_i{\lambda }_i>\sum _{j=1}^n\ell \left|e_{ij}\right|M_j}$ for all i. Moreover, the estimated condition of the settling time

$$\begin{array}{c}\hfill T\le t_0+\frac{ln(1+\frac{a}{\overline{r}\tilde{\rho }}{V}^{1-\mu }\left(t_0\right))}{a(1-\mu )},\end{array}$$

where $a=min\{\widehat{a},\check{a}\}$, ${\displaystyle \widehat{a}=\underset{1\le i\le n}{min}\{a_i+{\check{r}}_i{\iota }_i-1\}>0,}$ ${\displaystyle \check{a}=\underset{1\le i\le n}{min}\{{\check{r}}_i{\kappa }_i-b_i-\sum _{j=1}^n|c_{ji}|l_i\}>0}$ and $\tilde{\rho }=min\{\rho ,\varrho \}$.

Proof. 

We adopt a positive definite Lyapunov functions as follows:

$$\begin{array}{c}\hfill V\left(t\right)=\sum _{i=1}^n\left(\right|x_i\left(t\right)|+|x_i^{\prime }\left(t\right)\left|\right).\end{array}$$

Computing the time-derivative of $V\left(t\right)$, we have

$$\begin{array}{cc}\hfill D^{+}V\left(t\right)=& \sum _{i=1}^{n}sign\left(x_i\left(t\right)\right)x_i^{\prime }\left(t\right)+\sum _{i=1}^{n}sign\left(x_i^{\prime }\left(t\right)\right)[-a_i{x}_i^{\prime }\left(t\right)-b_i{x}_i\left(t\right)+\sum _{j=1}^n{c}_{ij}{f}_j\left(x_j\left(t\right)\right)\hfill \\ \hfill & +\sum _{j=1}^n{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)+\sum _{j=1}^n{e}_{ij}{\int }_{t-{\ell }_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)ds]+\sum _{i=1}^{n}sign\left(x_i^{\prime }\left(t\right)\right)U_i^r\left(t\right).\hfill \end{array}$$

Based on Assumption 1, we obtain

$$\begin{array}{cc}\hfill D^{+}V\left(t\right)\le & \sum _{i=1}^n[-(a_i-1)|x_i^{\prime }\left(t\right)|+b_i|x_i\left(t\right)|+\sum _{j=1}^n|c_{ij}|l_j|x_j\left(t\right)|+\sum _{j=1}^n|d_{ij}|l_j|x_j(t-{\tau }_j\left(t\right))|\hfill \\ \hfill & +\sum _{j=1}^n\ell \left|e_{ij}\right|M_j]+\sum _{i=1}^{n}sign\left(x_i^{\prime }\left(t\right)\right)U_i^r\left(t\right)\hfill \\ \hfill \le & -\sum _{i=1}^n(a_i+{\check{r}}_i{\iota }_i-1)|x_i^{\prime }\left(t\right)|-\sum _{i=1}^n\left({\check{r}}_i{\kappa }_i-b_i-\sum _{j=1}^n\left|c_{ji}\right|l_i\right)\left|x_i\left(t\right)\right|\hfill \\ \hfill & -\varrho \sum _{i=1}^n{r}_i\left(t\right)|x_i{\left(t\right)|}^{\mu }-\rho \sum _{i=1}^n{r}_i\left(t\right){\left|x_i^{\prime }\left(t\right)\right|}^{\mu }\hfill \\ \hfill \le & -aV\left(t\right)-\varrho \sum _{i=1}^n{r}_i\left(t\right)|x_i{\left(t\right)|}^{\mu }-\rho \sum _{i=1}^n{r}_i\left(t\right){\left|x_i^{\prime }\left(t\right)\right|}^{\mu }.\hfill \end{array}$$

(14)

where $a=min\{\widehat{a},\check{a}\}$, ${\displaystyle \widehat{a}=\underset{1\le i\le n}{min}\{a_i+{\check{r}}_i{\iota }_i-1\}>0,}$ ${\displaystyle \check{a}=\underset{1\le i\le n}{min}\{{\check{r}}_i{\kappa }_i-b_i-\sum _{j=1}^n|c_{ji}|l_i\}>0}$. As $0<\mu <1,$ it follows from Lemma 4 that

$$\begin{array}{c}\hfill \sum _{i=1}^n|x_i{\left(t\right)|}^{\mu }\ge (\sum _{i=1}^n|x_i{\left(t\right)|)}^{\mu },\sum _{i=1}^n|x_i^{\prime }{\left(t\right)|}^{\mu }\ge (\sum _{i=1}^n|x_i^{\prime }\left(t\right){|)}^{\mu }.\end{array}$$

(15)

Then, we can see from (14) that

$$\begin{array}{cc}\hfill V^{\prime }\left(t\right)\le & -aV\left(t\right)-\overline{r}\varrho (\sum _{i=1}^n|x_i{\left(t\right)|)}^{\mu }-\overline{r}\rho (\sum _{i=1}^n|x_i^{\prime }\left(t\right){|)}^{\mu }.\hfill \end{array}$$

(16)

where ${\displaystyle \overline{r}=\underset{1\le i\le n}{min}\left\{{\check{r}}_i\right\}>0.}$ On the basis of Lemma 4, we can obtain that

$$\begin{array}{cc}\hfill V^{\prime }\left(t\right)\le & -aV\left(t\right)-\overline{r}\tilde{\rho }V{\left(t\right)}^{\mu }.\hfill \end{array}$$

(17)

where $\tilde{\rho }=min\{\rho ,\varrho \}$.

Using Lemma 3, we can observe that the INNs (3) show finite-time stabilization under the controller (13). Furthermore, the estimated value of the settling time is as follows:

$$\begin{array}{c}\hfill T\le t_0+\frac{ln(1+\frac{a}{\overline{r}\tilde{\rho }}{V}^{1-\mu }\left(t_0\right))}{a(1-\mu )}.\end{array}$$

(18)

This proof is completed. □

Remark 5.

Compared with Theorem 2, we can clearly see that the controller (7) of Theorem 1 does not need to have an exponential term, and the INNs (1) can also achieve finite-time stabilization, which can effectively simplify the controller and save resources. In addition, the settling time in Theorem 1 is more concise and easier to test than that in Theorem 2.

Based on the proofs of Theorem 2, we can also obtain $V^{\prime }\left(t\right)\le -\overline{r}\tilde{\rho }2V{\left(t\right)}^{\mu }.$ Using Lemma 2, the FTS for the delayed INNs (3) can be realized under the delay-dependent feedback controller (13). Hence, we can directly acquire the following corollary.

Corollary 1.

If Assumption 1 holds, then FTS of the INNs (3) with MSTDs is achieved under the control mechanism (13) if the related parameters satisfy the conditions given by Theorem 2. Moreover, the estimated value of the settling time is as follows.

$$\begin{array}{c}\hfill T\le t_0+\frac{V^{1-\mu }\left(t_0\right)}{\overline{r}\tilde{\rho }(1-\mu )}.\end{array}$$

(19)

Remark 6.

Compared with Theorem 2, there are fewer parameters in Corollary 1, and the settling time is more concise and easier to test. However, Theorem 2 is a strengthened result, and Theorem 2 can be reduced to Corollary 1. Moreover, we can obtain

$$\begin{array}{c}\hfill \frac{ln\left(1+\frac{a}{\overline{r}\tilde{\rho }}{V}^{1-\mu }\left(t_0\right)\right)}{a(1-\mu )}/\frac{V^{1-\mu }\left(t_0\right)}{\overline{r}\tilde{\rho }(1-\mu )}=ln\left(1+\frac{a}{\overline{r}\tilde{\rho }}{V}^{1-\mu }\left(t_0\right)\right)/\frac{a}{\overline{r}\tilde{\rho }}{V}^{1-\mu }\left(t_0\right)<1,\end{array}$$

i.e., $ln\left(1+\frac{a}{\overline{r}\tilde{\rho }}V{\left(t_0\right)}^{1-\mu }\right)/a(1-\mu )<V^{1-\mu }\left(t_0\right)/\overline{r}\tilde{\rho }(1-\mu ).$ Theorem 2 can provide a tighter estimated settling value time than Corollary 1. Therefore, we can select Theorem 2 or Corollary 1 according to actual needs.

Remark 7.

To date, in the existing works [40,41,42,43,44,45,46] studying the dynamical properties of INNs, the criteria were basically obtained by utilizing the reduced-order method. Although the authors did not use the reduced-order method in [10,29], the acquired results are only related to asymptotical stability. To the best of our knowledge, the reduced-order method for the INNs requires the introduction of an intermediate variable to indirectly study its dynamic behaviors. Thus, a better way is to directly research the INNs themselves, which does not change the order of the second-order INNs, without using the reduced-order method. In this paper, by combining FTSTs with a novel NROD, finite-time stabilization for the INNs with MSTDs is achieved.

4. Illustrative Examples

In this section, two numerical simulations are carried out to show the effectiveness of the results proposed in this paper.

Example 1.

Consider the following two-neuron INNs with MSTDs:

$$\begin{array}{cc}\hfill \frac{d^2{x}_i\left(t\right)}{d{t}^2}=& -a_i\frac{d{x}_i\left(t\right)}{dt}-b_i{x}_i\left(t\right)+\sum _{j=1}^2{c}_{ij}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^2{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)\hfill \\ \hfill & +\sum _{j=1}^2{e}_{ij}{\int }_{t-{\ell }_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)ds,\hfill \end{array}$$

(20)

$i=1,2,$ where $a_1=0.76,$ $a_2=1.34,$ $b_1=1.05,$ $b_2=1.46,$ and $f_1(·)=f_2(·)=tanh(·),$ ${\tau }_1\left(t\right)={\tau }_2\left(t\right)={\ell }_1\left(t\right)={\ell }_2\left(t\right)=\frac{e^t}{1+e^t}.$ It is easy to check that $f_1$ and $f_2$ satisfy the given conditions with $M_1=M_2=1,$ $l_1=l_2=1,$ and $0\le {\tau }_1\left(t\right),{\tau }_2\left(t\right),{\ell }_1\left(t\right),{\ell }_2\left(t\right)\le 1.$ The connection weights take the following forms:

$$\begin{array}{cc}\hfill & C={\left(c_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}-1.72& -1.45\\ 1.78& 1.55\end{array}\right),D={\left(d_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}-1.93& -1.32\\ 1.64& -1.53\end{array}\right),\hfill \\ \hfill & E={\left(e_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}-1.82& 1.18\\ 1.25& -1.66\end{array}\right).\hfill \end{array}$$

The initial conditions of INNs (20) are $x_1\left(s\right)=-1.81,$ $x_1^{\prime }\left(s\right)=7.13,$ $x_2\left(s\right)=0.53,$ $x_2^{\prime }\left(s\right)=-3.11,$ $s\in (-1,0].$ $x_1\left(t\right)$, and $x_2\left(t\right)$ of INNs (20) without any control input are presented in Figure 1 and Figure 2, which clearly show that INNs (20) cannot converge to the Lyapunov equilibrium point in finite time.

Under the designed state-feedback control (7), we chose ${\kappa }_1={\kappa }_2=6$, ${\iota }_1={\iota }_2=1.25$ and ${\lambda }_1={\lambda }_2=10.$ The fault functions are given as $r_i\left(t\right)=0.9-0.1sin\left(t\right)$, so ${\check{r}}_i=0.8,{\widehat{r}}_i=1,i=1,2.$ Based on the above conditions, $a=0.76$ and $\Lambda =3.67$ can easily be obtained. Figure 3 shows the state trajectories $x_1\left(t\right)$ and $x_2\left(t\right)$ of INNs (20) under the given condition (7). The FTS of INNs (20) is successfully realized. On the basis of Theorem 1, it is easy to obtain the settling time $T\le 1.4974.$ Hence, Theorem 1 is verified.

Example 2.

Consider the two-neuron INNs with MSTDs as follows:

$$\begin{array}{cc}\hfill \frac{d^2{x}_i\left(t\right)}{d{t}^2}=& -a_i\frac{d{x}_i\left(t\right)}{dt}-b_i{x}_i\left(t\right)+\sum _{j=1}^2{c}_{ij}{f}_j\left(x_j\left(t\right)\right)+\sum _{j=1}^2{d}_{ij}{f}_j\left(x_j(t-{\tau }_j\left(t\right))\right)\hfill \\ \hfill & +\sum _{j=1}^2{e}_{ij}{\int }_{t-{\ell }_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)ds,\hfill \end{array}$$

(21)

$i=1,2,$ where $a_1=5.34,$ $a_2=6.77,$ $b_1=6.28,$ $b_2=7.39,$ and $f_1(·)=f_2(·)=tanh(·),$ ${\tau }_1\left(t\right)={\tau }_2\left(t\right)={\ell }_1\left(t\right)={\ell }_2\left(t\right)=\frac{e^t}{1+e^t}.$ $f_1$ and $f_2$ satisfy the given conditions 1 with $M_1=M_2=1,$ $l_1=l_2=1,$ and $0\le {\tau }_1\left(t\right),{\tau }_2\left(t\right),{\ell }_1\left(t\right),{\ell }_2\left(t\right)\le 1.$ The connection weights take the following forms:

$$\begin{array}{cc}\hfill & C={\left(c_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}-6.48& -7.24\\ 7.87& 6.89\end{array}\right),D={\left(d_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}-6.66& -7.03\\ 7.69& -6.99\end{array}\right),\hfill \\ \hfill & E={\left(e_{ij}\right)}_{2\times 2}=\left(\begin{array}{cc}5.89& -6.97\\ -5.92& -7.58\end{array}\right).\hfill \end{array}$$

The initial conditions of INNs (21) are $x_1\left(s\right)=2.68,$ $x_1^{\prime }\left(s\right)=-27.55,$ $x_2\left(s\right)=-3.76,$ $x_2^{\prime }\left(s\right)=37.98,$ $s\in (-1,0].$ The dynamic behavior of INNs (21) without control inputs are given in Figure 4 and Figure 5, from which one can obtain that INNs (21) cannot converge to the Lyapunov equilibrium point in finite time.

Choose ${\kappa }_1=21,{\kappa }_2=22,$ ${\lambda }_1=27,$ ${\lambda }_2=29,$ $\rho =\sigma =2$ and $\mu =0.6.$ The fault functions $r_1\left(t\right)$ and $r_2\left(t\right)$ are assumed to be continuous random variables on $[0.9,1]$; therefore, the condition of Theorem 2 is satisfied. According to Theorem 2, the INNs (21) can achieve FTS with the controller (13) and the estimated settling time value $T\le 1.2166.$ Figure 6 shows the state trajectories of INNs (21) under the delay-dependent state-feedback controller (13). It is obvious that the INNs (21) are successfully FTS. Therefore, the correctness of Theorem 2 and Corollary 1 is certified.

From the numerical simulation shown above, we can see that the theoretical results obtained in this paper can effectively deal with the FTS problem of more general INNs. Numerical simulations in some existing articles can be used as a special case for the theoretical results of this article. For example, the authors of [7] only considered the exponential stabilization of the INNs and did not consider the distributed time delay and actuator failure. According to Theorem 1 in this paper, we can apply the controller designed in this paper; then, the system of Example 1 in [7] can achieve finite-time stabilization. In addition, compared to the INNs with discrete or constant delay, the INNs without actuator failure, and the NNs without an inertial term in previous works, the INNs proposed in this work provide a more general case.

5. Conclusions

In this work, novel theoretical criteria to ensure the finite-time stabilization for the concerned INNs were acquired using an improved settling-time estimation protocol and reliable control mechanism. The novel FTS proposed in this paper differs from the existing results and enriches the analytical tools used to study FTS problems. In addition, we researched the finite-time stabilization directly from the INNs themselves, which does not change the order of the second-order INNs. Finally, numerical examples have been given to illustrate the validity of the novel research results. However, the time-varying fault function considered in this paper is deterministic and may not apply to stochastic faults. In addition, the considered control techniques require real-time information transmission, which may incur substantial control costs. In our future work, we will further investigate the case of stochastic faults and event-triggered control mechanisms, and the theoretical results will be extended to other dynamical systems, such as fractional-order neural networks [53], T-S neural networks [54], and switched systems [55,56].