Further improvement of finite-time boundedness based nonfragile state feedback control for generalized neural networks with mixed interval time-varying delays via a new integral inequality

Abstract

This article investigates new delay-dependent finite-time boundedness for generalized neural networks (GNNs) with mixed-interval time-varying delays based on nonfragile feedback control to achieve the improved stability criterion. We also propose a new integral inequality with an exponential function to estimate the derivative of the Lyapunov–Krasovskii functional (LKF). Furthermore, the well-known Wirtinger’s inequality is a particular case of the new integral inequality. Using a toolbox optimization in MATLAB, we derive and solve new delay-dependent conditions in terms of linear matrix inequalities (LMIs). Additionally, we give three numerical examples to show the advantages of our obtained methods. The examples can apply the continuous time-varying delays that do not need to be differentiable. One of them presents the benchmark problem’s real-world application, which is a four-tank system.

1 Introduction

Neural networks (NNs) have a large capacity for information processing. NNs have been utilized in various applications such as combinatorial optimization, pattern recognition, associate memory, image processing, fixed-point computations, and signal processing [1–4]. There are two major classes of NNs [5–13]. The first of those are static neural networks (SNNs), which utilize the external states of neurons (neural states of neurons). The second are local-field neural networks (LFNNs), which are the internal states of neurons (local-field states). In recent years, Zhang and Han [14] first combined SNNs and LFNNs into a new unified system of NNs called generalized neural networks (GNNs). Throughout the implementation of NNs, a time delay can occur due to the communication time of neurons or the finite switching speed of the neuron amplifiers. Time delays may cause poor performance, instability, divergence, or even oscillation. Hence, the stability analysis of GNNs with time delay has attracted much attention [14–21]. The delayed GNNs can be classified as constant delay, distributed delay, time-varying delay, interval time-varying delay, and mixed delays.

Finite-time stability can apply to real-life problems such as an industrial weight scale. The system of an industrial weight scale needs to attain its state value within a certain threshold for a finite time, so the system uses a magnetic force to reach the system’s equilibrium point faster. Stability analysis of the preceding situation is called finite-time stability. Finite-time stability was introduced in 1961 by Dorato [22]. In 2001, Amato [23] presented finite-time boundedness by extending finite-time stability with the external disturbance. The finite-time stability problem for delayed NNs has received much attention [10, 13, 19, 20, 24–26]. For example, Vadivel et al. [13] investigated the finite-time stability of the recurrent NNs with time-varying delays and leakage terms under the event-triggered controller. Later, Phanlert et al. [19] studied the problem of finite-time mixed $H_{\infty}$/passivity of the GNNs with mixed interval time-varying delays. After that, the problem of finite-time-based reliable dissipative control for neutral-type artificial NNs with time-varying delays has been presented by Saravanakumar et al. [25]. Past studies on finite-time stability for NNs with time-varying delays mostly assume that delays are in the form of differentiable functions. Consequently, this article focuses on the GNNs with continuous nondifferentiable time-varying delays.

In the real world, the inaccuracy appearing in controller operation is inevitable since perturbations in the controller gain are frequent and may result from actuator deterioration. Thus, precision controllers that are insensitive to the controller’s acceptable operating faults are called nonfragile controllers and have been studied by many researchers [5, 9, 10, 18, 27–31]. For instance, Ali et al. [5] studied the problem of NNs for finite-time $H_{\infty}$ with mixed time-varying delays based on a nonfragile feedback controller. Later, Rajavel et al. presented the problem of finite-time stability and passivity performance for NNs with time-varying delay based on a nonfragile state feedback control [9]. After that, the problem of extended dissipative for GNNs under a nonfragile feedback controller with time-varying delay has been investigated by Manivannan et al. [18]. Recently, Kumar et al. [27] investigated the finite-time stability for a T–S fuzzy flexible spacecraft system with uncertainties and stochastic actuator faults under a sampled-data nonfragile controller.

To reduce the conservatism of the Lyapunov theory’s stability criterion, the estimation of the derivative of the presented LKF applied several techniques. For instance, the various inequalities used in the control field are Park’s inequality [32], Moon’s inequality [33], the free-weighting matrix method [34], and other inequalities [35, 36]. Some well-known inequalities are Jensen’s inequality [37] and Wirtinger’s inequality [38]. Recently, an integral inequality with an exponential function has been presented by Zamart et al. [39]. Moreover, they presented the novel delay-dependent criteria of finite-time stabilization for linear systems with fewer conservatism stability criteria. However, finite-time stability is an important and pertinent problem for developing integral inequalities to reduce conservatism.

Inspired by the previous discussion, we aim to develop a new integral inequality combined with the LKF technique to improve results. Furthermore, we investigate the finite-time stability for GNNs with mixed-interval time-varying delays via the state feedback controller with a nonfragile issue. The main features of this article are listed as:

• We propose a new inequality with an exponential function to estimate the single integral terms of the derivative of LKFs. The stability criteria in terms of LMIs are less conservatism. Moreover, the new inequality covers the well-known Wirtinger’s inequality.

• We can solve new delay-dependent conditions for guaranteeing finite-time stable and finite-time boundedness of the GNNs with mixed-interval time-varying delays that do not need to be differentiable.

• We compare minimum allowable lower bounds (MALBs) of $c_{2}$ from the new sufficient conditions between the new inequality, Wirtinger’s inequality [38] and the inequality in [39]. Those inequalities apply to improve stability criteria using the same LKFs.

• Our results show that the new inequality can reduce conservatism more than Wirtinger’s inequality [38] and the inequality in [39].

• We design the nonfragile state feedback controller for the GNNs with mixed-interval time-varying delays and present an example of a practical application that applies our results on a four-tank system.

The paper is organized as follows. We present the GNNs, some preliminaries, and the new integral inequality with an exponential function, in Sect. 2. Section 3 investigates the new sufficient conditions of finite-time stability, finite-time boundedness, and finite-time boundedness based on the state feedback controller with a nonfragile issue for the delayed GNNs. In Sect. 4, three numerical examples illustrate the effectiveness of our methods. Finally, we conclude and discuss our article in Sect. 5.

Notations

This article uses the notations as follows: $\mathbb{R}^{n}$ denotes the n-dimensional Euclidean space; $\lVert \cdot \rVert$ denotes the Euclidean vector norm of a matrix; I indicates the identity matrix; $\operatorname{diag}\lbrace \cdots \rbrace$ refers to a block-diagonal matrix; $Q^{T}$ and $Q^{-1}$ represent the matrix transport Q and matrix inverse Q, respectively; the notation $Q< 0$ (or $Q\leq 0$) denotes the real symmetric matrix Q is negative definite (or Q is negative-semidefinite); $\lambda _{\min}(Q)$ (or $\lambda _{\max}(Q)$) represents the minimum (or maximum) eigenvalue for real symmetric matrix Q; $\mathcal{L}_{2}[0,\infty )$ refers to a quadratically integrable function space over $[0,\infty )$; $\operatorname{Sym}\{ Q \}$ denotes $Q+Q^{T}$; ∗ refers to the elements below the main diagonal in a symmetric matrix.

2 Problem statement and preliminaries

This article presents the GNNs with distributed and interval time-varying delays as the following:

$$\begin{aligned}& \dot{x}(t) = -A x(t)+L_{0} f \bigl(W x(t) \bigr)+ L_{1} f \bigl(W x \bigl(t-h(t) \bigr) \bigr) \\ & \hphantom{\dot{x}(t) =}{}+ L_{2} \int ^{t-\eta _{1}(t)}_{t-\eta _{2}(t)} f \bigl(W x(u) \bigr)\,du + L_{3} \omega (t) +B u(t), \\ & y(t)= x(t), \\ & x(t) = \phi (t), \quad \forall t\in [-h_{M},0], \end{aligned}$$

(1)

where $x(t)\in \mathbb{R}^{n}$ represents the state vector at time t; n is the number of neurons; $A=\operatorname{diag}\{a_{1},a_{2},\ldots,a_{n}\}$ with $a_{i}>0$ denotes a positive diagonal matrix; $f(Wx(t))=[f_{1}(Wx_{1}(t)), f_{2}(Wx_{2}(t)),\ldots,f_{n}(Wx_{n}(t))]^{T}$ indicates the activation functions; W, $L_{0}$, $L_{1}$, and $L_{2}$ refer to connection weight matrices; $L_{3}$ and B refer to real constant matrices that are known; $\omega (t)$ refers to the external disturbance input; $u(t)\in \mathbb{R}^{m}$ denotes the control input; $y(t)\in \mathbb{R}^{n}$ is the output of the system; $h(t)$ and $\eta _{i}(t)$ ($i=1,2$) represent the interval time-varying and interval distributed time-varying delays. Both functions are called mixed-interval time-varying delays. The continuous functions $h(t)$ and $\eta _{i}(t)$ ($i=1,2$) satisfy conditions as follows:

$$0\leq h_{m}\leq h(t)\leq h_{M}\quad \text{and}\quad 0\leq \eta _{1}\leq \eta _{1}(t)\leq \eta _{2}(t)\leq \eta _{2},\quad t\in [0,T], $$

where $h_{m}$, $h_{M}$, $\eta _{1}$, $\eta _{2}\in \mathbb{R}$ refers to known real constants.

This article studies the state feedback controller with a nonfragile issue as the following:

$$ u(t)= \bigl(\mathcal{K}+\Delta \mathcal{K}(t) \bigr)x(t), $$

(2)

where $\mathcal{K}$ denotes the controller matrix that is gained and $\Delta \mathcal{K}(t)$ refers to a perturbed matrix, where it is assumed that the function satisfies $\Delta \mathcal{K}(t)=D_{1} F(t)D_{2}$ where $D_{1}$ and $D_{2}$ represent known real matrices with appropriate dimensions and the unknown time-varying matrix $F(t)$ satisfies $F^{T}(t)F(t)\leq I$.

Remark 1

Figure 1 shows the nonfragile controller mechanism which is a type of control system used in the neural-network model. The controller regulates the flow of information between neurons. It is designed to provide robustness and stability to the model, allowing it to handle unexpected inputs better. The implementation of the controller may obtain some perturbations due to the system faults or the controller gain readjustment circumstances. The nonfragile state feedback controller is the fault-insensitive controller.

Figure 1

Systematic diagram of the nonfragile controller

Assumption A1

For each $f_{i}(t)$, $i=1, 2, \ldots, n$ are continuous activation functions that are bounded and satisfy

$$ F^{-}_{i}\leq \frac{f_{i}(W\kappa _{1})-f_{i}(W\kappa _{2})}{W\kappa _{1}-W\kappa _{2}} \leq F^{+}_{i}, \quad \forall \kappa _{1}, \kappa _{2} \in \mathbb{R}, \kappa _{1}\neq \kappa _{2}, $$

where $F^{-}_{i}$ and $F_{i}^{+}$ are known real constants.

Assumption A2

$\omega (t)$ is the external disturbance and satisfies

$$ \int _{0}^{T}\omega ^{T}(t)\omega (t)\,dt \leq d, \quad d\geq 0, T \text{ is a time constant}. $$

Definition 2.1

([23])

Given positive constants $c_{1}$, $c_{2}$, and T with $0< c_{1}< c_{2}$ and H is a symmetric positive-definite matrix. The GNNs (1) are finite-time bounded with respect to $(c_{1}, c_{2}, H, T)$, if $\forall t\in [0,T]$

$$ \sup_{-h_{M}\leq s\leq 0} \bigl\lbrace x^{T}(s)Hx(s), \dot{x}^{T}(s)H \dot{x}(s) \bigr\rbrace \leq c_{1} \quad \Longrightarrow \quad x^{T}(t)Hx(t) < c_{2}. $$

(3)

Remark 2

When including the external disturbance term, the finite-time stable can be extended to the finite-time boundedness. Thus, the finite-time stability is a particular case of finite-time boundedness. The finite-time stability and finite-time boundedness problems for NNs with time-varying delay have attracted considerable attention [10, 13, 19, 20, 24–26]. Consequently, this article investigates both the finite-time stability and finite-time boundedness for the GNNs with mixed-interval time-varying delays and an external disturbance. Hence, our results are genuinely general.

Lemma 2.2

(Jensen’s inequality [37])

For any scalars $d_{1}$ and $d_{2}$, any symmetric matrix $M\in \mathbb{R}^{m\times m}$, $M=M^{T}>0$, the inequality holds as follows:

$$ (d_{2}-d_{1}) \int _{d_{1}}^{d_{2}}x^{T}(u)Mx(u)\,du \geq \biggl( \int _{d_{1}}^{d_{2}}x(u)\,du \biggr)^{T} M \biggl( \int _{d_{1}}^{d_{2}}x(u)\,du \biggr). $$

Lemma 2.3

(Wirtinger’s inequality [38])

) For any symmetric matrix $M\in \mathbb{R}^{m\times m}$, $M=M^{T}>0$, any scalars $d_{1}$, $d_{2}$ and continuously differentiable function $x : [d_{1},d_{2}]\rightarrow \mathbb{R}^{n}$, the inequality holds as follows:

$$\begin{aligned} \int _{d_{1}}^{d_{2}}\dot{x}^{T}(u)M\dot{x}(u) \,du \geq \frac{1}{d_{2}-d_{1}}\Omega _{0}^{T}M\Omega _{0}+ \frac{3}{d_{2}-d_{1}}\Omega _{1}^{T}M\Omega _{1}, \end{aligned}$$

where $\Omega _{0}=x(d_{2})-x(d_{1})$, $\Omega _{1}=x(d_{2})+x(d_{1})-\frac{2}{d_{2}-d_{1}}\int _{d_{1}}^{d_{2}}x(u)\,du$.

Lemma 2.4

([39])

For any symmetric matrix $M=M^{T}>0$, $M\in \mathbb{R}^{n\times n}$ and scalars $\varrho >0$, $d_{1}, d_{2}\geq 0$ with $d=d_{2}-d_{1}>0$, the inequality holds as follows:

$$-{\int }_{t-d_2}^{t-d_1}{e}^{\varrho (t-u)}{\dot{x}}^T(u)M\dot{x}(u)du\le {\left[\begin{array}{c}{x}^T(t-d_1)\\ x^T(t-d_2)\end{array}\right]}^T\left[\begin{array}{cc}-k_{1}M& k_{2}M\\ k_{2}M& -k_{3}M\end{array}\right]\left[\begin{array}{c}x(t-d_1)\\ x(t-d_2)\end{array}\right],$$

where

$$\begin{aligned}& k_{1} = 2 \biggl(\frac{\varrho}{2}+\frac{1}{d} \biggr)e^{\varrho d_{1}}- \varepsilon \biggl(\frac{\varrho}{2}+\frac{1}{d} \biggr)^{2} e^{2 \varrho d_{1}}, \\ & k_{2} = \biggl(\frac{\varrho}{2}+\frac{1}{d} \biggr)e^{\varrho d_{1}}- \biggl(\frac{\varrho}{2}-\frac{1}{d} \biggr)e^{\varrho d_{2}}+ \varepsilon \biggl(\frac{\varrho ^{2}}{4}-\frac{1}{d^{2}} \biggr) e^{ \varrho (d_{1}+d_{2})}, \\ & k_{3} = -2 \biggl(\frac{\varrho}{2}-\frac{1}{d} \biggr)e^{\varrho d_{2}}- \varepsilon \biggl(\frac{\varrho}{2}-\frac{1}{d} \biggr)^{2} e^{2 \varrho d_{2}}, \\ & \varepsilon = \int _{t-d_{2}}^{t-d_{1}} e^{-\varrho (t-u)}\,du = \frac{e^{-\varrho d_{1}}-e^{-\varrho d_{2}}}{\varrho}. \end{aligned}$$

Lemma 2.5

For any symmetric matrix $M=M^{T}>0$, $M\in \mathbb{R}^{n\times n}$ and positive scalars $a, b>a$, and ϱ, the inequality holds as follows:

$$\begin{aligned} \int _{a}^{b}e^{\varrho (t-u)}\dot{x}^{T}(u)M \dot{x}(u)\,du\geq \frac{1}{\Phi _{0}}\Sigma _{0}^{T}M\Sigma _{0}+\frac{1}{\Phi _{1}} \Sigma _{1}^{T}M\Sigma _{1}, \end{aligned}$$

where

$$\begin{aligned}& \Sigma _{0}= x(b)-x(a), \qquad \Sigma _{1}=\varepsilon _{1} x(a)+ \varepsilon _{2} x(b)- \int _{a}^{b} x(u)\,du, \\& \varepsilon _{1}= \frac{(b-a)e^{-\varrho (t-b)}}{e^{-\varrho (t-b)} -e^{-\varrho (t-a)}}- \frac{1}{\varrho}, \qquad \varepsilon _{2}=\frac{1}{\varrho}- \frac{(b-a)e^{-\varrho (t-a)}}{e^{-\varrho (t-b)}-e^{-\varrho (t-a)}}, \\& \Phi _{0}= \int _{a}^{b} e^{-\varrho (t-u)}\,du= \frac{1}{\varrho} \bigl( e^{-\varrho (t-b)}-e^{-\varrho (t-a)} \bigr), \\& \Phi _{1}= \int _{a}^{b} e^{-\varrho (t-u)}l^{2}_{1}(u) \,du \\& \hphantom{\Phi _{1}}= \frac{e^{-2\varrho (t-a)}-(2+\varrho ^{2}(b-a)^{2})e^{-\varrho (2t-a-b)}+e^{-2\varrho (t-b)}}{\varrho ^{3} ( e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}, \\& l_{1}(u)= u- \biggl( \int _{a}^{b} e^{-\varrho (t-u)}\,du \biggr)^{-1} \biggl( \int _{a}^{b} e^{-\varrho (t-u)}u \,du \biggr). \end{aligned}$$

Proof

Define the function z as

$$\begin{aligned} z(u) =&e^{\varrho (t-u)}\dot{x}(u)- \biggl( \int _{a}^{b} e^{-\varrho (t-s)}\,ds \biggr)^{-1} \bigl(x(b)-x(a) \bigr) \\ &{}-l_{1}(u) \biggl( \int _{a}^{b} e^{-\varrho (t-s)}l^{2}_{1}(s) \,ds \biggr)^{-1} \biggl( \int _{a}^{b} l_{1}(s)\dot{x}(s)\,ds \biggr). \end{aligned}$$

Since $M>0$, we have $0\leq \int _{a}^{b}e^{-\varrho (t-u)}z^{T}(u)Mz(u)\,du$ and reinjecting $z(u)$ into the integral, we obtain

$$\begin{aligned} 0 \leq & \int _{a}^{b}e^{\varrho (t-u)}\dot{x}^{T}(u)M \dot{x}(u)\,du \\ &{}+ \biggl( \int _{a}^{b} e^{-\varrho (t-s)}\,ds \biggr)^{-1} \bigl(x(b)-x(a) \bigr)^{T} M \bigl(x(b)-x(a) \bigr) \\ &{}+ \biggl( \int _{a}^{b} e^{-\varrho (t-s)} l^{2}_{1}(s) \,ds \biggr)^{-1} \biggl( \int _{a}^{b} l_{1}(s)\dot{x}(s)\,ds \biggr)^{T} M \biggl( \int _{a}^{b} l_{1}(s)\dot{x}(s)\,ds \biggr) \\ &{}-2 \biggl( \int _{a}^{b} e^{-\varrho (t-s)}\,ds \biggr)^{-1} \bigl(x(b)-x(a) \bigr)^{T} M \bigl(x(b)-x(a) \bigr) \\ &{}-2 \biggl( \int _{a}^{b} e^{-\varrho (t-s)} l^{2}_{1}(s) \,ds \biggr)^{-1} \biggl( \int _{a}^{b} l_{1}(u)\dot{x}(u)\,du \biggr)^{T} M \biggl( \int _{a}^{b} l_{1}(s)\dot{x}(s)\,ds \biggr) \\ &{}+2 \biggl( \int _{a}^{b} e^{-\varrho (t-s)}\,ds \biggr)^{-1} \biggl( \int _{a}^{b} e^{-\varrho (t-s)}l^{2}_{1}(s) \,ds \biggr)^{-1} \bigl(x(b)-x(a) \bigr)^{T} \end{aligned}$$

$$\begin{aligned} &\times M \biggl( \int _{a}^{b} l_{1}(s)\dot{x}(s)\,ds \biggr) \biggl( \int _{a}^{b}e^{-\varrho (t-u)}l_{1}(u) \,du \biggr). \end{aligned}$$

(4)

By simple integral calculus and integration by parts, we find that

$$\begin{aligned}& \int _{a}^{b}e^{-\varrho (t-u)}l_{1}(u) \,du = \int _{a}^{b}e^{-\varrho (t-u)} \biggl(u- \frac{\int _{a}^{b} e^{-\varrho (t-u)}u\,du}{\int _{a}^{b} e^{-\varrho (t-u)}\,du} \biggr)\,du \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-u)}l_{1}(u)\,du}= \int _{a}^{b}e^{-\varrho (t-u)}u\,du- \int _{a}^{b} e^{-\varrho (t-u)}\,du \biggl( \frac{\int _{a}^{b} e^{-\varrho (t-u)}u\,du}{\int _{a}^{b} e^{-\varrho (t-u)}\,du} \biggr) \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-u)}l_{1}(u)\,du} =0, \end{aligned}$$

(5)

$$\begin{aligned}& \int _{a}^{b} e^{-\varrho (t-s)}\,ds = \frac{1}{\varrho} \bigl( e^{- \varrho (t-b)}-e^{-\varrho (t-a)} \bigr)=\Phi _{0}, \end{aligned}$$

(6)

$$\begin{aligned}& \int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds= \int _{a}^{b}e^{- \varrho (t-s)} \biggl(s- \frac{\int _{a}^{b} e^{-\varrho (t-s)}s\,ds}{\int _{a}^{b} e^{-\varrho (t-s)}\,ds} \biggr)^{2}\,ds \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds}= \int _{a}^{b}e^{-\varrho (t-s)}s^{2}\,ds \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds=}{} -2 \biggl[ \frac{ (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )}{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}- \frac{1}{\varrho} \biggr] \int _{a}^{b}e^{-\varrho (t-s)}s\,ds \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds=}{}+ \biggl[ \frac{ (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )}{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}- \frac{1}{\varrho} \biggr]^{2} \int _{a}^{b}e^{-\varrho (t-s)}\,ds \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds}= \biggl[\frac{1}{\varrho} \bigl(b^{2} e^{\varrho (t-b)}-a^{2} e^{ \varrho (t-a)} \bigr)-\frac{2}{\varrho ^{2}} \bigl(b e^{-\varrho (t-b)}-a e^{\varrho (t-a)} \bigr) \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds=}{} + \frac{2}{\varrho ^{3}} \bigl(e^{-\varrho (t-b)}-e^{-\varrho (t-a)} \bigr) \biggr]- \frac{2}{\varrho} \biggl[ \frac{ (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )^{2}}{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )} \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds=}{} - \frac{2 (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )}{\varrho}+ \frac{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}{\varrho ^{2}} \biggr] \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds=}{} +\frac{1}{\varrho} \biggl[ \frac{ (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )^{2}}{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}- \frac{2 (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )}{\varrho} \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds=}{}+ \frac{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}{\varrho ^{2}} \biggr] \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds}= \frac{ (b^{2}e^{-\varrho (t-b)}-a^{2}e^{-\varrho (t-a)} )}{\varrho}- \frac{ (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )^{2}}{\varrho (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )^{2}} \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds=}{} + \frac{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}{\varrho ^{3}} \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds}= \frac{-\varrho ^{2} (b^{2}-2ab+a^{2})e^{-\varrho (2t-a-b)}+ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )^{2}}{\varrho ^{3} (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )} \\& \hphantom{\int _{a}^{b}e^{-\varrho (t-s)}l_{1}^{2} (s)\,ds}= \frac{e^{-2\varrho (t-a)}-(2+\varrho ^{2}(b-a)^{2})e^{-\varrho (2t-a-b)}+e^{-2\varrho (t-b)}}{\varrho ^{3} ( e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}= \Phi _{1}, \end{aligned}$$

(7)

$$\begin{aligned}& \int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds= \int _{a}^{b} \biggl(s- \frac{\int _{a}^{b} e^{-\varrho (t-s)}s\,ds}{\int _{a}^{b} e^{-\varrho (t-s)}\,ds} \biggr) \dot{x}(s)\,ds \\& \hphantom{\int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds}= \int _{a}^{b} s \dot{x}(s)\,ds - \int _{a}^{b} \dot{x}(s)\,ds \biggl( \frac{\int _{a}^{b} e^{-\varrho (t-s)}s\,ds}{\int _{a}^{b} e^{-\varrho (t-s)}\,ds} \biggr) \\& \hphantom{\int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds}= bx(b)-ax(a)- \int _{a}^{b}x(s)\,ds \\& \hphantom{\int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds=}{} - \biggl[ \frac{ (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )}{\varrho}- \frac{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}{\varrho ^{2}} \biggr] \\& \hphantom{\int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds=}{}\times \frac{\varrho (x(b)-x(a))}{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )} \\& \hphantom{\int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds}= bx(b)-ax(a)- \int _{a}^{b}x(s)\,ds \\& \hphantom{\int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds=}{} - \frac{ (be^{-\varrho (t-b)}-ae^{-\varrho (t-a)} )(x(b)-x(a))}{ (e^{-\varrho (t-b)}-e^{-\varrho (t-a)} )}+ \frac{(x(b)-x(a))}{\varrho} \\& \hphantom{\int _{a}^{b} l_{1}(s) \dot{x}(s)\,ds}= \varepsilon _{1} x(a)+\varepsilon _{2} x(b) - \int _{a}^{b}x(s)\,ds= \Sigma _{1}, \end{aligned}$$

(8)

From inequality (4) and the above integral, we obtain

$$\begin{aligned} \int _{a}^{b}e^{\varrho (t-u)}\dot{x}^{T}(u)M \dot{x}(u)\,du\geq \frac{1}{\Phi _{0}}\Sigma _{0}^{T}M\Sigma _{0}+\frac{1}{\Phi _{1}} \Sigma _{1}^{T}M\Sigma _{1}. \end{aligned}$$

Therefore, the proof is complete. □

Remark 3

If $\varrho =0$, then $\Phi _{0}=b-a$, $\Phi _{1}=\frac{(b-a)^{3}}{12}$, $\Sigma _{0}=x(b)-x(a)$ and $\Sigma _{1}=\frac{b-a}{2} [x(a)+x(b)-\frac{2}{b-a}\int _{a}^{b} x(u)\,du ]$. That is, Lemma 2.3 or the well-known Wirtinger’s inequality is a particular case of Lemma 2.5.

Lemma 2.6

([40])

For any real matrices of appropriate dimensions $D_{1}$, $D_{2}$, and $F(t)$ satisfying $F^{T}(t)F(t)\leq I$, then, for any scalar $\varepsilon >0$,

$$ D_{1} F(t) D_{2} + D_{2}^{T} F^{T}(t) D_{1}^{T} \leq \varepsilon ^{-1} D_{1} D_{1}^{T} +\varepsilon D_{2}^{T} D_{2}. $$

Lemma 2.7

(Schur complement [41])

Given X, Y, and Z are constant matrices with appropriate dimensions and satisfying $X=X^{T}$, $Y=Y^{T}>0$, then $X+Z^{T}Y^{-1}Z<0$ if and only if

$$\left[\begin{array}{cc}X& Z^T\\ Z& -Y\end{array}\right]<0\mathit{\text{or}}\left[\begin{array}{cc}-Y& Z\\ Z^T& X\end{array}\right]<0.$$

3 Main results

This section proposes new delay-dependent conditions for delayed GNNs of the main theorems. Our first and foremost condition is defining parameters as follows: $h_{Mm}=h_{M}-h_{m}$, $h_{Mm}\neq 0$,

$$\begin{array}{c}{\gamma }_{1a}=\frac{1}{\varrho }(1-e^{-\varrho h_m}),{\gamma }_{1b}=\frac{e^{-2\varrho h_m}-(2+{\varrho }^2{h}_m^2)e^{-\varrho h_m}+1}{{\varrho }^3(1-e^{-\varrho h_m})},\hfill \\ {\gamma }_{2a}=\frac{1}{\varrho }(e^{-\varrho h_m}-e^{-\varrho h(t)}),\hfill \\ {\gamma }_{2b}=\frac{e^{-2\varrho h(t)}-(2+{\varrho }^2{(h(t)-h_m)}^2)e^{-\varrho (h(t)+h_m)}+e^{-2\varrho h_m}}{{\varrho }^3(e^{-\varrho h_m}-e^{-\varrho h(t)})},\hfill \\ {\gamma }_{3a}=\frac{1}{\varrho }(e^{-\varrho h(t)}-e^{-\varrho h_M}),\hfill \\ {\gamma }_{3b}=\frac{e^{-2\varrho h_M}-(2+{\varrho }^2{(h_M-h(t))}^2)e^{-\varrho (h(t)+h_M)}+e^{-2\varrho h(t)}}{{\varrho }^3(e^{-\varrho h(t)}-e^{-\varrho h_M})},\hfill \\ {\gamma }_{4a}=\frac{1}{\varrho }(1-e^{-\varrho h_M}),{\gamma }_{4b}=\frac{e^{-2\varrho h_M}-(2+{\varrho }^2{h}_M^2)e^{-\varrho h_M}+1}{{\varrho }^3(1-e^{-\varrho h_M})},\hfill \\ {\epsilon }_{11}=\frac{h_m}{1-e^{-\varrho h_m}}-\frac{1}{\varrho },{\epsilon }_{12}=\frac{1}{\varrho }-\frac{h_m{e}^{-\varrho h_m}}{1-e^{-\varrho h_m}},\hfill \\ {\epsilon }_{21}=\frac{(h(t)-h_m)e^{-\varrho h_m}}{e^{-\varrho h_m}-e^{-\varrho h(t)}}-\frac{1}{\varrho },{\epsilon }_{22}=\frac{1}{\varrho }-\frac{(h(t)-h_m)e^{-\varrho h(t)}}{e^{-\varrho h_m}-e^{-\varrho h(t)}},\hfill \\ {\epsilon }_{31}=\frac{(h_M-h(t))e^{-\varrho h(t)}}{e^{-\varrho h(t)}-e^{-\varrho h_M}}-\frac{1}{\varrho },{\epsilon }_{32}=\frac{1}{\varrho }-\frac{(h_M-h(t))e^{-\varrho h_M}}{e^{-\varrho h(t)}-e^{-\varrho h_M}},\hfill \\ {\epsilon }_{41}=\frac{h_M}{1-e^{-\varrho h_M}}-\frac{1}{\varrho },{\epsilon }_{42}=\frac{1}{\varrho }-\frac{h_M{e}^{-\varrho h_M}}{1-e^{-\varrho h_M}},\hfill \\ {\mathrm{\Gamma }}_1={\left[\begin{array}{ccc}{e}_1^T& h_m{e}_{10}^T& h_M{e}_{11}^T\end{array}\right]}^T,{\mathrm{\Gamma }}_2={\left[\begin{array}{ccc}{e}_7^T& e_1^T-e_2^T& e_1^T-e_4^T\end{array}\right]}^T,\hfill \\ {\mathrm{\Gamma }}_3={\left[\begin{array}{cc}{e}_1^T& e_7^T\end{array}\right]}^T,{\mathrm{\Gamma }}_4={\left[\begin{array}{cc}{e}_2^T& e_8^T\end{array}\right]}^T,{\mathrm{\Gamma }}_5={\left[\begin{array}{cc}{e}_4^T& e_9^T\end{array}\right]}^T,{\mathrm{\Gamma }}_6={\left[\begin{array}{c}{e}_1^T-e_2^T\end{array}\right]}^T,\hfill \\ {\mathrm{\Gamma }}_7={[{\epsilon }_{11}{e}_2^T+{\epsilon }_{12}{e}_1^T-h_m{e}_{10}^T]}^T,{\mathrm{\Gamma }}_8={[e_2^T-e_3^T]}^T,\hfill \\ {\mathrm{\Gamma }}_9={[{\epsilon }_{21}{e}_3^T+{\epsilon }_{22}{e}_2^T-h_{Mm}{e}_{12}^T]}^T,{\mathrm{\Gamma }}_{10}={[e_3^T-e_4^T]}^T,\hfill \\ {\mathrm{\Gamma }}_{11}={[{\epsilon }_{31}{e}_4^T+{\epsilon }_{32}{e}_3^T-h_{Mm}{e}_{13}^T]}^T,{\mathrm{\Gamma }}_{12}={[e_1^T-e_4^T]}^T,\hfill \\ {\mathrm{\Gamma }}_{13}={[{\epsilon }_{41}{e}_4^T+{\epsilon }_{42}{e}_1^T-h_M{e}_{11}^T]}^T,{\mathrm{\Gamma }}_{14}={[e_5^T-e_1^T{W}^T{F}_M^T]}^T,\hfill \\ {\mathrm{\Gamma }}_{15}=[F_{P}W{e}_1-e_5],{\mathrm{\Gamma }}_{16}={[e_6^T-e_3^T{W}^T{F}_M^T]}^T,{\mathrm{\Gamma }}_{17}=[F_{P}W{e}_3-e_6],\hfill \\ {\mathrm{\Gamma }}_{18}={[e_5^T-e_6^T-e_1^T{W}^T{F}_M^T+e_3^T{W}^T{F}_M^T]}^T,\hfill \\ {\mathrm{\Gamma }}_{19}=[F_{P}W{e}_1-F_{P}W{e}_3-e_5+e_6],\hfill \\ F_M=diag\{F_1^{-},\dots ,F_n^{-}\},F_P=diag\{F_1^{+},\dots ,F_n^{+}\},\hfill \\ {\delta }_1=\frac{e^{\varrho h_m}-1}{\varrho },{\delta }_2=\frac{e^{\varrho h_M}-1}{\varrho },{\delta }_3=\frac{e^{\varrho h_m}-\varrho h_m-1}{{\varrho }^2},\hfill \\ {\delta }_4=\frac{e^{\varrho h_M}-e^{\varrho h_m}-\varrho h_{Mm}}{{\varrho }^2},{\delta }_5=\frac{e^{\varrho h_M}-\varrho h_M-1}{{\varrho }^2},\hfill \\ {\delta }_6=\frac{e^{\varrho {\eta }_2}-e^{\varrho {\eta }_1}-\varrho {\eta }_{21}}{{\varrho }^2},{\eta }_{21}={\eta }_2-{\eta }_1\hfill \end{array}$$

and we define vectors as follows:

$$\begin{array}{c}\zeta (t)=[x^T(t),x^T(t-h_m),x^T(t-h(t)),x^T(t-h_M),f^T(Wx(t)),\hfill \\ \phantom{\zeta (t)=}{f}^T\left(Wx(t-h(t))\right),{\dot{x}}^T(t),{\dot{x}}^T(t-h_m),{\dot{x}}^T(t-h_M),\hfill \\ \phantom{\zeta (t)=}\frac{1}{h_m}{\int }_{t-h_m}^t{x}^T(u)du,\frac{1}{h_M}{\int }_{t-h_M}^t{x}^T(u)du,\frac{1}{h(t)-h_m}{\int }_{t-h(t)}^{t-h_m}{x}^T(u)du,\hfill \\ \phantom{\zeta (t)=}\frac{1}{h_M-h(t)}{\int }_{t-h_M}^{t-h(t)}{x}^T(u)du,{\int }_{t-{\eta }_2(t)}^{t-{\eta }_1(t)}{f}^T(Wx(u))du,{\omega }^T(t){]}^T,\hfill \\ e_j=\left[\begin{array}{ccc}{0}_{n\times (j-1)n}& I_n& 0_{n\times (15-j)n}\end{array}\right],j=1,2,\dots ,15.\hfill \end{array}$$

3.1 Analysis of finite-time boundedness

We first obtain new delay-dependent conditions for the problem of finite-time boundedness of the GNNs (9) with mixed-interval time-varying delays as the following:

$$\begin{aligned} &\dot{x}(t) = -A x(t)+L_{0} f \bigl(W x(t) \bigr)+ L_{1} f \bigl(W x \bigl(t-h(t) \bigr) \bigr) \\ &\hphantom{\dot{x}(t) =}{}+L_{2} \int _{t-\eta _{2}(t)}^{t-\eta _{1}(t)}f \bigl(W x(u) \bigr) \,du+L_{3} \omega (t), \\ &x(t) = \phi (t), \quad \forall t\in [-h_{M},0]. \end{aligned}$$

(9)

Theorem 3.1

Given positive scalars $h_{M}$ and ϱ then the delayed GNNs (9) are finite-time bounded regarding $(c_{1},c_{2},T,H,d)$, if there exist symmetric positive-definite matrices $P\in \mathbb{R}^{3n\times 3n}$, $Q_{i}\in \mathbb{R}^{2n\times 2n}$, $R_{j}\in \mathbb{R}^{n\times n}$ ($i=1, 2$, $j=1, 2, 3$), $Z, X\in \mathbb{R}^{n\times n}$, any matrices $N_{1}$, $N_{2}$, and positive diagonal matrices $S_{1}$, $S_{2}$, $S_{3}$, such that the conditions hold as follows:

$$\begin{aligned}& \Xi < 0, \end{aligned}$$

(10)

$$\begin{aligned}& \lambda _{0} I\leq \overline{P}_{1} \leq \lambda _{1} I, \qquad 0 \leq \overline{P}_{2}\leq \lambda _{2} I, \qquad 0 \leq \overline{P}_{3} \leq \lambda _{3} I, \qquad 0\leq \overline{Q}_{11}\leq \lambda _{4} I , \\& 0 \leq \overline{Q}_{12}\leq \lambda _{5} I, \qquad 0 \leq \overline{Q}_{13} \leq \lambda _{6} I, \qquad 0 \leq \overline{Q}_{21}\leq \lambda _{7} I, \qquad 0 \leq \overline{Q}_{22}\leq \lambda _{8} I, \end{aligned}$$

(11)

$$\begin{aligned}& 0 \leq \overline{Q}_{23}\leq \lambda _{9} I,\qquad 0 \leq \overline{R}_{1} \leq \lambda _{10} I, \qquad 0 \leq \overline{R}_{2}\leq \lambda _{11} I, \qquad 0 \leq \overline{R}_{3}\leq \lambda _{12} I, \\& 0 \leq \overline{Z}\leq \lambda _{13} I, \qquad 0 \leq \overline{X} \leq \lambda _{14} I, \\& e^{\varrho T} \bigl[\Xi _{\lambda}c_{1}+d \lambda _{14} \bigl(1-e^{-\varrho T} \bigr) \bigr]< \lambda _{0} c_{2}, \end{aligned}$$

(12)

where

$$\begin{array}{c}P=\left[\begin{array}{ccc}{P}_1& 0& 0\\ 0& P_2& 0\\ 0& 0& P_3\end{array}\right],Q_1=\left[\begin{array}{cc}{Q}_{11}& Q_{12}\\ \ast & Q_{13}\end{array}\right],Q_2=\left[\begin{array}{cc}{Q}_{21}& Q_{22}\\ \ast & Q_{23}\end{array}\right],\hfill \\ \mathrm{\Xi }=\sum _{i=1}^7{\mathrm{\Xi }}_i,\hfill \\ {\mathrm{\Xi }}_1=Sym\left\{{\mathrm{\Gamma }}_1^{T}P{\mathrm{\Gamma }}_2\right\}-\varrho {\mathrm{\Gamma }}_1^{T}P{\mathrm{\Gamma }}_1,\hfill \\ {\mathrm{\Xi }}_2={\mathrm{\Gamma }}_3^T(Q_1+Q_2){\mathrm{\Gamma }}_3-{\mathrm{\Gamma }}_4^T\left(e^{\varrho h_m}{Q}_1\right){\mathrm{\Gamma }}_4-{\mathrm{\Gamma }}_5^T\left(e^{\varrho h_M}{Q}_2\right){\mathrm{\Gamma }}_5,\hfill \\ {\mathrm{\Xi }}_3=h_m^2{e}_7^T{R}_1{e}_7+h_{Mm}^2{e}_7^T{R}_2{e}_7+h_M^2{e}_7^T{R}_3{e}_7-\frac{h_m}{{\gamma }_{1a}}{\mathrm{\Gamma }}_6^T{R}_1{\mathrm{\Gamma }}_6-\frac{h_m}{{\gamma }_{1b}}{\mathrm{\Gamma }}_7^T{R}_1{\mathrm{\Gamma }}_7\hfill \\ \phantom{{\mathrm{\Xi }}_3=}-\frac{h_{Mm}}{{\gamma }_{2a}}{\mathrm{\Gamma }}_8^T{R}_2{\mathrm{\Gamma }}_8-\frac{h_{Mm}}{{\gamma }_{2b}}{\mathrm{\Gamma }}_9^T{R}_2{\mathrm{\Gamma }}_9-\frac{h_{Mm}}{{\gamma }_{3a}}{\mathrm{\Gamma }}_{10}^T{R}_2{\mathrm{\Gamma }}_{10}-\frac{h_{Mm}}{{\gamma }_{3b}}{\mathrm{\Gamma }}_{11}^T{R}_2{\mathrm{\Gamma }}_{11}\hfill \\ \phantom{{\mathrm{\Xi }}_3=}-\frac{h_M}{{\gamma }_{4a}}{\mathrm{\Gamma }}_{12}^T{R}_3{\mathrm{\Gamma }}_{12}-\frac{h_M}{{\gamma }_{4b}}{\mathrm{\Gamma }}_{13}^T{R}_3{\mathrm{\Gamma }}_{13},\hfill \\ {\mathrm{\Xi }}_4={\eta }_{21}^2{e}_5^{T}Z{e}_5-e^{\varrho {\eta }_2}{e}_{14}^{T}Z{e}_{14},\hfill \\ {\mathrm{\Xi }}_5=2{\mathrm{\Gamma }}_{14}^T{S}_1{\mathrm{\Gamma }}_{15}^T+2{\mathrm{\Gamma }}_{16}^T{S}_2{\mathrm{\Gamma }}_{17}^T+2{\mathrm{\Gamma }}_{18}^T{S}_3{\mathrm{\Gamma }}_{19}^T,\hfill \\ {\mathrm{\Xi }}_6=-\varrho e_{15}^{T}X{e}_{15},\hfill \\ {\mathrm{\Xi }}_7=Sym\{[e_1^T{N}_1+e_7^T{N}_2][-e_7-A{e}_1+L_0{e}_5+L_1{e}_6+L_2{e}_{14}+L_3{e}_{15}]\},\hfill \\ {\mathrm{\Xi }}_{\lambda }={\lambda }_1+h_m{\lambda }_2+h_M{\lambda }_3+{\delta }_1({\lambda }_4+2{\lambda }_5+{\lambda }_6)+{\delta }_2({\lambda }_7+2{\lambda }_8+{\lambda }_9)\hfill \\ \phantom{X{i}_{\lambda }=}+h_m{\delta }_3{\lambda }_{10}+h_{Mm}{\delta }_4{\lambda }_{11}+h_M{\delta }_5{\lambda }_{12}+{\eta }_{21}{\delta }_6{\lambda }_{13},\hfill \\ {\lambda }_0={\lambda }_{min}({\bar{P}}_1),{\lambda }_1={\lambda }_{max}({\bar{P}}_1),{\lambda }_2={\lambda }_{max}({\bar{P}}_2),{\lambda }_3={\lambda }_{max}({\bar{P}}_3),\hfill \\ {\lambda }_4={\lambda }_{max}({\bar{Q}}_{11}),{\lambda }_5={\lambda }_{max}({\bar{Q}}_{12}),{\lambda }_6={\lambda }_{max}({\bar{Q}}_{13}),{\lambda }_7={\lambda }_{max}({\bar{Q}}_{21}),\hfill \\ {\lambda }_8={\lambda }_{max}({\bar{Q}}_{22}),{\lambda }_9={\lambda }_{max}({\bar{Q}}_{23}),{\lambda }_{10}={\lambda }_{max}({\bar{R}}_1),{\lambda }_{11}={\lambda }_{max}({\bar{R}}_2),\hfill \\ {\lambda }_{12}={\lambda }_{max}({\bar{R}}_3),{\lambda }_{13}={\lambda }_{max}(\bar{Z}),{\lambda }_{14}={\lambda }_{max}(\bar{X}).\hfill \end{array}$$

Proof

We construct the LKFs as the following:

$$ V \bigl(t,x(t) \bigr)=\sum_{i=1}^{4}V_{i} \bigl(t,x(t) \bigr), $$

(13)

where

$$\begin{array}{rl}& V_1(t,x(t))={\rho }_1^T(t)P{\rho }_1(t),\\ & V_2(t,x(t))={\int }_{t-h_m}^t{e}^{\varrho (t-u)}{\rho }_2^T(u)Q_1{\rho }_2(u)du\\ & \phantom{V_2(t,x(t))=}+{\int }_{t-h_M}^t{e}^{\varrho (t-u)}{\rho }_2^T(u)Q_2{\rho }_2(u)du,\\ & V_3(t,x(t))=h_m{\int }_{-h_m}^0{\int }_{t+u}^t{e}^{\varrho (t-s)}{\dot{x}}^T(s)R_1\dot{x}(s)dsdu\\ & \phantom{V_3(t,x(t))=}+h_{Mm}{\int }_{-h_M}^{-h_m}{\int }_{t+s}^t{e}^{\varrho (t-s)}{\dot{x}}^T(s)R_2\dot{x}(s)dsdu\\ & \phantom{V_3(t,x(t))=}+h_M{\int }_{-h_M}^0{\int }_{t+s}^t{e}^{\varrho (t-s)}{\dot{x}}^T(s)R_3\dot{x}(s)dsdu,\\ & V_4(t,x(t))={\eta }_{21}{\int }_{-{\eta }_2}^{-{\eta }_1}{\int }_{t+u}^t{e}^{\varrho (t-s)}{f}^T(Wx(s))Zf(Wx(s))dsdu,\\ & {\rho }_1(t)={\left[\begin{array}{ccc}{x}^T(t)& {\int }_{t-h_m}^t{x}^T(u)du& {\int }_{t-h_M}^t{x}^T(u)du\end{array}\right]}^T,\\ & {\rho }_2(t)={\left[\begin{array}{cc}{x}^T(t)& {\dot{x}}^T(t)\end{array}\right]}^T.\end{array}$$

Taking the derivative of (13) along the trajectory of the GNNs (9), we obtain

$$\begin{array}{rl}& {\dot{V}}_1(t,x(t))=2{\rho }_1^T(t)P\left[\begin{array}{c}\dot{x}(t)\\ x(t)-x(t-h_m)\\ x(t)-x(t-h_M)\end{array}\right]-\varrho {\rho }_1^T(t)P{\rho }_1(t)+\varrho V_1(t,x(t))\\ & \phantom{{\dot{V}}_1(t,x(t))}={\zeta }^T(t)\{Sym\left\{{\mathrm{\Gamma }}_1^{T}P{\mathrm{\Gamma }}_2\right\}-\varrho {\mathrm{\Gamma }}_1^{T}P{\mathrm{\Gamma }}_1\}\zeta (t)+\varrho V_1(t,x(t))\\ & \phantom{{\dot{V}}_1(t,x(t))}={\zeta }^T(t){\mathrm{\Xi }}_1\zeta (t)+\varrho V_1(t,x(t)),\end{array}$$

(14)

$$\begin{aligned} &\dot{V}_{2} \bigl(t,x(t) \bigr) = \rho _{2}^{T}(t) Q_{1} \rho _{2}(t) - e^{ \varrho h_{m}} \rho _{2}^{T}(t-h_{m}) Q_{1} \rho _{2}(t-h_{m}) +\rho ^{T}_{2}(t)Q_{2} \rho _{2}(t) \\ &\hphantom{\dot{V}_{2} (t,x(t) ) =}{}-e^{\varrho h_{M}} \rho _{2}^{T}(t-h_{M}) Q_{2} \rho _{2}(t-h_{M}) + \varrho V_{2} \bigl(t,x(t) \bigr) \\ &\hphantom{\dot{V}_{2} (t,x(t) ) }=\zeta ^{T}(t) \bigl\{ \Gamma ^{T}_{3} (Q_{1} + Q_{2} ) \Gamma _{3} - \Gamma _{4}^{T} \bigl(e^{\varrho h_{m}} Q_{1} \bigr) \Gamma _{4}-\Gamma ^{T}_{5} \bigl(e^{ \varrho h_{M}} Q_{2} \bigr) \Gamma _{5} \bigr\} \zeta (t) \\ &\hphantom{\dot{V}_{2} (t,x(t) ) =}{}+\varrho V_{2} \bigl(t,x(t) \bigr) \\ &\hphantom{\dot{V}_{2} (t,x(t) ) }= \zeta ^{T}(t) \Xi _{2} \zeta (t)+\varrho V_{2} \bigl(t,x(t) \bigr), \end{aligned}$$

(15)

$$\begin{aligned} &\dot{V}_{3} \bigl(t,x(t) \bigr) = h_{m}^{2} \dot{x}^{T}(t) R_{1} \dot{x}(t) - h_{m} \int ^{t}_{t-h_{m}} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{1} \dot{x}(u)\,du \\ & \hphantom{\dot{V}_{3} (t,x(t) ) =}{}+h_{Mm}^{2} \dot{x}^{T}(t) R_{2} \dot{x}(t) - h_{Mm} \int ^{t-h_{m}}_{t-h_{M}} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{2} \dot{x}(u)\,du \\ &\hphantom{\dot{V}_{3} (t,x(t) ) =} {}+h_{M}^{2} \dot{x}^{T}(t) R_{3} \dot{x}(t) - h_{M} \int ^{t}_{t-h_{M}} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{3} \dot{x}(u)\,du \\ &\hphantom{\dot{V}_{3} (t,x(t) )=}{}+\varrho V_{3} \bigl(t,x(t) \bigr), \end{aligned}$$

(16)

$$\begin{aligned} &\dot{V}_{4} \bigl(t,x(t) \bigr)=\eta _{21}^{2}f^{T} \bigl(Wx(t) \bigr)Zf \bigl(Wx(t) \bigr) \\ &\hphantom{\dot{V}_{4} (t,x(t) ) =}{}-(\eta _{2}-\eta _{1}) \int _{t-\eta _{2}}^{t-\eta _{1}}e^{\varrho (t-u)}f^{T} \bigl(Wx(u) \bigr)Zf \bigl(Wx(u) \bigr)\,du+ \varrho V_{4} \bigl(t,x(t) \bigr) \\ &\hphantom{\dot{V}_{4} (t,x(t) ) }\leq \eta _{21}^{2}f^{T} \bigl(Wx(t) \bigr)Zf \bigl(Wx(t) \bigr) \\ &\hphantom{\dot{V}_{4} (t,x(t) ) =}{}- \bigl(\eta _{2}(t)-\eta _{1}(t) \bigr)e^{\varrho \eta _{2}} \int _{t-\eta _{2}(t)}^{t- \eta _{1}(t)}f^{T} \bigl(Wx(u) \bigr)Zf \bigl(Wx(u) \bigr)\,du \\ &\hphantom{\dot{V}_{4} (t,x(t) ) =}{}+\varrho V_{4} \bigl(t,x(t) \bigr). \end{aligned}$$

(17)

Applying Lemma 2.5 to the integral, we obtain

$$\begin{aligned}& -h_{m} \int ^{t}_{t-h_{m}} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{1} \dot{x}(u)\,du \\& \quad \leq \zeta ^{T}(t) \biggl\{ -\frac{h_{m}}{\gamma _{1a}} (e_{1} - e_{2})^{T} R_{1} (e_{1} - e_{2}) \\& \qquad {}-\frac{h_{m}}{\gamma _{1b}}(\varepsilon _{11} e_{2} + \varepsilon _{12} e_{1} -h_{m} e_{10})^{T} R_{1} (\varepsilon _{11} e_{2} + \varepsilon _{12} e_{1} - h_{m} e_{10}) \biggr\} \zeta (t) \\& \quad = \zeta ^{T}(t) \biggl\{ -\frac{h_{m}}{\gamma _{1a}} \Gamma _{6}^{T} R_{1} \Gamma _{6} - \frac{h_{m}}{\gamma _{1b}} \Gamma _{7}^{T} R_{1} \Gamma _{7} ) \biggr\} \zeta (t), \end{aligned}$$

(18)

$$\begin{aligned}& -h_{Mm} \int ^{t-h_{m}}_{t-h_{M}} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{2} \dot{x}(u)\,du \\& \quad = {}-h_{Mm} \int ^{t-h_{m}}_{t-h(t)} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{2} \dot{x}(u)\,du \\& \qquad {}-h_{Mm} \int ^{t-h(t)}_{t-h_{M}} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{2} \dot{x}(u)\,du \\& \quad \leq \zeta ^{T}(t) \biggl\{ -\frac{h_{Mm}}{\gamma _{2a}} (e_{2} -e_{3})^{T} R_{2} (e_{2} -e_{3}) \\& \qquad {}- \frac{h_{Mm}}{\gamma _{2b}} (\varepsilon _{21}e_{3} + \varepsilon _{22}e_{2}-h_{Mm}e_{12})^{T} R_{2} (\varepsilon _{21}e_{3} + \varepsilon _{22}e_{2}-h_{Mm}e_{12}) \\& \qquad {}-\frac{h_{Mm}}{\gamma _{3a}} (e_{3} -e_{4})^{T} R_{2} (e_{3} -e_{4}) \\& \qquad {}- \frac{h_{Mm}}{\gamma _{3b}} (\varepsilon _{31}e_{4} + \varepsilon _{32}e_{3}-h_{Mm}e_{13})^{T} R_{2}(\varepsilon _{31}e_{4} + \varepsilon _{32}e_{3}-h_{Mm}e_{13}) \biggr\} \zeta (t) \\& \quad = \zeta ^{T}(t) \biggl\{ -\frac{h_{Mm}}{\gamma _{2a}} \Gamma _{8}^{T} R_{2} \Gamma _{8} - \frac{h_{Mm}}{\gamma _{2b}} \Gamma _{9}^{T} R_{2} \Gamma _{9} -\frac{h_{Mm}}{\gamma _{3a}} \Gamma _{10}^{T} R_{2} \Gamma _{10} \\& \qquad {}- \frac{h_{Mm}}{\gamma _{3b}} \Gamma _{11}^{T} R_{2} \Gamma _{11} \biggr\} \zeta (t), \end{aligned}$$

(19)

$$\begin{aligned}& -h_{M} \int ^{t}_{t-h_{M}} e^{\varrho (t - u)} \dot{x}^{T}(u) R_{3} \dot{x}(u)\,du \\& \quad \leq \zeta ^{T}(t) \biggl\{ -\frac{h_{M}}{\gamma _{4a}} (e_{1} -e_{4})^{T} R_{3} (e_{1} -e_{4}) \\& \qquad {}- \frac{h_{M}}{\gamma _{4b}} (\varepsilon _{41}e_{4} + \varepsilon _{42}e_{1}-h_{M}e_{11})^{T} R_{3} (\varepsilon _{41}e_{4} + \varepsilon _{42}e_{1}-h_{M}e_{11}) \biggr\} \zeta (t) \\& \quad = \zeta ^{T}(t) \biggl\{ -\frac{h_{M}}{\gamma _{4a}} \Gamma _{12}^{T} R_{3} \Gamma _{12} - \frac{h_{M}}{\gamma _{4b}} \Gamma _{13}^{T} R_{3} \Gamma _{13} \biggr\} \zeta (t). \end{aligned}$$

(20)

Applying Lemma 2.2, we obtain

$$\begin{aligned}& - \bigl(\eta _{2}(t)-\eta _{1}(t) \bigr)e^{\varrho \eta _{2}} \int _{t-\eta _{2}(t)}^{t- \eta _{1}(t)}f^{T} \bigl(Wx(u) \bigr)Z f \bigl(Wx(u) \bigr)\,du \\& \quad \leq -e^{\varrho \eta _{2}} \biggl( \int _{t-\eta _{2}(t)}^{t-\eta _{1}(t)}f^{T} \bigl(Wx(u) \bigr)\,du \biggr)^{T} Z \biggl( \int _{t-\eta _{2}(t)}^{t-\eta _{1}(t)}f^{T} \bigl(Wx(u) \bigr)\,du \biggr) \\& \quad = \zeta ^{T}(t) \bigl\{ -e^{T}_{14} \bigl(e^{\varrho \eta _{2}}Z \bigr)e_{14} \bigr\} \zeta (t). \end{aligned}$$

(21)

From Assumption A1, it can be inferred that for any $\beta _{1i}, \beta _{2i}, \beta _{3i} > 0$, $i=1,2,\ldots,n$, we have

$$\begin{aligned}& 2 \bigl[f_{i} \bigl(W_{i} x(t) \bigr) - F^{-}_{i} W_{i} x(t) \bigr] \beta _{1i} \bigl[F^{+}_{i}W_{i} x(t)-f_{i} \bigl(W_{i} x(t) \bigr) \bigr] \geq 0, \end{aligned}$$

(22)

$$\begin{aligned}& 2 \bigl[f_{i} \bigl(W_{i} x \bigl(t-h(t) \bigr) \bigr) -F^{-}_{i}W_{i}x \bigl(t-h(t) \bigr) \bigr] \beta _{2i} \\& \quad {}\times \bigl[F^{+}_{i}W_{i} x \bigl(t-h(t) \bigr) -f_{i} \bigl(W_{i}x \bigl(t-h(t) \bigr) \bigr) \bigr] \geq 0, \end{aligned}$$

(23)

$$\begin{aligned}& 2 \bigl[f_{i} \bigl(W_{i}x(t) \bigr) - f_{i} \bigl(W_{i} x \bigl(t-h(t) \bigr) \bigr) - F^{-}_{i} \bigl(W_{i} x(t)-W_{i}x \bigl(t-h(t) \bigr) \bigr) \bigr] \beta _{3i} \\& \quad {}\times \bigl[F^{+}_{i} \bigl(W_{i}x(t) -W_{i} x \bigl(t-h(t) \bigr) \bigr) - f_{i} \bigl(W_{i}x(t) \bigr) + f_{i} \bigl(W_{i}x \bigl(t-h(t) \bigr) \bigr) \bigr] \geq 0, \end{aligned}$$

(24)

which imply

$$\begin{aligned}& 2\zeta ^{T}(t) \Gamma ^{T}_{14}S_{1} \Gamma _{15} \zeta (t) \geq 0, \end{aligned}$$

(25)

$$\begin{aligned}& 2\zeta ^{T}(t) \Gamma ^{T}_{16}S_{2} \Gamma _{17} \zeta (t) \geq 0, \end{aligned}$$

(26)

$$\begin{aligned}& 2\zeta ^{T}(t) \Gamma ^{T}_{18}S_{3} \Gamma _{19} \zeta (t) \geq 0. \end{aligned}$$

(27)

Thus, we have

$$\begin{aligned} 0\leq {}& \zeta ^{T}(t) \bigl\{ 2\Gamma ^{T}_{14}S_{1} \Gamma _{15}+2 \Gamma ^{T}_{16}S_{2} \Gamma _{17} +2\Gamma ^{T}_{18}S_{3} \Gamma _{19} \bigr\} \zeta (t) \\ ={}&\zeta ^{T}(t) \Xi _{5} \zeta (t), \end{aligned}$$

where $S_{1}=\operatorname{diag}\{\beta _{11}, \beta _{12}, \ldots, \beta _{1n}\}$, $S_{2}=\operatorname{diag}\{\beta _{21}, \beta _{22}, \ldots, \beta _{2n}\}$, and $S_{3}=\operatorname{diag}\{\beta _{31}, \beta _{32}, \ldots, \beta _{3n}\}$.

Moreover, for any appropriate dimensions matrices $N_{1}$, $N_{2}$, we obtain

$$\begin{aligned} 0={}&2 \bigl[x^{T}(t)N_{1}+\dot{x}(t)N_{2} \bigr] \biggl[-\dot{x}(t)-Ax(t)+L_{0}f \bigl(Wx(t) \bigr) \\ &{}+L_{1}f \bigl(Wx \bigl(t-h(t) \bigr) \bigr)+L_{2} \int _{t-\eta _{2}(t)}^{t-\eta _{1}(t)}f \bigl(Wx(u) \bigr) \,du+L_{3} \omega (t) \biggr] \\ ={}& \zeta ^{T}(t) \bigl\{ \operatorname{Sym} \bigl\{ \bigl[e_{1}^{T}N_{1} + e^{T}_{7}N_{2} \bigr] \\ &{}\times [-e_{7} -A e_{1} + L_{0} e_{5} + L_{1} e_{6} + L_{2} e_{14} + L_{3} e_{15}] \bigr\} \bigr\} \zeta (t) \\ ={}& \zeta ^{T}(t) \Xi _{7} \zeta (t). \end{aligned}$$

(28)

Combining (14)–(28), we obtain

$$\begin{aligned} \dot{V} \bigl(t,x(t) \bigr) - \varrho V \bigl(t,x(t) \bigr) - \varrho \omega ^{T}(t) X \omega (t) \leq \zeta ^{T}(t) \Xi \zeta (t). \end{aligned}$$

From the conditions (10) and (12), we have

$$\begin{aligned} \dot{V} \bigl(t,x(t) \bigr) - \varrho V \bigl(t,x(t) \bigr) - \varrho \omega ^{T}(t) X \omega (t) < 0. \end{aligned}$$

(29)

Multiplying (29) by $e^{-\varrho t}$, we can derive that

$$ \frac{d}{dt} \bigl( e^{-\varrho t} V \bigl(t,x(t) \bigr) \bigr) < \varrho e^{- \varrho t}\omega ^{T}(t) X \omega (t). $$

(30)

Using assumption A2 and integrating (30) from 0 to t with $t\in [0,T]$, we have

$$\begin{aligned} V \bigl(t,x(t) \bigr) &< e^{\varrho T} \biggl[V \bigl(0,x(0) \bigr)+\varrho \int _{0}^{T}e^{- \varrho u}\omega ^{T}(u)X\omega (u)\,du \biggr] \\ &< e^{\varrho T} \bigl[V \bigl(0,x(0) \bigr)+d\lambda _{14} \bigl(1-e^{-\varrho T} \bigr) \bigr]. \end{aligned}$$

(31)

Considering $V(0, x(0))$, we can derive that

$$\begin{aligned} V \bigl(0, x(0) \bigr) ={}& \rho ^{T}_{1}(0) P \rho _{1} (0) + \int ^{0}_{-hm} e^{- \varrho u} \rho _{2}(u) Q_{1} \rho _{2}(u)\,du \\ &{}+ \int ^{0}_{h_{M}} e^{-\varrho u} \rho _{2}^{T}(u) Q_{2}\rho _{2}(u)\,du \\ &{}+ h_{m} \int ^{0}_{-h_{m}} \int ^{0}_{u} e^{-\varrho s} \dot{x}(s) R_{1} \dot{x}(s)\,ds \,du \\ &{}+h_{Mm} \int ^{-h_{m}}_{-h_{M}} \int ^{0}_{u} e^{-\varrho s} \dot{x}(s) R_{2} \dot{x}(s)\,ds \,du \\ &{}+h_{M} \int ^{0}_{-h_{M}} \int ^{0}_{u} e^{-\varrho s} \dot{x}(s) R_{3} \dot{x}(s)\,ds \,du \\ &{}+\eta _{21} \int _{-\eta _{2}}^{-\eta _{1}} \int _{u}^{0} e^{-\varrho s}f^{T} \bigl(Wx(s) \bigr)Zf \bigl(Wx(s) \bigr)\,ds \,du \\ \leq {}& \rho ^{T}_{1}(0) P \rho _{1} (0) + \int ^{0}_{-hm} e^{- \varrho u} \rho _{2}(u) Q_{1} \rho _{2}(u)\,du \\ &{}+ \int ^{0}_{h_{M}} e^{-\varrho u} \rho _{2}^{T}(u) Q_{2}\rho _{2}(u)\,du \\ &{}+ h_{m} \int ^{0}_{-h_{m}} \int ^{0}_{u} e^{-\varrho s} \dot{x}(s) R_{1} \dot{x}(s)\,ds \,du \\ &{}+h_{Mm} \int ^{-h_{m}}_{-h_{M}} \int ^{0}_{u} e^{-\varrho s} \dot{x}(s) R_{2} \dot{x}(s)\,ds \,du \\ &{}+h_{M} \int ^{0}_{-h_{M}} \int ^{0}_{u} e^{-\varrho s} \dot{x}(s) R_{3} \dot{x}(s)\,ds \,du \\ &{}+\eta _{21} \int _{-\eta _{2}}^{-\eta _{1}} \int _{u}^{0} e^{-\varrho s}x^{T}(s) F_{w} Z F_{w} x(s)\,ds \,du, \end{aligned}$$

where $F_{w}=\operatorname{diag} \{ F^{+}_{1}, F^{+}_{2}, \ldots, F^{+}_{n} \}W$.

Letting $\overline{P}_{i} = H^{-\frac{1}{2}}P_{i} H^{-\frac{1}{2}}$, $\overline{Q}_{1i} = H^{-\frac{1}{2}}Q_{1i} H^{-\frac{1}{2}}$, $\overline{Q}_{2i} = H^{-\frac{1}{2}}Q_{2i} H^{-\frac{1}{2}}$, $\overline{R}_{i} = H^{-\frac{1}{2}}R_{i} H^{-\frac{1}{2}}$, $\overline{Z} = H^{-\frac{1}{2}}F_{w} Z F_{w} H^{-\frac{1}{2}}$, $i=1,2,3$, we have

$$\begin{aligned} V \bigl(0, x(0) \bigr) \leq {}& \rho ^{T}_{1}(0) H^{\frac{1}{2}}\overline{P} H^{ \frac{1}{2}} \rho _{1} (0) + \int ^{0}_{-hm} e^{\varrho u} \rho _{2}(u) H^{\frac{1}{2}}\overline{Q}_{1} H^{\frac{1}{2}} \rho _{2}(u)\,du \\ &{}+ \int ^{0}_{h_{M}} e^{-\varrho u} \rho _{2}^{T}(u) H^{\frac{1}{2}} \overline{Q}_{2} H^{\frac{1}{2}}\rho _{2}(u)\,du \\ &{}+ h_{m} \int ^{0}_{-h_{m}} \int ^{0}_{u} e^{-\varrho s} \dot{x}^{T}(s) H^{\frac{1}{2}}\overline{R}_{1} H^{\frac{1}{2}} \dot{x}(s)\,ds \,du \\ &{}+ h_{Mm} \int ^{-h_{m}}_{-h_{M}} \int ^{0}_{u} e^{-\varrho s} \dot{x}^{T}(s) H^{\frac{1}{2}}\overline{R}_{2} H^{\frac{1}{2}} \dot{x}(s)\,ds \,du \\ &{}+ h_{M} \int ^{0}_{-h_{M}} \int ^{0}_{u} e^{-\varrho s} \dot{x}^{T}(s) H^{\frac{1}{2}}\overline{R}_{3} H^{\frac{1}{2}} \dot{x}^{T}(s)\,ds \,du \\ &{}+\eta _{21} \int _{-\eta _{2}}^{-\eta _{1}} \int _{u}^{0} e^{-\varrho s}x^{T}(s) H^{\frac{1}{2}} \overline{Z} H^{\frac{1}{2}} x(s)\,ds \,du \\ \leq {}& \bigl\{ \lambda _{\max}(\overline{P}_{1})+ h_{m} \lambda _{\max}( \overline{P}_{2})+ h_{M}\lambda _{\max}(\overline{P}_{3}) \\ &{}+ \delta _{1} \bigl[\lambda _{\max}(\overline{Q}_{11}) + 2\lambda _{\max}( \overline{Q}_{12}) + \lambda _{\max}(\overline{Q}_{13}) \bigr] \\ &{}+\delta _{2} \bigl[\lambda _{\max}(\overline{Q}_{21}) + 2\lambda _{\max}( \overline{Q}_{22}) + \lambda _{\max}(\overline{Q}_{23}) \bigr] +h_{m} \delta _{3}\lambda _{\max}(\overline{R}_{1}) \\ &{} + h_{Mm}\delta _{4}\lambda _{\max}( \overline{R}_{2}) + h_{M}\delta _{5} \lambda _{\max}(\overline{R}_{3})+\eta _{21}\delta _{6} \lambda _{ \max}(\overline{Z}) \bigr\} \\ &{}\times \sup_{-h_{M}\leq u\leq 0} \bigl\{ x^{T}(u)Hx(u), \dot{x}^{T}(u)H \dot{x}(u) \bigr\} \\ \leq {}& \bigl\{ \lambda _{1} + h_{m} \lambda _{2} + h_{M} \lambda _{3}+ \delta _{1} (\lambda _{4} + 2 \lambda _{5} + \lambda _{6}) \\ &{}+ \delta _{2}(\lambda _{7} + 2\lambda _{8} + \lambda _{9}) + h_{m} \delta _{3}\lambda _{10} + h_{Mm}\delta _{4}\lambda _{11} \\ &{}+ h_{M}\delta _{5} \lambda _{12}+ \eta _{21}\delta _{6} \lambda _{13} \bigr\} c_{1} \\ ={}& \Xi _{\lambda }c_{1}. \end{aligned}$$

Moreover, from (13), we obtain

$$\begin{aligned} V \bigl(t,x(t) \bigr) \geq x^{T}(t) P_{1} x(t) \geq \lambda _{\min}(\overline{P}_{1})x^{T}(t) H x(t) = \lambda _{0} x^{T}(t) H x(t). \end{aligned}$$

(32)

Then, from (31), (32), and LMI (12), we obtain

$$\begin{aligned} x^{T}(t) H x(t) \leq & \frac{e^{\varrho T}}{\lambda _{0}} \bigl[\Xi _{ \lambda }c_{1} + d\lambda _{14} \bigl(1-e^{-\varrho T} \bigr) \bigr] < c_{2}. \end{aligned}$$

Therefore, the delayed GNNs (9) are finite-time bounded respecting $(c_{1},c_{2},T,H,d)$. The proof is complete. □

Remark 4

The activation function in Assumption A1 does not need to be nonmonotonic and differentiable since the constants $F^{-}_{i}$ and $F^{+}_{i}$, $i=1, 2, \ldots , n$ can be either positive, zero, or negative. Since Assumption A1 has been considered in (22)–(24) of this article, not only $F^{-}_{i}\leq \frac{f_{i}(Wx(t))}{Wx(t)}\leq F^{+}_{i}$ and $F^{-}_{i}\leq \frac{f_{i}(Wx(t-h(t)))}{Wx(t-h(t))}\leq F^{+}_{i}$ but also $F^{-}_{i}\leq \frac{f_{i}(Wx_{1})-f_{i}(Wx_{2})}{Wx_{1}-Wx_{2}}\leq F^{+}_{i}$. Thus, the assumption is weaker and more general than the usual Lipschitz condition ($|f(Wx_{1})-f(Wx_{2})|\leq F|Wx_{1}-Wx_{2}|$).

Moreover, we derive the new sufficient conditions of finite-time boundedness of the GNNs (9) with mixed-interval time-varying delays by applying the Wirtinger-based integral inequality as the following:

Corollary 3.2

Given positive scalars $h_{M}$ and ϱ then the delayed GNNs (9) are finite-time bounded regarding $(c_{1},c_{2},T,H,d)$, if there exist symmetric positive-definite matrices $P\in \mathbb{R}^{3n\times 3n}$, $Q_{i}\in \mathbb{R}^{2n\times 2n}$, $R_{j}\in \mathbb{R}^{n\times n}$ ($i=1, 2$, $j=1, 2, 3$), Z, X, any matrices $N_{1}$, $N_{2}$, and positive diagonal matrices $S_{1}$, $S_{2}$, $S_{3}$ satisfying LMIs (11) and (12) and

$$\begin{aligned} \widetilde{\Xi}< 0, \end{aligned}$$

(33)

where $\widetilde{\Xi}=\Xi _{1}+\Xi _{2}+\widetilde{\Xi}_{3}+\Xi _{4}+\Xi _{5}+ \Xi _{6}+\Xi _{7}$,

$$\begin{aligned}& \widetilde{\Xi}_{3}= h_{m}^{2} e^{T}_{7} R_{1} e_{7} +h_{Mm}^{2} e^{T}_{7} R_{2} e_{7} + h_{M}^{2} e^{T}_{7} R_{3} e_{7} -\Gamma ^{T}_{6} R_{1} \Gamma _{6} -3\widetilde{\Gamma}^{T}_{7} R_{1} \widetilde{\Gamma}_{7} \\& \hphantom{\widetilde{\Xi}_{3}=}{}-\Gamma ^{T}_{8} R_{2} \Gamma _{8}-3 \widetilde{\Gamma}^{T}_{9} R_{2} \widetilde{ \Gamma}_{9}-\Gamma ^{T}_{10} R_{2} \Gamma _{10}-3 \widetilde{\Gamma}^{T}_{11} R_{2} \widetilde{\Gamma}_{11}-\Gamma ^{T}_{12} R_{3} \Gamma _{12} \\& \hphantom{\widetilde{\Xi}_{3}=}{}-3\widetilde{\Gamma}^{T}_{13} R_{3} \widetilde{ \Gamma}_{13}, \\& \widetilde{\Gamma}_{7}= \bigl[e^{T}_{2}+e^{T}_{1}-2e^{T}_{10} \bigr], \qquad \widetilde{\Gamma}_{9}= \bigl[e^{T}_{3}+e^{T}_{2}-2e^{T}_{12} \bigr],\qquad \widetilde{\Gamma}_{11}= \bigl[e^{T}_{4}+e^{T}_{3}-2e^{T}_{13} \bigr], \\& \widetilde{\Gamma}_{13}= \bigl[e^{T}_{4}+e^{T}_{1}-2e^{T}_{11} \bigr], \end{aligned}$$

and the others as given in Theorem 3.1.

Proof

We apply the similarity of proof as in Theorem 3.1, except that we apply Lemma 2.3 (Wirtinger’s inequality) to the single integral in Equation (16). Therefore, it is omitted here. □

Furthermore, we derive the finite-time boundedness of the delayed GNNs (9) by applying the inequality in Lemma 2.4 [39], and we define vectors as follows:

$$\begin{array}{rl}& \zeta (t)=[x^T(t),x^T(t-h_m),x^T(t-h(t)),x^T(t-h_M),f^T(Wx(t)),\\ & \phantom{\zeta (t)=}{f}^T\left(Wx(t-h(t))\right),{\dot{x}}^T(t),{\dot{x}}^T(t-h_m),{\dot{x}}^T(t-h_M),\\ & \phantom{\zeta (t)=}\frac{1}{h_m}{\int }_{t-h_m}^t{x}^T(u)du,\frac{1}{h_M}{\int }_{t-h_M}^t{x}^T(u)du,\\ & \phantom{\zeta (t)=}{\int }_{t-{\eta }_2(t)}^{t-{\eta }_1(t)}{f}^T(Wx(u))du,{\omega }^T(t){]}^T,\\ & e_j=\left[\begin{array}{ccc}{0}_{n\times (j-1)n}& I_n& 0_{n\times (13-j)n}\end{array}\right],j=1,2,\dots ,13.\end{array}$$

Corollary 3.3

Given positive scalars $h_{M}$ and ϱ then the delayed GNNs (9) are finite-time bounded regarding $(c_{1},c_{2},T,H,d)$, if there exist symmetric positive-definite matrices $P\in \mathbb{R}^{3n\times 3n}$, $Q_{i}\in \mathbb{R}^{2n\times 2n}$, $R_{j}\in \mathbb{R}^{n\times n}$ ($i=1, 2$, $j=1, 2, 3$), Z, X, any matrices $N_{1}$, $N_{2}$, and positive diagonal matrices $S_{1}$, $S_{2}$, $S_{3}$ satisfying LMIs (11) and (12) and

$$\begin{aligned} \overline{\Xi}< 0, \end{aligned}$$

(34)

where $\overline{\Xi}=\Xi _{1}+\Xi _{2}+\overline{\Xi}_{3}+\Xi _{4}+\Xi _{5}+ \Xi _{6}+\Xi _{7}$,

$$\begin{array}{rl}& {\bar{\mathrm{\Xi }}}_3=h_m^2{e}_7^T{R}_1{e}_7+h_{Mm}^2{e}_7^T{R}_2{e}_7+h_M^2{e}_7^T{R}_3{e}_7\\ & \phantom{{\bar{\mathrm{\Xi }}}_3=}+h_m{\left[\begin{array}{c}{e}_1^T\\ e_2^T\end{array}\right]}^T\left[\begin{array}{cc}-u_1{R}_1& u_2{R}_1\\ u_2{R}_1& -u_3{R}_1\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_2\end{array}\right]\\ & \phantom{{\bar{\mathrm{\Xi }}}_3=}+h_{Mm}{\left[\begin{array}{c}{e}_2^T\\ e_3^T\end{array}\right]}^T\left[\begin{array}{cc}-m_1{R}_2& m_2{R}_2\\ m_2{R}_2& -m_3{R}_2\end{array}\right]\left[\begin{array}{c}{e}_2\\ e_3\end{array}\right]\\ & \phantom{{\bar{\mathrm{\Xi }}}_3=}+h_{Mm}{\left[\begin{array}{c}{e}_3^T\\ e_4^T\end{array}\right]}^T\left[\begin{array}{cc}-m_1{R}_2& m_2{R}_2\\ m_2{R}_2& -m_3{R}_2\end{array}\right]\left[\begin{array}{c}{e}_3\\ e_4\end{array}\right]\\ & \phantom{{\bar{\mathrm{\Xi }}}_3=}+h_M{\left[\begin{array}{c}{e}_1^T\\ e_4^T\end{array}\right]}^T\left[\begin{array}{cc}-v_1{R}_3& v_2{R}_3\\ v_2{R}_3& -v_3{R}_3\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_4\end{array}\right],\\ & u_1=2(\frac{\varrho }{2}+\frac{1}{h_m})-{\rho }_1{(\frac{\varrho }{2}+\frac{1}{h_m})}^2,\\ & u_2=(\frac{\varrho }{2}+\frac{1}{h_m})-(\frac{\varrho }{2}-\frac{1}{h_m})e^{\varrho h_m}+{\rho }_1(\frac{{\varrho }^2}{4}-\frac{1}{h_m^2})e^{\varrho h_m},\\ & u_3=-2(\frac{\varrho }{2}-\frac{1}{h_m})e^{\varrho h_m}-{\rho }_1{(\frac{\varrho }{2}-\frac{1}{h_m})}^2{e}^{2\varrho h_m},\\ & v_1=2(\frac{\varrho }{2}+\frac{1}{h_M})-{\rho }_2{(\frac{\varrho }{2}+\frac{1}{h_M})}^2,\\ & v_2=(\frac{\varrho }{2}+\frac{1}{h_M})-(\frac{\varrho }{2}-\frac{1}{h_M})e^{\varrho h_M}+{\rho }_2(\frac{{\varrho }^2}{4}-\frac{1}{h_M^2})e^{\varrho h_M},\\ & v_3=-2(\frac{\varrho }{2}-\frac{1}{h_M})e^{\varrho h_M}-{\rho }_2{(\frac{\varrho }{2}-\frac{1}{h_M})}^2{e}^{2\varrho h_M},\\ & m_1=2(\frac{\varrho }{2}+\frac{1}{h_{Mm}})e^{\varrho h_m}-{\rho }_3{(\frac{\varrho }{2}+\frac{1}{h_{Mm}})}^2{e}^{2\varrho h_m},\\ & m_2=(\frac{\varrho }{2}+\frac{1}{h_{Mm}})e^{\varrho h_m}-(\frac{\varrho }{2}-\frac{1}{h_{Mm}})e^{\varrho h_M}+{\rho }_3(\frac{{\varrho }^2}{4}-\frac{1}{h_{Mm}^2})e^{\varrho (h_m+h_M)},\\ & m_3=-2(\frac{\varrho }{2}-\frac{1}{h_{Mm}})e^{\varrho h_M}-{\rho }_3{(\frac{\varrho }{2}-\frac{1}{h_{Mm}})}^2{e}^{2\varrho h_M},\\ & {\rho }_1=\frac{1-e^{-\varrho h_m}}{\varrho },{\rho }_2=\frac{1-e^{-\varrho h_M}}{\varrho },{\rho }_3=\frac{e^{-\varrho h_m}-e^{-\varrho h_M}}{\varrho },\end{array}$$

and the others as given in Theorem 3.1.

Proof

We follow the similarity of proof as in Theorem 3.1, except that we apply Lemma 2.4 [39] to the single integral in Equation (16). Therefore, it is omitted here. □

3.2 Analysis of finite-time stability

This part presents the new delay-dependent criteria for guaranteeing the finite-time stability of the GNNs (35) with interval time-varying delay. If we let $L_{2}=0$ and $L_{3}=0$ in the GNNs (9), the GNNs (9) can be written as

$$ \begin{aligned} &\dot{x}(t)= -A x(t)+L_{0} f \bigl(Wx(t) \bigr)+ L_{1} f \bigl(Wx \bigl(t-h(t) \bigr) \bigr), \\ &x(t)= \phi (t), \quad \forall t\in [-h_{M},0], \end{aligned} $$

(35)

which in (35) is a particular case for the delayed GNNs (9) and they can be encountered as in [12, 15, 20].

Furthermore, we define vectors as follows:

$$\begin{array}{rl}& \zeta (t)=[x^T(t),x^T(t-h_m),x^T(t-h(t)),x^T(t-h_M),f^T(Wx(t)),\\ & \phantom{\zeta (t)=}{f}^T\left(Wx(t-h(t))\right),{\dot{x}}^T(t),{\dot{x}}^T(t-h_m),{\dot{x}}^T(t-h_M),\\ & \phantom{\zeta (t)=}\frac{1}{h_m}{\int }_{t-h_m}^t{x}^T(u)du,\frac{1}{h_M}{\int }_{t-h_M}^t{x}^T(u)du,\frac{1}{h(t)-h_m}{\int }_{t-h(t)}^{t-h_m}{x}^T(u)du,\\ & \phantom{\zeta (t)=}\frac{1}{h_M-h(t)}{\int }_{t-h_M}^{t-h(t)}{x}^T(u)du{]}^T,\\ & e_j=\left[\begin{array}{ccc}{0}_{n\times (j-1)n}& I_n& 0_{n\times (13-j)n}\end{array}\right],j=1,2,\dots ,13.\end{array}$$

We derive the new delay-dependent conditions of the finite-time stability for the GNNs (9) with interval time-varying delay by using the new integral inequality in Lemma 2.5 as the following.

Corollary 3.4

Given positive scalars $h_{M}$ and ϱ then the delayed GNNs (35) are finite-time stable regarding $(c_{1},c_{2},T,H)$, if there exist symmetric positive-definite matrices $P\in \mathbb{R}^{3n\times 3n}$, $Q_{i}\in \mathbb{R}^{2n\times 2n}$, $R_{j}\in \mathbb{R}^{n\times n}$ ($i=1, 2$, $j=1, 2, 3$), any matrices $N_{1}$, $N_{2}$, and positive diagonal matrices $S_{1}$, $S_{2}$, $S_{3}$, such that the following conditions hold:

$$\begin{aligned}& \Xi _{s}< 0, \end{aligned}$$

(36)

$$\begin{aligned}& \lambda _{0} I \leq \overline{P}_{1} \leq \lambda _{1} I, \qquad 0 \leq \overline{P}_{2}\leq \lambda _{2} I, \qquad 0 \leq \overline{P}_{3} \leq \lambda _{3} I, \qquad 0\leq \overline{Q}_{11}\leq \lambda _{4} I , \\& 0 \leq \overline{Q}_{12}\leq \lambda _{5} I,\qquad 0 \leq \overline{Q}_{13} \leq \lambda _{6} I, \qquad 0 \leq \overline{Q}_{21}\leq \lambda _{7} I, \qquad 0 \leq \overline{Q}_{22}\leq \lambda _{8} I, \end{aligned}$$

(37)

$$\begin{aligned}& 0 \leq \overline{Q}_{23}\leq \lambda _{9} I,\qquad 0 \leq \overline{R}_{1} \leq \lambda _{10} I,\qquad 0 \leq \overline{R}_{2}\leq \lambda _{11} I, \qquad 0 \leq \overline{R}_{3}\leq \lambda _{12} I, \\& e^{\varrho T}\Xi _{s\lambda}c_{1}< \lambda _{0} c_{2}, \end{aligned}$$

(38)

where

$$\begin{aligned}& \Xi _{s}= \Xi _{1} + \Xi _{2} + \Xi _{3} + \Xi _{5} + \Xi _{7s} \\& \Xi _{7s}= \operatorname{Sym} \bigl\{ \bigl[e^{T}_{1} N_{1}+e^{T}_{7} N_{2} \bigr] [-e_{7}-A e_{1} +L_{0} e_{5} +L_{1} e_{6}] \bigr\} , \\& \Xi _{s\lambda}= \lambda _{1}+h_{m} \lambda _{2}+h_{M} \lambda _{3}+ \delta _{1}( \lambda _{4}+2\lambda _{5}+\lambda _{6})+\delta _{2}( \lambda _{7}+2\lambda _{8}+\lambda _{9}) \\& \hphantom{\Xi _{s\lambda}=}{}+h_{m}\delta _{3}\lambda _{10}+h_{Mm} \delta _{4} \lambda _{11}+h_{M} \delta _{5} \lambda _{12}, \end{aligned}$$

and the others as given in Theorem 3.1.

Proof

We follow the similarity of proof as in Theorem 3.1, except $Z=0$ or $V_{4}=0$. This corollary uses the new inequality to improve stability criteria. Therefore, it is omitted here. □

In addition, we derive the new sufficient conditions of the finite-time stability for the delayed GNNs (35) that apply the Wirtinger-based integral inequality as the following:

Corollary 3.5

Given positive scalars $h_{M}$ and ϱ then the delayed GNNs (35) are finite-time stable regarding $(c_{1},c_{2},T,H)$, if there exist symmetric positive-definite matrices $P\in \mathbb{R}^{3n\times 3n}$, $Q_{i}\in \mathbb{R}^{2n\times 2n}$, $R_{j}\in \mathbb{R}^{n\times n}$ ($i=1, 2$, $j=1, 2, 3$), any matrices $N_{1}$, $N_{2}$, and positive diagonal matrices $S_{1}$, $S_{2}$, $S_{3}$ satisfying LMIs (37) and (38) and

$$\begin{aligned} \widetilde{\Xi}_{s}< 0, \end{aligned}$$

(39)

where $\widetilde{\Xi}_{s}=\Xi _{1}+\Xi _{2}+\widetilde{\Xi}_{3}+\Xi _{5}+ \Xi _{7s}$,and the others as given in Theorem 3.1, and Corollaries 3.2and 3.4.

Proof

We follow the same proof as in Corollary 3.4, except that we apply Lemma 2.3 (Wirtinger’s inequality) to the single integral in Equation (16). Therefore, it is omitted here. □

Furthermore, we derive the finite-time stability for the GNNs (35) with interval time-varying delay by applying the inequality in Lemma 2.4 [39], and we define the following vectors:

$$\begin{array}{rl}& \zeta (t)=[x^T(t),x^T(t-h_m),x^T(t-h(t)),x^T(t-h_M),f^T(Wx(t)),\\ & \phantom{\zeta (t)=}{f}^T\left(Wx(t-h(t))\right),{\dot{x}}^T(t),{\dot{x}}^T(t-h_m),{\dot{x}}^T(t-h_M),\\ & \phantom{\zeta (t)=}\frac{1}{h_m}{\int }_{t-h_m}^t{x}^T(u)du,\frac{1}{h_M}{\int }_{t-h_M}^t{x}^T(u)du,{]}^T,\\ & e_j=\left[\begin{array}{ccc}{0}_{n\times (j-1)n}& I_n& 0_{n\times (11-j)n}\end{array}\right],j=1,2,\dots ,11.\end{array}$$

Corollary 3.6

Given positive scalars $h_{M}$ and ϱ then the delayed GNNs (35) are finite-time stable regarding $(c_{1},c_{2},T,H)$, if there exist symmetric positive-definite matrices $P\in \mathbb{R}^{3n\times 3n}$, $Q_{i}\in \mathbb{R}^{2n\times 2n}$, $R_{j}\in \mathbb{R}^{n\times n}$ ($i=1, 2$, $j=1, 2, 3$), any matrices $N_{1}$, $N_{2}$, and positive diagonal matrices $S_{1}$, $S_{2}$, $S_{3}$ satisfying LMIs (37) and (38) and

$$\begin{aligned} \overline{\Xi}_{s}< 0, \end{aligned}$$

(40)

where $\overline{\Xi}_{s}=\Xi _{1}+\Xi _{2}+\overline{\Xi}_{3}+\Xi _{5}+ \Xi _{7s}$,and the others as given in Theorem 3.1, and Corollaries 3.3and 3.4.

Proof

We follow the same proof as in Corollary 3.4, except that we apply Lemma 2.4 [39] to the single integral in Equation (16). Therefore, it is omitted here. □

3.3 Analysis of nonfragile finite-time boundedness

This part presents the new delay-dependent criteria of the finite-time boundedness for the GNNs under a nonfragile feedback controller with delays like the following:

$$\begin{aligned} & \dot{x}(t) = \bigl(-A+B \bigl(\mathcal{K}+\Delta \mathcal{K}(t) \bigr) \bigr) x(t)+L_{0} f \bigl(W x(t) \bigr)+ L_{1} f \bigl(W x \bigl(t-h(t) \bigr) \bigr) \\ &\hphantom{\dot{x}(t) =}{}+L_{2} \int ^{t-\eta _{1}(t)}_{t-\eta _{2}(t)} f \bigl(W x(u) \bigr)\,du + L_{3} \omega (t) +B u(t), \\ &y(t)=x(t), \\ &x(t) =\phi (t), \quad \forall t\in [-h_{M},0]. \end{aligned}$$

(41)

Theorem 3.7

Given positive scalars $h_{M}$ and ϱ then the delayed GNNs (41) are finite-time bounded respecting $(c_{1},c_{2},T,H,d)$, if there exist positive symmetric definite matrices $\widetilde{P}, \widetilde{Q}_{i}, \widetilde{R}_{j}$ ($i=1, 2$, $j=1, 2, 3$), Z̃, X̃, and positive diagonal matrices $\widetilde{S}_{1}$, $\widetilde{S}_{2}$, $\widetilde{S}_{3}$, such that the conditions hold as follows:

$$\begin{aligned} &\Xi _{u}< 0, \end{aligned}$$

(42)

$$\begin{aligned} &e^{\varrho T} \bigl[\Xi _{\lambda}c_{1}+d\lambda _{14} \bigl(1-e^{- \varrho T} \bigr) \bigr]< \lambda _{0} c_{2}, \end{aligned}$$

(43)

where

$$\begin{aligned}& \Xi _{u}= \sum^{8}_{i=1}\Xi _{ui}, \\& \Xi _{u1}= \operatorname{Sym} \bigl\{ \Gamma ^{T}_{1} \widetilde{P} \Gamma ^{T}_{2} \bigr\} - \varrho \Gamma ^{T}_{1}\widetilde{P} \Gamma _{1}, \\& \Xi _{u2}= \Gamma ^{T}_{3} ( \widetilde{Q}_{1} + \widetilde{Q}_{2}) \Gamma _{3}-\Gamma ^{T}_{4} \bigl(e^{\varrho h_{m}} \widetilde{Q}_{1} \bigr) \Gamma _{4} -\Gamma ^{T}_{5} \bigl(e^{\varrho h_{M}}\widetilde{Q}_{2} \bigr) \Gamma _{5}, \\& \Xi _{u3}= h_{m}^{2}e^{T}_{7} \widetilde{R}_{1}e_{1} + h^{2}_{Mm}e^{T}_{7} \widetilde{R}_{2}e_{7}+h^{2}_{M} e^{T}_{7}R_{3}e_{7} - \frac{h_{m}}{\gamma _{1a}}\Gamma ^{T}_{6}\widetilde{R}_{1} \Gamma _{6}- \frac{h_{m}}{\gamma _{1b}}\Gamma ^{T}_{7} \widetilde{R}_{2}\Gamma _{7} \\& \hphantom{\Xi _{u3}=}{}- \frac{h_{Mm}}{\gamma _{2a}}\Gamma ^{T}_{8} \widetilde{R}_{2} \Gamma _{8} - \frac{h_{Mm}}{\gamma _{2b}}\Gamma ^{T}_{9} \widetilde{R}_{2}\Gamma _{9} - \frac{h_{Mm}}{\gamma _{3a}}\Gamma ^{T}_{10} \widetilde{R}_{2}\Gamma _{10} - \frac{h_{Mm}}{\gamma _{3b}}\Gamma ^{T}_{11}\widetilde{R}_{2} \Gamma _{11} \\& \hphantom{\Xi _{u3}=} {}- \frac{h_{M}}{\gamma _{4a}}\Gamma ^{T}_{12} \widetilde{R}_{2} \Gamma _{12} - \frac{h_{M}}{\gamma _{4b}}\Gamma ^{T}_{13} \widetilde{R}_{3}\Gamma _{13}, \\& \Xi _{u4}= \eta ^{2}_{21}e^{T}_{5} \widetilde{Z}e_{5} - e^{\varrho \eta _{2}} e^{T}_{14} \widetilde{Z} e_{14}, \\& \Xi _{u5}= 2\Gamma ^{T}_{14} \widetilde{S}_{1}\Gamma _{15}+ 2\Gamma ^{T}_{16} \widetilde{S}_{2}\Gamma _{17} +2\Gamma ^{T}_{18} \widetilde{S}_{3} \Gamma _{19},\qquad \Xi _{u6}= - \varrho e^{T}_{15} \widetilde{X} e_{15}, \\& \Xi _{u7}= \operatorname{Sym} \bigl\{ -e_{1}^{T}U^{T}e_{7} -e^{T}_{1}AU^{T}e_{1} +e_{1}^{T}BYe_{1} +e^{T}_{1}L_{o}U^{T}e_{5} +e^{T}_{1}L_{1}U^{T}e_{6} \\& \hphantom{\Xi _{u7}=} {}+ e^{T}_{1}L_{2}U^{T}e_{14}+e_{1}^{T}L_{3}U^{T}e_{15} - e^{T}_{7}Ue_{7} -e^{T}_{7} AU^{T}e_{1}+ e^{T}_{7}BYe_{1} \\& \hphantom{\Xi _{u7}=}{}+e^{T}_{7}L_{0}U^{T}e_{5} + e^{T}_{7}L_{1}U^{T}e_{6}+e^{T}_{7}L_{2}U^{T}e_{14}+e^{T}_{7}L_{3}U^{T}e_{15} \bigr\} , \\& \Xi _{u8}= \operatorname{Sym} \bigl\{ e^{T}_{1} B D_{1} e_{16}+e^{T}_{7} B D_{1} e_{16}+ \alpha e^{T}_{1} U D_{2}^{T} e_{17} \bigr\} -e^{T}_{16}( \alpha I)e_{16}-e^{T}_{17}( \alpha I)e_{17}. \end{aligned}$$

Additionally, the gain matrix $\mathcal{K}$ of the feedback controller with a nonfragile issue can be created as $\mathcal{K}=YU^{-1}$.

Proof

We follow the method of proof as in Theorem 3.1 and replace A by $A-B(\mathcal{K}+\Delta \mathcal{K}(t))$ in Ξ, and we obtain

$$ \widetilde{\Xi} + \Pi _{1} F(t) \Pi _{2}+\Pi ^{T}_{2}F^{T}(t)\Pi ^{T}_{1}< 0, $$

where

$$\begin{aligned}& \widetilde{\Xi} = \sum^{6}_{i=1} \Xi _{i} + \widetilde{\Xi}_{7}, \\& \widetilde{\Xi}_{7} = \operatorname{Sym} \bigl\{ -e_{1}^{T}N_{1}e_{7}-e^{T}_{1}N_{1}Ae_{1} + e^{T}_{1} N_{1} B \mathcal{K} e_{1} +e^{T}_{1}N_{1}L_{0}e_{5} \\& \hphantom{\widetilde{\Xi}_{7} =} {}+e^{T}_{1} N_{1} L_{1} e_{6} + e^{T}_{1}N_{2}L_{2}e_{14}+e^{T}_{1} N_{1} L_{3} e_{15} -e^{T}_{7} N_{2} e_{7} - e^{T}_{7}N_{2}Ae_{1} \\& \hphantom{\widetilde{\Xi}_{7} =} {}+e^{T}_{7}N_{2}B \mathcal{K} e_{1}+e^{T}_{7}N_{2}L_{0}e_{5}+e^{T}_{7}N_{2}L_{1}e_{6} + e^{T}_{7}N_{2}L_{2}e_{14}+e^{T}_{7}N_{2}L_{3}e_{15} \bigr\} , \\& \Pi _{1} = \bigl[(N_{1}BD_{1})^{T}, \underbrace{0 \cdots 0}_{5 \text{ times}}, (N_{2}BD_{1})^{T}, \underbrace{0 \cdots 0}_{8 \text{ times}} \bigr]^{T}, \\& \Pi _{2} = [D_{2}, \underbrace{0 \cdots 0}_{14 \text{ times}}]^{T}. \end{aligned}$$

Applying Lemma 2.6, there exists $\alpha >0$, so that

$$ \widetilde{\Xi} + \alpha ^{-1}\Pi _{1}\Pi _{1}^{T} +\alpha \Pi _{2}^{T} \Pi _{2}< 0. $$

Applying Lemma 2.7 (the Schur complement), we obtain

$$\left[\begin{array}{ccc}\tilde{\mathrm{\Xi }}& {\mathrm{\Pi }}_1& \alpha {\mathrm{\Pi }}_2^T\\ {\mathrm{\Pi }}_1^T& -\alpha I& 0\\ \alpha {\mathrm{\Pi }}_2& 0& -\alpha I\end{array}\right]<0.$$

(44)

It can be seen that the LMI conditions in (44) cannot be directly applied to the controllers. Hence, we need to convert the conditions to be the LMI terms. We let $N_{1}=N_{2}=U^{-1}$. Then, the criteria (44) are pre- and postmultiplied by $\operatorname{diag}\{\underbrace{U \cdots U}_{15 \text{ times}},I,I\}$ and its transpose, describe variables as the following:

$$\begin{aligned} &\widetilde{P} = UPU^{T},\qquad \widetilde{Q}_{1}=UQ_{1}U^{T}, \qquad \widetilde{Q}_{2}=UQ_{2}U^{T},\qquad \widetilde{R}_{1}=UR_{1}U^{T}, \\ &\widetilde{R}_{2}=UR_{2}U^{T},\qquad \widetilde{R}_{3} = UR_{3}U^{T},\qquad \widetilde{Z}=UZU^{T},\qquad \widetilde{X}=UXU^{T}, \\ &\widetilde{S}_{1}=US_{1}U^{T},\qquad \widetilde{S}_{2}=US_{2}U^{T},\qquad \widetilde{S}_{3}=US_{3}U^{T},\qquad Y= \mathcal{K}U. \end{aligned}$$

Thus, we can obtain $\Xi _{u}<0$. The proof is complete. □

4 Numerical examples

This section demonstrates the effectiveness of our approaches with three numerical examples.

Example 4.1

Consider the following parameters for GNNs (9):

$$\begin{array}{c}A=\left[\begin{array}{cc}2& 0\\ 0& 1.5\end{array}\right],L_0=\left[\begin{array}{cc}-1& 1\\ 0.5& -1\end{array}\right],L_1=\left[\begin{array}{cc}-0.5& 0.6\\ 0.7& 0.8\end{array}\right],\hfill \\ L_2=\left[\begin{array}{cc}0.15& 0.1\\ 0& -0.3\end{array}\right],L_3=\left[\begin{array}{cc}0.05& 0.2\\ 0.2& 0.1\end{array}\right],W=\left[\begin{array}{cc}1.28& 0.35\\ 0.28& 0.35\end{array}\right],\hfill \\ F_M=diag\{0,0\}\text{and}{F}_P=diag\{0.6,0.8\}.\hfill \end{array}$$

Let $f(x)=\tanh (x(t))$, $\eta _{1}(t)=0.4+0.3\sin (t)$, $\eta _{2}(t)=0.5+0.2\sin (t)$, and $\omega (t)=\sqrt{0.5}\cos (t)$. Given scalars $c_{1}=0.72$, $d=0.5$, $\eta _{1}=0.1$, $\eta _{2}=0.7$ and matrix $H=I$. From the parameters as mentioned above, we solve the LMIs in Theorem 3.1, and Corollaries 3.2 and 3.3 to obtain the feasible solution guaranteeing finite-time boundedness regarding $(c_{1},c_{2},T,H,d)$ for comparing the minimum allowable lower bounds (MALBs) of $c_{2}$. In this example, we investigate four cases to demonstrate the effectiveness of our results as follows:

Case I: Let $h(t)=1.25|\sin (t)|+1$, which means $[h_{m},h_{M}]=[1.0,2.25]$ for $t\in [0,T]$, $T=10,20,30,40,50$. We solve the LMIs in Theorem 3.1, and Corollaries 3.2 and 3.3 to obtain the MALBs of $c_{2}$ for different values of final time $T=10,20,30,40,50$, as displayed in Table 1. Theorem 3.1 provides the smallest MALBs of $c_{2}$ for various values of T, which are smaller than those from Corollaries 3.2 and 3.3. By applying the new integral inequality, Theorem 3.1 is less conservative than Corollaries 3.2 and 3.3.

Table 1 MALBs of $c_{2}$ for different values of T in Example 4.1 (Case I). Numbers in parentheses are ϱ

Figures 2 and 3 display the state responses of $x(t)$ and the time history of $x^{T}(t)Hx(t)$ for the GNNs (9) in Example 4.1 with an initial condition $\varphi (t)={\left[\begin{array}{cc}-0.6cos(t)& 0.6cos(t)\end{array}\right]}^T$. From Fig. 3, the time history of $x^{T}(t)Hx(t)$ does not exceed the MALBs of $c_{2}$ in Table 1. Thus, the accuracy of the proposed results is confirmed.

Figure 2

State responses of $x(t)$ for the GNNs (9) in Example 4.1

Figure 3

Time history of $x^{T}(t)Hx(t)$ for the GNNs (9) in Case I Example 4.1, $T=50$

Case II: Let $h(t)=0.25l|\sin (t)|+1$, $l=2,3,4,5,6$, for $t\in [0,10]$. In this case, we investigate the effect of ranges of time delay $[h_{m},h_{M}]$ for $h_{Mm}=h_{M}-h_{m}=0.5,0.75,1.0,1.25,1.5$. For fixed lower bounds $h_{m}=1.0$ and different upper bounds $h_{M}=1.5, 1.75, 2.0, 2.25, 2.50$, we solve the LMIs in Theorem 3.1, and Corollaries 3.2 and 3.3 to obtain the MALBs of $c_{2}$, as shown in Table 2. From the table, we observe that the MALBs of $c_{2}$ from Theorem 3.1 are smaller than those from Corollaries 3.2 and 3.3 in the ranges of delay $h_{Mm}=1.25, 1.5$. In the range of delay $h_{Mm}=1.0$, the MALBs of $c_{2}$ from Theorem 3.1 are equal to those from Corollary 3.3 but smaller than those from Corollary 3.2. On the other hand, the MALBs of $c_{2}$ from Theorem 3.1 are greater than those from Corollary 3.3 but smaller than those from Corollary 3.2 in the ranges of delay $h_{Mm}=0.5, 0.75$. Moreover, the MALBs of $c_{2}$ from our results increase as $h_{Mm}$ increases.

Table 2 MALBs of $c_{2}$ for $T=10$, $\varrho =0.3$ and different values of $[h_{m},h_{M}]$ in Example 4.1 (Case II)

Case III: Let $h(t)=0.5|\sin (t)|+1+0.1l$, $l=1,2,3,4,5$, for $t\in [0,10]$. In this case, the effect of changing the interval time-delay range $[h_{m}, h_{M}]$ for $h_{Mm}=0.5$ is investigated. We solve the LMIs in Theorem 3.1, Corollaries 3.2 and 3.3 to obtain the MALBs of $c_{2}$ with a fixed range of interval time delay $h_{Mm}=0.5$ and various lower bounds $h_{m}=1.1, 1.2, 1.3, 1.4, 1.5$. From Table 3, the MALBs of $c_{2}$ from Theorem 3.1 are greater than those from Corollary 3.3 but smaller than those from Corollary 3.2 for the delay range of $h_{Mm}=0.5$. Furthermore, the MALBs of $c_{2}$ from our results decrease as the lower bound $h_{m}$ increases.

Table 3 MALBs of $c_{2}$ for $T=10$, $\varrho =0.3$ and different values of $[h_{m},h_{M}]$ ($h_{Mm}=0.5$) in Example 4.1 (Case III)

Case IV: Let $h(t)=1.5|\sin (t)|+1+0.1l$, $l=1,2,3,4,5$, for $t\in [0,10]$. In this case, the effect of changing the interval time-delay range $[h_{m}, h_{M}]$ for $h_{Mm}=1.5$ is analyzed. We solve the LMIs in Theorem 3.1, and Corollaries 3.2 and 3.3 to obtain the MALBs of $c_{2}$ with a defined interval time delay $h_{Mm}=1.5$ and various lower bounds $h_{m}=1.1, 1.2, 1.3, 1.4, 1.5$, as shown in Table 4. In the delay range $h_{Mm}=1.5$, the MALBs of $c_{2}$ from Theorem 3.1 are smaller than those from Corollaries 3.2 and 3.3. Additionally, the MALBs of $c_{2}$ decrease, when the lower bound $h_{m}$ increases.

Table 4 MALBs of $c_{2}$ for $T=10$, $\varrho =0.3$ and different values of $[h_{m},h_{M}]$ ($h_{Mm}=1.5$) in Example 4.1 (Case IV)

Example 4.2

Consider the following parameters for GNNs (35):

$$\begin{array}{c}A=\left[\begin{array}{cc}1.5& 0\\ 0& 1.7\end{array}\right],L_0=\left[\begin{array}{cc}0.2& -0.1\\ -0.5& 0.1\end{array}\right],L_1=\left[\begin{array}{cc}-0.5& 0\\ -0.3& -0.2\end{array}\right],\hfill \\ W=\left[\begin{array}{cc}1.5& 0.25\\ 0.68& 0.66\end{array}\right],F_M=diag\{0,0\}\text{and}{F}_P=diag\{0.6,1.6\}.\hfill \end{array}$$

The activation function is given by $f(x)=\tanh (x(t))$. Given scalars $c_{1}=1.28$ and matrix $H=I$. From the parameters as mentioned above, we solve the LMIs in Corollaries 3.4, 3.5, and 3.6 to obtain the feasible solution guaranteeing finite-time stability regarding $(c_{1},c_{2},T,H,d)$ for comparing the MALBs of $c_{2}$. In this example, we investigate four cases to demonstrate the effectiveness of our results as follows:

Case I: Let $h(t)=1.25|\sin (t)|+1.5$, which mean $[h_{m},h_{M}]=[1.5,2.75]$ for $t\in [0,T]$, $T=10,20,30,40,50$. We solve the LMIs in Corollaries 3.4, 3.5, and 3.6 to obtain the MALBs of $c_{2}$ for various values of final time $T=10,20,30,40,50$, as displayed in Table 5. Corollary 3.4 provides the smallest MALBs of $c_{2}$ for different values of T, which are smaller than those from Corollaries 3.5 and 3.6. By applying the new integral inequality, Corollary 3.4 is less conservative than Corollaries 3.5 and 3.6.

Table 5 MALBs of $c_{2}$ for different values of T in Example 4.2 (Case I). Numbers in parentheses are ϱ

Figures 4 and 5 illustrate the state responses of $x(t)$ and time history of $x^{T}(t)Hx(t)$ for the GNNs (35) in Example 4.2 with an initial condition $\varphi (t)={[\begin{array}{cc}-0.8cos(t)& 0.8cos(t)\end{array}]}^T$. From Fig. 5, the time history of $x^{T}(t)Hx(t)$ does not exceed the MALBs of $c_{2}$ in Table 5. Thus, the correctness of the proposed results is confirmed.

Figure 4

State responses of $x(t)$ for the GNNs (35) in Example 4.2

Figure 5

Time history of $x^{T}(t)Hx(t)$ for the GNNs (35) in Case I Example 4.2, $T=30$

Case II: Let $h(t)=0.25l|\sin (t)|+1.5$, $l=2,3,4,5,6$, for $t\in [0,30]$. In this case, we analyze the effect of ranges of time delays $[h_{m},h_{M}]$ for $h_{Mm}=h_{M}-h_{m}=0.5,0.75,1.0,1.25,1.5$. For fixed lower bound $h_{m}=1.5$ and various upper bounds $h_{M}=2.0, 2.25, 2.5, 2.75, 3.0$, we solve the LMIs in Corollaries 3.4, 3.5, and 3.6 to obtain the MALBs of $c_{2}$, as displayed in Table 6. From the table, we can see that the MALBs of $c_{2}$ from Corollary 3.4 are smaller than those from Corollaries 3.5 and 3.6 in the ranges of delay $h_{Mm}=1.0, 1.25, 1.5$. However, the MALBs of $c_{2}$ from Corollary 3.4 are greater than those from Corollary 3.6 but smaller than those from Corollary 3.5 in the ranges of delay $h_{Mm}=0.5, 0.75$. Additionally, the MALBs of $c_{2}$ from our results increase as $h_{Mm}$ increases.

Table 6 MALBs of $c_{2}$ for $T=30$, $\varrho =0.45$ and different values of $[h_{m},h_{M}]$ in Example 4.2 (Case II)

Case III: Let $h(t)=0.25|\sin (t)|+1.5+0.1l$, $l=1,2,3,4,5$, for $t\in [0,30]$. In this case, the effect of changing the interval time-delay range $[h_{m}, h_{M}]$ for $h_{Mm}=0.25$ is examined. We solve the LMIs in Corollaries 3.4, 3.5, and 3.6 to obtain the MALBs of $c_{2}$ with a fixed range of interval time delay $h_{Mm}=0.25$ and different lower bounds $h_{m}=1.6, 1.7, 1.8, 1.9, 2.0$. From Table 7, the MALBs of $c_{2}$ from Corollary 3.4 are greater than those from Corollary 3.6 but smaller than those from Corollary 3.5 for the delay range of $h_{Mm}=0.25$. Moreover, the MALBs of $c_{2}$ from our results increase as the lower bound $h_{m}$ increases.

Table 7 MALBs of $c_{2}$ for $T=30$, $\varrho =0.45$ and different values of $[h_{m},h_{M}]$ ($h_{Mm}=0.25$) in Example 4.2 (Case III)

Case IV: Let $h(t)=1.25|\sin (t)|+1.5+0.1l$, $l=1,2,3,4,5$, for $t\in [0,30]$. In this case, the effect of changing the interval time-delay range $[h_{m}, h_{M}]$ for $h_{Mm}=1.25$ is investigated. We solve the LMIs in Corollaries 3.4, 3.5, and 3.6 to obtain the MALBs of $c_{2}$ with a fixed interval time delay $h_{Mm}=1.25$ and different lower bounds $h_{m}=1.6, 1.7, 1.8, 1.9, 2.0$, as shown in Table 8. In the delay range $h_{Mm}=1.25$, the MALBs of $c_{2}$ from Corollary 3.4 are smaller than those from Corollaries 3.5 and 3.6. Furthermore, the MALBs of $c_{2}$ increase, when the lower bound $h_{m}$ increases.

Table 8 MALBs of $c_{2}$ for $T=30$, $\varrho =0.45$ and different values of $[h_{m},h_{M}]$ ($h_{Mm}=1.25$) in Example 4.2 (Case IV)

Remark 5

In Lemma 2.5, we derive the new integral inequality with the exponential function to estimate the single integral terms of the derivative of LKFs in Theorem 3.1 and Corollary 3.4. In contrast, we use the approximation $-e^{\varrho (t-u)}\leq -e^{\varrho d_{2}}$, $t-d_{1}\leq u \leq t-d_{2}$ and Wirtinger’s integral inequality without the exponential term such as $-\int _{t-d_{1}}^{t-d_{2}}e^{\varrho (t-u)}\dot{x}^{T}(t)M\dot{x}(t)\,du \leq -e^{\varrho d_{2}}\int _{t-d_{1}}^{t-d_{2}}\dot{x}^{T}(t)M \dot{x}(t)\,du$ in Corollaries 3.2 and 3.5. In Examples 4.1 and 4.2, the MALBs of $c_{2}$ from the new integral inequality are smaller than those from Wirtinger’s integral inequality in all cases. Thus, the results obtained by the new inequality are less conservative than those obtained by Wirtinger’s inequality.

Remark 6

Note that the similarities between Lemma 2.4 [39] and the new inequality are in the form of exponential functions. In contrast, the differences are dimension-free matrices from an estimate of the single integral terms of the derivative of LKFs. While Lemma 2.4 [39] desires only a 2-dimensional-free matrix ${[\begin{array}{cc}{x}^T(t-h_m)& x^T(t-h_M)\end{array}]}^T$, the new inequality requires a 3-dimensional-free matrix ${[\begin{array}{ccc}{\epsilon }_1{x}^T(t-h_m)& {\epsilon }_2{x}^T(t-h_M)& {\int }_{t-h_M}^{t-h_m}{x}^T(u)du\end{array}]}^T$. In Case III of Examples 4.1 and 4.2, the MALBs of $c_{2}$ obtained by Lemma 2.4 [39] are smaller than those from Lemma 2.5 (New inequality), where $h_{Mm}=0.25, 0.50$. However, the MALBs of $c_{2}$ obtained by Lemma 2.4 [39] are greater than those from the new inequality, where $h_{Mm}=1.25, 1.50$ in Case IV. Similar to Case II of Examples 4.1 and 4.2, we observe that the MALBs of $c_{2}$ by applying the new inequality are smaller than those from Lemma 2.4 [39], where $h_{Mm} > 1.0$. Thus, the results obtained by the new integral inequality can open up the possibility of overcoming results obtained by Lemma 2.4 [39].

Remark 7

We use a specific form of the Lyapunov–Krasovskii functional (13) with exponential functions to obtain finite-time stability results, simplifying the application of our results to analyze the finite-time stability and finite-time boundedness in practice. The proofs of Theorem 3.1 and Corollary 3.4 show that the new inequality and the given Lyapunov–Krasovskii functional with both exponential functions can be used to quickly derive the derivative condition of finite-time stability and finite-time boundedness, which is an advantage of the approach used in this work. However, our method is theoretically difficult to determine the upper bound of the delay of settling time T except for certain particular cases. Furthermore, our new integral inequality is complex in practice, making it difficult to solve and requiring much time and effort to find a solution. These are the downsides of our method.

Remark 8

If we choose $L_{0}=0$, $L_{1}=1$, $L_{2}=0$ and the external disturbance input is equal to zero in the GNNs (9), we obtain

$$ \begin{aligned} &\dot{x}(t)= -A x(t) + f \bigl(x \bigl(t-h(t) \bigr) \bigr), \\ &x(t)= \phi (t), \quad \forall t\in [-h_{M},0], \end{aligned} $$

(45)

then (45) is a particular case of the GNNs (9) and can be found as in [11, 42–44].

Example 4.3

Consider the following parameters for the system (45):

$$\begin{array}{rl}& A=diag\{0.8,5.3\},W_0=\left[\begin{array}{cc}0.1& 0.3\\ 0.9& 0.1\end{array}\right],\\ & h_m=1,F_m=diag\{0,0\}\text{and}{F}_M=diag\{1,1\}.\end{array}$$

This example examines the stability criterion of the system (45). Using the parameters mentioned above, we solve the method of this paper with the new integral inequality to obtain the maximum allowable bounds of $h_{M}$ for different values of μ, as shown in Table 9. Our method provides the greatest maximum allowable bounds of $h_{M}$ for different values of μ, which are greater than those in [11, 42–44]. Therefore, our approach in this paper is less conservative than those in [11, 42–44].

Table 9 Maximum allowable bounds of $h_{M}$ for different values of μ in Example 4.3

Example 4.4

Neural Networks have been widely applied in several applications. In particular, the four-tank system is a fascinating neural-network application. In 2000, Johansson [45] first proposed the four-tank system. Johansson’s four-tank system consists of 4 correspondingly connected water tanks with valves and two batches of pumps, as illustrated in Fig. 6. The purpose of the four-tank system is to manage the water level with two pumps. The voltages to the water pumps are the process inputs. Pump 1 is responsible for adding water to Tanks 1 and 4. Pump 2 is responsible for filling water to Tanks 2 and 3. Water flows from Tank 3 to Tank 1 and Tank 4 to Tank 2 by gravity. The water levels for Tank 1 ($q_{1}$) and Tank 2 ($q_{2}$) are evaluated together as outputs. The four-tank system has received a great deal of attention, as in [13, 17, 18, 26, 46]. The four-tank system and the present controller can be written as:

$$ \dot{\widetilde{x}}(t)=\widetilde{{A}}_{0} \widetilde{x}(t)+ \widetilde{{A}}_{1} \widetilde{x}(t- \widetilde{d}_{1})+ \widetilde{{B}}_{0} \widetilde{u}(t- \widetilde{d}_{2})+ \widetilde{{B}}_{1}\widetilde{u}(t- \widetilde{d}_{3}), $$

(46)

where

$$\begin{array}{c}{\tilde{A}}_0=\left[\begin{array}{cccc}-0.0021& 0& 0& 0\\ 0& -0.0021& 0& 0\\ 0& 0& -0.0424& 0\\ 0& 0& 0& -0.0424\end{array}\right],\hfill \\ {\tilde{A}}_1=\left[\begin{array}{cccc}0& 0& 0.0424& 0\\ 0& 0& 0& 0.0424\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\hfill \\ {\tilde{B}}_0={\left[\begin{array}{cccc}0.1113{\gamma }_1& 0& 0& 0\\ 0& 0.1042{\gamma }_2& 0& 0\end{array}\right]}^T,\hfill \\ {\tilde{B}}_1={\left[\begin{array}{cccc}0& 0& 0& 0.1113(1-{\gamma }_1)\\ 0& 0& 0.1042(1-{\gamma }_2)& 0\end{array}\right]}^T,\hfill \\ {\gamma }_1=0.333,{\gamma }_2=0.307,\tilde{u}(t)=\tilde{\mathcal{K}}\tilde{x}(t)\text{and}\hfill \\ \tilde{\mathcal{K}}=\left[\begin{array}{cccc}-0.1609& -0.1765& -0.0795& -0.2073\\ -0.1977& -0.1579& -0.2288& -0.0772\end{array}\right].\hfill \end{array}$$

Figure 6

Schematic representation of the four-tank system. Source: Johansson [45]

In addition, this example illustrates how transportation delays occur between the tanks and the valves, which are interval time-delay signals. Suppose the controller $\widetilde{u}(t)$ is the water quantity from the pumps and $\widetilde{d}_{1}=0$, $\widetilde{d}_{2}=0$, $\widetilde{d}_{3}=h(t)$ ($h_{m}< h(t)< h_{M}$). Hence, $\widetilde{u}(t)$ is a nonlinear function as the following: $\widetilde{u}(t)=\widetilde{{\mathcal{K}}}\widetilde{f}( \widetilde{x}(t))$, $\widetilde{u}(t-h(t))=\widetilde{{\mathcal{K}}}\widetilde{f}( \widetilde{x}(t-h(t)))$, $\widetilde{f}(\widetilde{x}(t))=[\widetilde{f}(\widetilde{x}_{1}(t)), \ldots, \widetilde{f}(\widetilde{x}_{4}(t))]^{T}$, $\widetilde{f}_{i}(\widetilde{x}_{i}(t))=0.01(|\widetilde{x}_{i}(t)+1|-| \widetilde{x}_{i}(t)-1|)$, $\forall i=1,2,3,4$.

We modify the four-tank system (46) as the following delayed NNs (47) as follows:

$$ \dot{x}(t)= \bigl(-A+B \bigl(\mathcal{K}+\Delta \mathcal{K}(t) \bigr) \bigr) x(t)+L_{0} f \bigl(x(t) \bigr)+ L_{1} f \bigl(x \bigl(t-h(t) \bigr) \bigr)+ L_{3} \omega (t), $$

(47)

where $A=-\widetilde{{A}}_{0}-\widetilde{{A}}_{1}$, $L_{0}=\widetilde{{B}}_{0}\widetilde{\mathcal{K}}$, $L_{1}=\widetilde{{B}}_{1}\widetilde{{\mathcal{K}}}$, $f(\cdot )=\widetilde{f}(\cdot )$, $W=I$, $D_{1}=D_{2}=-0.5I$, $B=-I$, $F_{M}=0$, $F_{P}=0.5I$. Given $h(t)=1.0+0.8\cos (t)$, which means $h_{m}=0.2$, $h_{M}=1.8$. Given scalars $d=0.1$, $\varrho =0.01$, $c_{1}=0.5$, $c_{2}=5$, $T=30$ and matrix $H=I$. From the parameters mentioned above, we can compute the gain matrix $\mathcal{K}$ of the state feedback controller with a nonfragile issue by Theorem 3.7 as the following:

$$\mathcal{K}=\left[\begin{array}{cccc}2.0236& -0.0028& 0.0408& -0.0034\\ -0.0028& 2.0240& -0.0032& 0.0410\\ -0.0008& -0.0010& 1.9876& 0.0019\\ -0.0010& -0.0007& 0.0019& 1.9875\end{array}\right].$$

(48)

We show the effectiveness of our results in Example 4.4. Figure 7 illustrates the state responses of $x(t)$ for the four-tank system (47) without $u(t)$. Figure 8 shows the state responses for $x(t)$ of the four-tank system (47) with $u(t)$. Moreover, we present the control inputs in Fig. 9. The proposed controller internally stabilizes the four-tank system (47) with external disturbance. Thus, our results are of a consistently high effectiveness nonfragile feedback control scheme while maintaining state-response stability.

Figure 7

State responses of $x(t)$ against $u(t)=0$ of the GNNs (47) in Example 4.4

Figure 8

State responses of $x(t)$ for the GNNs with $u(t)$ (47) in Example 4.4

Figure 9

Control inputs for the GNNs (47) in Example 4.4

Remark 9

We select the time-delay functions $h(t)$ that are continuous functions and satisfy the condition $h_{m}\leq h(t) \leq h_{M}$. In Examples 4.1, 4.2, and 4.4, we use $h(t)=1.25|\sin (t)|+1.0$, $h(t)=1.25|\sin (t)|+1.5$, and $h(t)=1.0 +0.8\cos (t)$ for $t\in [0,50]$, respectively (see Fig. 10). From Fig. 10, our time-delay functions $h(t)$ are continuous functions and satisfy $h_{m}\leq h(t) \leq h_{M}$. However, our delay functions do not need to be differentiable. In contrast to other previous studies [9, 10, 19], the time-delay function is always differentiable.

Figure 10

Continuous time-varying delay functions for $t\in [0,50]$

5 Conclusions

The conservatism of the finite-time stability criterion in Lyapunov theory is an important topic. Developing integral inequalities leads to reduced conservatism. Hence, this article proposes the new integral inequality with an exponential function to estimate the derivative of the LKFs. The well-known Wirtinger’s inequality is a particular case of the new integral inequality. Furthermore, we investigate the new delay dependence for guaranteeing finite-time stability of the GNNs with mixed-interval time-varying delays that do not need to be differentiable and design the state feedback controller with a nonfragile issue. Numerical examples show the MALBs of $c_{2}$ obtained by several inequalities, including our new inequality, Wirtinger’s inequality [38], and the inequality with an exponential function in [39]. Our new inequality efficiently reduces conservatism more than using Wirtinger’s inequality [38] and the inequality in [39]. Moreover, one of the examples presents a practical implementation that applies our results on the four-tank system. For future work, this article can be applied to various dynamical systems such as T–S fuzzy NNs [26], neutral-type NNs [46], uncertain NNs [47], and neutral high-order Hopfield NNs [48] or several time delays, such as additive time delay [49–51] and leakage time-delay [13, 46, 48]. Additionally, future work could potentially design a sample-data nonfragile controller [30] for the delayed dynamical systems.