Robust stability of interactional genetic regulatory networks with reaction-diffusion terms

Abstract

We have analyzed the stability of interactional genetic regulatory networks with reaction-diffusion terms under Dirichlet boundary conditions in this article. Corresponding to interaction between unstable genetic regulatory networks and stable genetic regulatory networks, the model is given, and a stability criterion is proposed through construction of appropriate Lyapunov-Krasovskii functions and linear matrix inequalities (LMI). By means of a numerical simulation, we have proved the effectiveness and correctness of the theorem, and we analyzed the factors that influence the stability for interactional genetic regulatory networks.

1 Introduction

Since 2002, the Alon research group has proposed the network module [1–3], network modules with several nodes become a hot research topic. Modeling and dynamic analysis of these genetic regulatory networks (GRNs), which are considered as important sub-module of complex biology network, because a GRN can clarify the mechanism of biological network. At present, the models of genetic regulatory network include directed graphs [4–6], the Boolean model [7–11], the Bayesian model [12–16] and the differential equations model [17–27]. In these models, the differential equations model has obvious advantages, for example, the differential equations model is more accurate than the Boolean model in describing GRNs, and it has less computational complexity than the Bayesian model. The differential equations model is an open model, because a great deal about dynamic systems can be directly applied to this model, which has attracted a large number of other field experts to join the relevant research, and reaped rich fruits. A real biological network contains tens of thousands of nodes, while the simulations of these studies are based on a few nodes as the research object, the theoretical basis of its simplification is that the number of molecules involved in the chemical reaction is usually very low at a given moment.

Although there are many outstanding achievements in the study of differential equations as a model of the GRN, there are still some problems that needed further research.

On the one hand, the spatial diffusion phenomenon exists widely in the fields of physics, chemistry, biology, and so on, but most of the current studies on GRNs in terms of the spatial homogeneity of concentrations are for cell components. This proposition can lead to missing a lot of space information, but there are only five articles [19, 20, 23, 26, 27] about the study of GRNs with reaction-diffusion terms, and the five articles have only discussed the influence of reaction-diffusion terms on the time-delay conservatism,but they did not explore the essence of reaction-diffusion terms.

On the other hand, corresponding to the different biological signals, the change of the expression of a protein will affect the expression of gene, which makes the network achieve stability ultimately. Lest life activity is only the result of a local GRN, it also has extensive connections with the surrounding GRN, and the dynamical properties of these GRNs also directly affect the survival of the living body. For example, a virus cannot survive independently, and its replication and transmission must be completed by the synthesis system, the replication system and the protein transport system of the host cell. A large number of studies showed that, after the host is infected by the virus, when the virus is active, enzymes and mRNA concentrations for viral replication will be unstable; when the virus is repressed or dormant, the virus replication related enzymes and mRNA concentrations will tend to stability [28–31]. In recent years, it was found that there is a wide and complex relation between the virus and the host at the molecular level based on the proteins atlas of interaction between viral and host [32–35].

In this article we have proposed a model of interactional genetic regulatory networks (GRNs), and we analyzed the stability of interactional GRNs, the theoretical support for the above method is given from the aspect of dynamics. By numerical simulation, three significant conclusions are obtained.

2 Problem formulation

Two different nonlinear delayed GRNs are described by equations (1) and (2), respectively:

$$\begin{aligned}& \textstyle\begin{cases} \frac{\mathrm{d} u_{1i} ( t)}{\mathrm{d} t} =- a_{1i} u_{1i} ( t) + \sum_{j=1}^{n} \omega_{ij} f_{j} ( v_{1j} ( t-\sigma ( t))), \\ \frac{\mathrm{d} v_{1i} ( t)}{\mathrm{d} t} =- c_{1i} v_{1i} ( t) + b_{1i} u_{1i} ( t-\tau ( t)),\quad i=1,2,\ldots, n_{1}, \end{cases}\displaystyle \end{aligned}$$

(1)

$$\begin{aligned}& \textstyle\begin{cases} \frac{\mathrm{d} u_{2 p} ( t)}{\mathrm{d} t} =- a_{2 p} u_{2 p} ( t) + \sum_{q =1}^{n} \varpi_{pq} g_{q} ( v_{2 q} ( t - \sigma ' ( t))), \\ \frac{\mathrm{d} v_{2 p} ( t)}{\mathrm{d} t} =- c_{2 p} v_{2 p} ( t) + b_{2 p} u_{2 p} ( t - \tau ' ( t)),\quad p =1,2,\ldots, n_{2}, \end{cases}\displaystyle \end{aligned}$$

(2)

where GRN (1) is stable, and GRN (2) is unstable, $u_{1i} ( t), v_{1i} ( t) \in R^{n_{1}}$ and $u_{2p} (t), v_{2p} ( t) \in R^{n_{2}}$ are the concentrations of mRNA and protein of the ith and the pth nodes at the time t respectively; the parameters $a_{1i}$ and $a_{2p}$ are the degradation rates of the mRNA, $c_{1i}$ and $c_{2p}$ are the degradation rates of the protein, $b_{1i}$ and $b_{2p}$ are the translation rate; $f_{i} (x)$ and $g_{p} (y)$ are the Hill form regulatory functions, which represent the feedback regulation of the protein on the transcription, their forms are described by equation (3)

$$ \textstyle\begin{cases} f_{j} ( x) = \frac{( \frac{x}{m_{j}})^{H_{j}}}{1+( \frac{x}{m_{j}})^{H_{j}}},\\ g_{q} ( y) = \frac{( \frac{y}{n_{q}})^{H_{q}}}{1+( \frac{y}{n_{q}})^{H_{q}}}, \end{cases} $$

(3)

where $H_{j}$ and $H_{q}$ are the Hill coefficients, $m_{j}$ and $n_{q}$ are positive constants, $\tau(t)$, $\tau ' (t)$, $\sigma(t)$ and $\sigma ' (t)$ are time-varying delays satisfying

$$ \textstyle\begin{cases} 0\leq \tau_{1} \leq \tau ( \mathrm{t} ) \leq \tau_{2},\quad 0\leq \sigma_{1} \leq \sigma ( \mathrm{t} ) \leq \sigma_{2}, \\ \lambda_{1} \leq \dot{\tau} \leq \lambda_{2},\qquad \lambda_{3} \leq \dot{\sigma} \leq \lambda_{4}, \\ 0\leq \tau_{1} ' \leq \tau ' ( \mathrm{t} ) \leq \tau_{2} ',\qquad 0\leq \sigma_{1} ' \leq \sigma ' ( t) \leq \sigma_{2} ',\\ \eta_{1} \leq \dot{\tau} ' \leq \eta_{2},\qquad \eta_{3} \leq \dot{\sigma} ' \leq \eta_{4}, \end{cases} $$

(4)

where $W_{1} =( \omega_{ij})\in R^{n_{1} \times n_{1}}$ and $W_{2} = ( \varpi_{pq} ) \in R^{n_{2} \times n_{2}}$ are described as equations (5) and (6), $\alpha_{ij}$ and $\beta_{pq}$ are the dimensionless transcriptional rates of transcriptional factor j to gene i and transcriptional factor q to gene p respectively.

$$\begin{aligned}& \omega_{ij} = \textstyle\begin{cases} \alpha_{ij} &\mbox{if transcription factor } j \mbox{ is an activator of gene } i,\\ 0 &\mbox{if there is no link from node } j \mbox{ to } i, \\ - \alpha_{ij} &\mbox{if transcription factor } j \mbox{ is a repressor of gene } i, \end{cases}\displaystyle \end{aligned}$$

(5)

$$\begin{aligned}& \varpi_{pq} =\textstyle\begin{cases} \beta_{pq} & \mbox{if transcription factor } q \mbox{ is an activator of gene } p,\\ 0 &\mbox{if there is no link from node } q \mbox{ to } p, \\ - \beta_{pq} &\mbox{if transcription factor } q \mbox{ is a repressor of gene } p. \end{cases}\displaystyle \end{aligned}$$

(6)

Considering the diffusion term, equations (1) and (2) can be rewrite as

$$\begin{aligned}& \textstyle\begin{cases} \frac{\partial u_{1i} ( t,l)}{\partial t} = \sum_{k=1}^{L} \frac{\partial}{\partial l_{k}} ( D_{ik} \frac{\partial u_{1i} ( t,l)}{\partial l_{k}} ) - a_{1i} u_{1i} ( t,l) + \sum_{j=1}^{n} \omega_{ij} f_{j} ( v_{1j} ( t-\sigma ( t),l)), \\ \frac{\partial v_{1i} ( t,l)}{\partial t} = \sum_{k=1}^{L} \frac{\partial}{\partial l_{k}} ( D_{ik}^{*} \frac{\partial v_{1i} ( t,l)}{\partial l_{k}} ) - c_{1i} p_{1i} ( t) + b_{1i} u_{1i} ( t-\tau ( t)),\quad i=1,2,\ldots, n_{1} \end{cases}\displaystyle \end{aligned}$$

(7)

$$\begin{aligned}& \textstyle\begin{cases} \frac{\partial u_{2p} ( t,l)}{\partial t} = \sum_{k=1}^{L} \frac{\partial}{\partial l_{k}} ( d_{pk} \frac{\partial u_{2p} ( t,l)}{\partial l_{k}} ) - a_{2p} u_{2p} ( t,l) + \sum_{q=1}^{n} \varpi_{pq} g_{q} ( v_{1q} ( t- \sigma ' ( t),l)), \\ \frac{\partial v_{2p} ( t,l)}{\partial t} = \sum_{k=1}^{L} \frac{\partial}{\partial l_{k}} ( d_{pk}^{*} \frac{\partial v_{2p} ( t,l)}{\partial l_{k}} ) - c_{2p} v_{2p} ( t,l) + b_{2p} u_{2p} ( t- \tau ' ( t),l),\\ \quad p=1,2,\ldots, n_{2} \end{cases}\displaystyle \end{aligned}$$

(8)

where $l = ( l_{1}, l_{2},\ldots, l_{L} )^{\mathrm{T}} \in \Sigma \subset R^{c}$, $\Sigma= \{ l \vert \vert l_{k} \vert \leq L_{k} \}$, $L_{k}$ is constant, $k =1,2,\ldots L$, $D_{ik} = D_{ik} ( t,l) >0$, $D_{ik}^{*} = D_{ik}^{*} ( t,l) >0$, denote the transmission diffusion operator along the ith gene of mRNA and protein, respectively, $d_{pk} = d_{pk} ( t,l) >0$, $d_{pk}^{*} = d_{pk}^{*} ( t,l) >0$, denote the transmission diffusion operator along the pth gene of mRNA and protein, respectively.

The initial conditions are given by

$$\begin{aligned}& \textstyle\begin{cases} u_{1i} ( s,l) = \psi_{1i} ( s,l),&s\in ( -\infty,0 ],i=1,2,\ldots, n_{1}\\ v_{1i} ( s,l) = \psi_{1i}^{*} ( s,l),&s\in ( -\infty,0 ],i=1,2,\ldots, n_{1} \end{cases}\displaystyle \end{aligned}$$

(9)

$$\begin{aligned}& \textstyle\begin{cases} u_{2p} ( s,l) = \psi_{2p} ( s,l),&s\in ( -\infty,0 ],p=1,2,\ldots, n_{2}\\ v_{2p} ( s,l) = \psi_{2p}^{*} ( s,l),&s\in ( -\infty,0 ],p=1,2,\ldots, n_{2} \end{cases}\displaystyle \end{aligned}$$

(10)

Here $\psi_{1i} ( s,l)$, $\psi_{1i}^{*} ( s,l)$, $\psi_{2p} ( s,l)$ and $\psi_{2p}^{*} ( s,l)$ are bounded and continuous on $( -\infty,0 ] \times \Sigma$.

Dirichlet boundary condition is considered:

$$\begin{aligned}& \textstyle\begin{cases} u_{1i} ( t,l) =0,& l\in \partial\Sigma,t\in [ -\kappa,+\infty),\\ v_{1i} ( t,l) =0,& l\in \partial\Sigma,t\in [ -\kappa,+\infty), \end{cases}\displaystyle \end{aligned}$$

(11)

$$\begin{aligned}& \textstyle\begin{cases} u_{2p} ( t,l) =0, &l\in \partial\Sigma,t\in [ -\kappa,+\infty),\\ v_{2p} ( t,l) =0, &l\in \partial\Sigma,t\in [ -\kappa,+\infty). \end{cases}\displaystyle \end{aligned}$$

(12)

The system (7) and system (8) can be rewritten in vector-matrix form:

$$\begin{aligned}& \textstyle\begin{cases} \frac{\partial u_{1} ( t, l)}{\partial t} = \sum_{k =1}^{l} \frac{\partial}{\partial l_{k}} ( D_{k} \frac{\partial u_{1} ( t, l)}{\partial x_{k}} ) - A_{1} u_{1} ( t, l) + W_{1} F ( v_{1} ( t - \sigma ( t), l)), \\ \frac{\partial v_{1} ( t, l)}{\partial t} = \sum_{k =1}^{l} \frac{\partial}{\partial l_{k}} ( D_{k}^{*} \frac{\partial v_{1} ( t, l)}{\partial l_{k}} ) - C_{1} v_{1} ( t, l) + B_{1} u_{1} ( t - \tau ( t), l), \quad i =1,2,\ldots, n_{1} \end{cases}\displaystyle \end{aligned}$$

(13)

$$\begin{aligned}& \textstyle\begin{cases} \frac{\partial u_{2} ( t, l)}{\partial t} = \sum_{k =1}^{l} \frac{\partial}{\partial l_{k}} ( d_{k} \frac{\partial u_{2} ( t, l)}{\partial x_{k}} ) - A_{2} u_{2} ( t, l) + W_{2} G ( v_{2} ( t - \sigma ' ( t), l)), \\ \frac{\partial v_{2} ( t, l)}{\partial t} = \sum_{k =1}^{l} \frac{\partial}{\partial l_{k}} ( d_{k}^{*} \frac{\partial v_{2} ( t, l)}{\partial l_{k}} ) - C_{2} v_{2} ( t) + B_{2} u_{2} ( t - \tau ' ( t), l),\quad p =1,2,\ldots, n_{2} \end{cases}\displaystyle \\& A_{1} =\operatorname{diag}( a_{11}, a_{12},\ldots, a_{1 n_{1}}), \\& B_{1} =\operatorname{diag}( b_{11}, b_{12},\ldots, b_{1 n_{1}}), \\& C_{1} =\operatorname{diag}( c_{11}, c_{12},\ldots, c_{1 n_{1}}), \\& A_{2} = \operatorname{diag}( a_{21}, a_{22},\ldots, a_{2 n_{2}}), \\& B_{2} =\operatorname{diag}( b_{21}, b_{22},\ldots, b_{2 n_{2}}), \\& C_{2} =\operatorname{diag}( c_{21}, c_{22},\ldots, c_{2 n_{2}}), \\& D_{k} =\operatorname{diag}( D_{1 k}, D_{2 k},\ldots, D_{n_{1} k}), \\& D_{k}^{*} =\operatorname{diag}\bigl( D_{1 k}^{*}, D_{2 k}^{*},\ldots, D_{n_{1} k}^{*} \bigr), \\& d_{k} =\operatorname{diag}( d_{1 k}, d_{2 k},\ldots, d_{n_{2} k}), \\& d_{k}^{*} =\operatorname{diag}\bigl( d_{1 k}^{*}, d_{2 k}^{*},\ldots, d_{n_{2} k}^{*} \bigr), \\& u_{1} ( t, l) = \bigl( u_{11} ( t, l), u_{12} ( t, l),\ldots, u_{1 n_{1}} ( t, l) \bigr)^{\mathrm{T}}, \\& v_{1} ( t, l) = \bigl( v_{11} ( t, l), v_{12} ( t, l),\ldots, v_{1 n_{1}} ( t, l) \bigr)^{\mathrm{T}}, \\& u_{2} ( t, l) = \bigl( u_{21} ( t, l), u_{22} ( t, l),\ldots, u_{2 n_{2}} ( t, l) \bigr)^{\mathrm{T}}, \\& v_{2} ( t, l) = \bigl( v_{21} ( t, l), v_{22} ( t, l),\ldots, v_{2 n_{2}} ( t, l) \bigr)^{\mathrm{T}}, \\& F( v_{1j} \bigl(t-\sigma(t),l \bigr) =( f_{1} (v_{11} \bigl(t-\sigma(t),l \bigr), f_{2} ( v_{12} \bigl( t-\sigma ( t),l \bigr), \ldots, f_{n} \bigl(v_{1 n_{1}} \bigl( t-\sigma ( t),l \bigr) \bigr)^{\mathrm{T}}, \\& G( v_{2q} \bigl(t- \sigma ' (t),l \bigr) =( g_{1} (v_{21} \bigl(t- \sigma ' (t),l \bigr), g_{2} ( v_{22} \bigl( t- \sigma ' ( t),l \bigr),\ldots, g_{n} \bigl(v_{2 n_{2}} \bigl( t- \sigma ' ( t),l \bigr) \bigr)^{\mathrm{T}}. \end{aligned}$$

(14)

$f_{i} (\boldsymbol{\cdot})$ and $g_{p} (\boldsymbol{\cdot})$ satisfy inequality (15) and inequality (16), respectively, because $f_{i} (\boldsymbol{\cdot})$ and $g_{p} (\boldsymbol{\cdot})$ are monotonically increase functions with saturation

$$\begin{aligned}& 0\leq \frac{f_{i} ( x_{i} )}{x_{i}} \leq \varsigma_{i},\quad \forall x_{i} \neq0,i=1,2,\ldots, n_{1} \end{aligned}$$

(15)

$$\begin{aligned}& 0\leq \frac{g_{p} ( y_{p} )}{y_{p}} \leq \chi_{p},\quad \forall y_{p} \neq0,p=1,2,\ldots, n_{2} \end{aligned}$$

(16)

i.e.

$$\begin{aligned}& f^{\mathrm{T}} ( x) \bigl( f ( x) - K_{1} x \bigr) \leq0, \end{aligned}$$

(17)

$$\begin{aligned}& g^{\mathrm{T}} ( y) \bigl( g ( y) - K_{2} y \bigr) \leq0, \end{aligned}$$

(18)

where $K_{1} =\operatorname{diag}( \varsigma_{1}, \varsigma_{2},\ldots, \varsigma_{n_{1}} ) >0$, $K_{2} =\operatorname{diag}( \chi_{1}, \chi_{2},\ldots, , \chi_{n_{2}} )^{\mathrm{T}} >0$, $x = [ x_{1}, x_{2},\ldots, x_{n_{1}} ]^{\mathrm{T}}$ and $y = [ y_{1}, y_{2},\ldots, y_{n_{2}} ]^{\mathrm{T}}$.

Lemma 1

Let $f ( v)$ be a real-valued function defined on $[ a, b ]\subset R$, with $f ( a) =f ( b) =0$. If $f ( v)\in C^{1} [a,b]$, then

$$ \int_{a}^{b} f^{2} ( v) \,\mathrm{d} v \leq \frac{( b - a)^{2}}{\pi^{2}} \int_{a}^{b} \bigl[ f ' ( v) \bigr]^{2} \,\mathrm{d} v. $$

(19)

Lemma 2

If Σ is a bounded $C^{1}$ open set in $R^{n}$ and $\eta, \varphi \in C^{2} ( \overline{\Sigma})$, then

$$ \int_{\Sigma} \eta \Delta \varphi \,\mathrm{d} x = \int_{\Sigma} \eta \Delta \varphi \,\mathrm{d} x + \int_{\partial \Sigma} \biggl( \eta \frac{\partial\varphi}{\partial \overline{n}} - \varphi \frac{\partial\eta}{\partial \overline{n}} \biggr) \,\mathrm{d} S, $$

(20)

where $\frac{\partial\eta}{\partial \overline{n}}$ and $\frac{\partial\varphi}{\partial \overline{n}}$ are the directional derivatives of η and φ in the direction of the outward pointing normal n̅ to the surface element dS, respectively. $\sum_{k=1}^{l} \frac{\partial}{\partial x_{k}} ( D_{k} \frac{\partial}{\partial x_{k}})$ can be regarded as a Laplacian operator which is formally self-adjoint and a differential; in Lemma 2 we have an inner product for a function with Dirichlet boundary.

Lemma 3

[20]

From the Green formula, under Dirichlet boundary conditions, and by using Lemma 1 and Lemma 2, we can obtain

$$ 2 \int_{\Sigma} \mu^{\mathrm{T}} \sum_{k =1}^{l} \frac{\partial}{\partial x_{k}} \biggl( \frac{\partial\mu}{\partial x_{k}} \biggr) \,\mathrm{d} x \leq- \frac{\pi^{2}}{2} \int_{\Sigma} \mu^{\mathrm{T}} \mu \,\mathrm{d} x . $$

(21)

Lemma 4

Let $M >0\in R^{n\times n}$, a positive scalar $\vartheta>0$, vector function $x: [ 0,\vartheta ] \rightarrow R^{n}$ such that the integrations concerned are well defined, and they exist:

$$ \biggl( \int_{0}^{\vartheta} x ( s)\,\mathrm{d} s \biggr)^{\mathrm{T}} M \biggl( \int_{0}^{\vartheta} x ( s)\,\mathrm{d} s \biggr) \boldsymbol{\leq} \vartheta \biggl( \int_{0}^{\vartheta} x ( s) Mx ( s)\,\mathrm{d} s \biggr). $$

(22)

Lemma 5

For vectors $X, Y \in R^{n}$ are any positive definite matrix, and any scalar ε, there exists the following inequality:

$$ 2 X^{T} Y \leq \varepsilon X^{T} X + \varepsilon^{-1} Y^{T} Y. $$

(23)

Lemma 6

For any vectors $X, Y \in R^{n}$, and any scalar $\varepsilon >0$ are positive, there exists the following inequality:

$$ 2 X^{T} HY\leq \varepsilon X^{T} HX+ \varepsilon^{-1} Y^{T} HY. $$

(24)

3 Model of interactional GRNs

According to GRN (13) and (14), $W_{1}$ or $W_{2}$ express the interaction of genes in single GRN, we assumed that interaction of the different GRNs is like a single GRN. We have constructed a bidirectional coupling model for the type of interactional GRNs as equations (25) and (26), and we investigate a stability criterion under Dirichlet boundary condition.

$$\begin{aligned}& \textstyle\begin{cases} \frac{\mathrm{d} u_{1} ( t,l)}{\mathrm{d} t} = \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} ( D_{k} \frac{\partial u_{1} ( t,l)}{\partial x_{k}} ) - A_{1} u_{1} ( t,l) + W_{1} F ( v_{1} ( t-\sigma ( t),l)) \\ \hphantom{\frac{\mathrm{d} u_{1} ( t,l)}{\mathrm{d} t} =} {} + W_{1}^{*} G ( v_{2} ( t- \sigma ' ( t),l)),\\ \frac{\mathrm{d} v_{1} ( t,l)}{\mathrm{d} t} = \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} ( D_{k}^{*} \frac{\partial v_{1} ( t,l)}{\partial l_{k}} ) - C_{1} v_{1} ( t,l) + B_{1} u_{1} ( t-\tau ( t),l),\quad i=1,2,\ldots, n_{1} \end{cases}\displaystyle \end{aligned}$$

(25)

$$\begin{aligned}& \textstyle\begin{cases} \frac{\mathrm{d} u_{2} ( t,l)}{\mathrm{d} t} = \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} ( d_{k} \frac{\partial u_{2} ( t,l)}{\partial x_{k}} ) - A_{2} u_{2} ( t,l) \\ \hphantom{\frac{\mathrm{d} u_{2} ( t,l)}{\mathrm{d} t} =} {} + W_{2} G ( v_{2} ( t- \sigma ' ( t),l)) + W_{2}^{*} F ( v_{1} ( t-\sigma ( t),l)),\\ \frac{\mathrm{d} v_{2} ( t,l)}{\mathrm{d} t} = \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} ( d_{k}^{*} \frac{\partial v_{2} ( t,l)}{\partial l_{k}} ) - C_{2} v_{2} ( t,l) + b_{2p} u_{2} ( t- \tau ' ( t),l),\quad p=1,2,\ldots, n_{2} \end{cases}\displaystyle \end{aligned}$$

(26)

where $W_{1}^{*} =( \omega_{iq})\in R^{n_{1} \times n_{2}}$, $W_{2}^{*} = ( \varpi_{pj}^{*} ) \in R^{n_{2} \times n_{1}}$ are described by equations (27) and (28), $\alpha_{iq}^{*}$ and $\beta_{pj}^{*}$ are the dimensionless transcriptional rates of transcriptional factor q of GRN (26) to gene i of GRN (25) and transcriptional factor j of GRN (25) to gene p of GRN (26), respectively.

$$\begin{aligned}& \omega_{iq} = \textstyle\begin{cases} \alpha_{iq}^{*} & \mbox{if transcription factor } q \mbox{ is an activator of gene } i, \\ 0 &\mbox{if there is no link from node } q \mbox{ to } i, \\ - \alpha_{iq}^{*} &\mbox{if transcription factor } q \mbox{ is a repressor of gene } i, \end{cases}\displaystyle \end{aligned}$$

(27)

$$\begin{aligned}& \varpi_{pj}^{*} = \textstyle\begin{cases} \beta_{pj}^{*}& \mbox{if transcription factor } j \mbox{ is an activator of gene } p,\\ 0& \mbox{if there is no link from node } q \mbox{ to } p, \\ - \beta_{pj}^{*} & \mbox{if transcription factor } j \mbox{ is a repressor of gene } p. \end{cases}\displaystyle \end{aligned}$$

(28)

Theorem

For given scalars $\tau_{2}$, $\sigma_{2}$, $\tau_{2} '$, $\sigma_{2} '$, $\lambda_{1}$, $\lambda_{2}$, $\eta_{1}$ and $\eta_{2}$ satisfying equation (4), GRN (25) and GRN (26) under a Dirichlet boundary condition are robust stable if there exist matrices $P_{i}^{\mathrm{T}} = P_{i} >0$ and $\Lambda_{i}^{\mathrm{T}} =\Lambda_{i} >0$ ($i =1,\ldots,4$); $R_{i}^{\mathrm{T}} = R_{i} >0$ ($i =1,\ldots12$); $Q_{i}^{\mathrm{T}} = Q_{i} >0$ ($i =1,\ldots,6$), such that the following linear matrices inequalities (LMIs) hold:

$$\begin{array}{c}{\mathrm{\Pi }}_1=-P_1{A}_1-\frac{{\pi }^2}{2}{P}_1{D}_L+R_1+{\sigma }_2^2{Q}_1,\hfill \\ {\mathrm{\Pi }}_2=-R_1+R_2,\hfill \\ {\mathrm{\Pi }}_3=({\lambda }_2-1)R_2+(1-{\lambda }_1)R_3+B_1^{\mathrm{T}}{P}_2{B}_1,\hfill \\ {\mathrm{\Pi }}_4=-2{\mathrm{\Lambda }}_2+K{\mathrm{\Lambda }}_{2}K+({\lambda }_4-1)Q_3,\hfill \\ {\mathrm{\Pi }}_5=-2P_3{A}_2-\frac{{\pi }^2}{2}{P}_3{d}_L+\epsilon P_3{W}_2+R_7+{\sigma }_2^{\prime 2}{Q}_4,\hfill \\ {\mathrm{\Pi }}_6=-R_7+R_8,\hfill \\ {\mathrm{\Pi }}_7=({\eta }_2-1)R_8+(1-{\eta }_1)R_9+B_2^{\mathrm{T}}{P}_4{B}_2,\hfill \\ {\mathrm{\Pi }}_8=({\eta }_4-1)Q_6-2{\mathrm{\Lambda }}_4+K{\mathrm{\Lambda }}_{4}K+\frac{P_3{W}_2}{\epsilon },\hfill \\ {\mathrm{\Omega }}_1=-P_2{C}_1-\frac{{\pi }^2}{2}{P}_2{D}_L^{\ast }+P_2+R_4+{\tau }_2^2{Q}_2,\hfill \\ {\mathrm{\Omega }}_2=-R_4+R_5,\hfill \\ {\mathrm{\Omega }}_3=({\lambda }_4-1)R_5+(1-{\lambda }_3)R_6+K_1{\mathrm{\Lambda }}_2{K}_1,{\mathrm{\Omega }}_4=Q_3-2{\mathrm{\Lambda }}_1,\hfill \\ {\mathrm{\Omega }}_5=-2P_4{C}_2-\frac{{\pi }^2}{2}{P}_4{d}_L^{\ast }+P_4+R_{10}+{\tau }_2^{\prime 2}{Q}_5,\hfill \\ {\mathrm{\Omega }}_6=-R_{10}+R_{11},\hfill \\ {\mathrm{\Omega }}_7=({\eta }_4-1)R_{11}+(1-{\eta }_3)R_{12}+K_2{\mathrm{\Lambda }}_4{K}_2,\hfill \\ {\mathrm{\Omega }}_8=Q_6-2{\mathrm{\Lambda }}_3,\hfill \\ {\mathrm{\Xi }}_1=\left[\begin{array}{cccccccccccc}{\mathrm{\Pi }}_1& 0& 0& 0& 0& P_1{W}_1& 0& 0& 0& 0& 0& P_1{W}_1^{\ast }\\ \ast & {\mathrm{\Pi }}_2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & -R_3& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & {\mathrm{\Pi }}_3& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & -Q_1& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & {\mathrm{\Pi }}_4& W_2^{\ast }{P}_3& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Pi }}_5& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Pi }}_6& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -R_9& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Pi }}_7& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -Q_4& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Pi }}_8\end{array}\right]\hfill \\ \phantom{{\mathrm{\Xi }}_1}<0,\hfill \end{array}$$

(29)

$$\begin{array}{c}{\mathrm{\Xi }}_2=\left[\begin{array}{cccccccccccc}{\mathrm{\Omega }}_1& 0& 0& 0& 0& K_1{\mathrm{\Lambda }}_1& 0& 0& 0& 0& 0& 0\\ \ast & {\mathrm{\Omega }}_2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & -R_6& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & {\mathrm{\Omega }}_3& 0& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & -Q_2& 0& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & {\mathrm{\Omega }}_4& 0& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Omega }}_5& 0& 0& 0& 0& K_2{\mathrm{\Lambda }}_3\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Omega }}_6& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -R_{12}& 0& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Omega }}_7& 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & -Q_5& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Omega }}_8\end{array}\right]\hfill \\ \phantom{{\mathrm{\Xi }}_2}<0.\hfill \end{array}$$

(30)

Proof

Define a Lyapunov-Krasovskii functional candidate for GRN (13) as

$$ V ( t,l) = \sum_{i=1}^{8} V_{i} (t,l) $$

(31)

where

$$\begin{aligned}& V_{1} ( t,l) = \int_{\Sigma} u_{1}^{T} (t,l) P_{1} u_{1} (t,l) \,\mathrm{d} l + \int_{\Sigma} v_{1}^{T} ( t,l) P_{2} v_{1} ( t,l) \,\mathrm{d} l, \end{aligned}$$

(32)

$$\begin{aligned}& \begin{aligned}[b] V_{2} ( t,l) ={}& \int_{\Sigma} \int_{t- \tau_{1}}^{t} u_{1}^{T} ( s,l) R_{1} u_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l+ \int_{\Sigma} \int_{t-\tau(t)}^{t-\tau_{1}} u_{1}^{T} ( s,l) R_{2} u_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\ &{}+ \int_{\Sigma} \int_{t- \tau_{2}}^{t-\tau(t)} u_{1}^{T} ( s,l) R_{3} u_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\ &{}+ \int_{\Sigma} \int_{t- \sigma_{1}}^{t} v_{1}^{T} ( s,l) R_{4} v_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l+ \int_{\Sigma} \int_{t-\sigma(t)}^{t- \sigma_{1}} v_{1}^{T} ( s,l) R_{5} v_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\ &{}+ \int_{\Sigma} \int_{t- \sigma_{2}}^{t-\sigma(t)} v_{1}^{T} ( s,l) R_{6} v_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l, \end{aligned} \end{aligned}$$

(33)

$$\begin{aligned}& \begin{aligned}[b] V_{3} ( t,l) ={}&\tau \int_{\Sigma} \int_{-\tau(t)}^{0} \int_{t+s}^{t} u_{1}^{T} ( s,l) Q_{1} u_{1} ( s,l) \,\mathrm{d} \theta \,\mathrm{d} s \, \mathrm{d} l \\ &{}+ \sigma \int_{\Sigma} \int_{-\sigma(t)}^{0} \int_{t+s}^{t} v_{1}^{T} ( s,l) Q_{2} v_{1} ( s,l) \,\mathrm{d} \theta \,\mathrm{d} s \, \mathrm{d} l, \end{aligned} \end{aligned}$$

(34)

$$\begin{aligned}& V_{4} ( t,m,p) = \int_{\Sigma} \int_{t-\sigma(t)}^{t} F^{T} \bigl( v_{1} ( s,l) \bigr) Q_{3} F \bigl( v_{1} ( s,l) \bigr) \,\mathrm{d} s \,\mathrm{d} l, \end{aligned}$$

(35)

$$\begin{aligned}& V_{5} ( t,l) = \int_{\Sigma} u_{2}^{T} (t,l) P_{3} u_{1} (t,l) \,\mathrm{d} l + \int_{\Sigma} v_{2}^{T} ( t,l) P_{4} v_{2} ( t,l) \,\mathrm{d} l, \end{aligned}$$

(36)

$$\begin{aligned}& \begin{aligned}[b] V_{6} ( t,l) ={}& \int_{\Sigma} \int_{t- \tau_{1} '}^{t} u_{2}^{T} ( s,l) R_{7} u_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l+ \int_{\Sigma} \int_{t- \tau ' (t)}^{t- \tau_{1} '} u_{2}^{T} ( s,l) R_{8} u_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\ &{}+ \int_{\Sigma} \int_{t- \tau_{2} '}^{t- \tau ' (t)} u_{2}^{T} ( s,l) R_{9} u_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\ &{}+ \int_{\Sigma} \int_{t- \sigma_{1} '}^{t} v_{2}^{T} ( s,l) R_{10} v_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l+ \int_{\Sigma} \int_{t- \sigma ' (t)}^{t- \sigma_{1} '} v_{2}^{T} ( s,l) R_{11} v_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\ &{} + \int_{\Sigma} \int_{t- \sigma_{2} '}^{t- \sigma '} v_{2}^{T} ( s,l) R_{12} v_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \end{aligned} \end{aligned}$$

(37)

$$\begin{aligned}& \begin{aligned}[b] V_{7} ( t,l) ={}& \tau ' \int_{\Sigma} \int_{- \tau ' (t)}^{0} \int_{t+s}^{t} u_{2}^{T} ( s,l) Q_{4} u_{2} ( s,l) \,\mathrm{d} \theta \,\mathrm{d} s \, \mathrm{d} l \\ &{} + \sigma ' \int_{\Sigma} \int_{- \sigma ' (t)}^{0} \int_{t+s}^{t} v_{2}^{T} ( s,l) Q_{5} v_{2} ( s,l) \,\mathrm{d} \theta \,\mathrm{d} s \, \mathrm{d} l, \end{aligned} \end{aligned}$$

(38)

$$\begin{aligned}& V_{8} ( t,l) = \int_{\Sigma} \int_{t- \sigma ' (t)}^{t} G^{\mathrm{T}} \bigl( v_{2} ( s,l) \bigr) Q_{6} G \bigl( v_{2} ( s,l) \bigr) \,\mathrm{d} s \,\mathrm{d} l, \end{aligned}$$

(39)

then, computing the derivatives of $V_{i} ( t,m,p)$ ($i=1,2,3$), we can get

$$\begin{aligned}& \frac{\partial V_{1} (t,l)}{\partial t} =2 \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) P_{1} \frac{\partial u_{1} (t,l)}{ \partial t} \,\mathrm{d} l+2 \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) P_{2} \frac{\partial v_{1} (t,l)}{\partial t} \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{1} (t,l)}{\partial t}} =2 \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) P_{1} \Biggl[ \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} \biggl( D_{k} \frac{\partial u_{1} (t,l)}{\partial l_{k}} \biggr) - A_{1} u_{1} (t,l) W_{1} F \bigl( v_{1} \bigl( t-\sigma ( t),l \bigr) \bigr) \\& \hphantom{\frac{\partial V_{1} (t,l)}{\partial t}=} {} + W_{1}^{*} G \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr) \Biggr] \\& \hphantom{\frac{\partial V_{1} (t,l)}{\partial t}=} {}+2 \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) P_{2} \Biggl[ \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} \biggl( D_{k}^{*} \frac{\partial u_{1} (t,l)}{\partial l_{k}} \biggr) \\& \hphantom{\frac{\partial V_{1} (t,l)}{\partial t}=} {} - C_{1} v_{1} (t,l)+ B_{1} u_{1} \bigl( t-\tau ( t),l \bigr) \Biggr], \end{aligned}$$

(40)

$$\begin{aligned}& \frac{\partial V_{2} (t,l)}{\partial t} = \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) R_{1} u_{1} (t,l) \,\mathrm{d} l- \int_{\Sigma} \bigl( u_{1}^{\mathrm{T}} ( t- \tau_{1},l) R_{1} u_{1} ( t- \tau_{1},l) \,\mathrm{d} l \bigr) \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{}+ \int_{\Sigma} u_{1}^{\mathrm{T}} ( t- \tau_{1},l) R_{2} u_{1} (t- \tau_{1},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{}- \bigl( 1- \dot{\tau} (t) \bigr) \int_{\Sigma} u_{1}^{\mathrm{T}} \bigl( t-\tau(t),l \bigr) R_{2} u_{1} \bigl( t-\tau(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + \bigl( 1- \dot{\tau} (t) \bigr) \int_{\Sigma} u_{1}^{\mathrm{T}} \bigl( t-\tau(t),l \bigr) R_{3} u_{1} \bigl( t-\tau(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{}- \int_{\Sigma} u_{1}^{\mathrm{T}} ( t- \tau_{2},l) R_{3} u_{1} (t- \tau_{2},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) R_{4} v_{1} (t,l) \,\mathrm{d} l - \int_{\Sigma} v_{1}^{\mathrm{T}} ( t- \sigma_{1},l) R_{4} v_{1} ( t- \sigma_{1},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + \int_{\Sigma} v_{1}^{\mathrm{T}} ( t- \sigma_{1},l) R_{5} v_{1} (t- \sigma_{1},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{}- \bigl( 1- \dot{\sigma} (t) \bigr) \int_{\Sigma} v_{1}^{\mathrm{T}} \bigl( t-\sigma(t),l \bigr) R_{5} v_{1} \bigl( t-\sigma(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + \bigl( 1- \dot{\sigma} (t) \bigr) \int_{\Sigma} v_{1}^{\mathrm{T}} \bigl( t-\sigma(t),l \bigr) R_{6} v_{1} \bigl( t-\sigma(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} - \int_{\Sigma} v_{1}^{\mathrm{T}} ( t- \sigma_{2},l) R_{6} v_{1} (t- \sigma_{2},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t}} \leq \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) R_{1} u_{1} (t,l) \,\mathrm{d} l- \int_{\Sigma} u_{1}^{\mathrm{T}} ( t- \tau_{1},l) R_{1} u_{1} ( t- \tau_{1},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + \int_{\Sigma} u_{1}^{\mathrm{T}} ( t- \tau_{1},l) R_{2} u_{1} (t- \tau_{1},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{}- ( 1- \lambda_{2}) \int_{\Sigma} u_{1}^{\mathrm{T}} \bigl( t-\tau(t),l \bigr) R_{2} u_{1} \bigl( t-\tau(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + ( 1- \lambda_{1}) \int_{\Sigma} u_{1}^{\mathrm{T}} \bigl( t-\tau(t),l \bigr) R_{3} u_{1} \bigl( t-\tau(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} - \int_{\Sigma} u_{1}^{\mathrm{T}} ( t- \tau_{2},l) R_{3} u_{1} (t- \tau_{2},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) R_{4} v_{1} (t,l) \,\mathrm{d} l - \int_{\Sigma} v_{1}^{\mathrm{T}} ( t- \sigma_{1},l) R_{4} v_{1} ( t- \sigma_{1},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + \int_{\Sigma} v_{1}^{\mathrm{T}} ( t- \sigma_{1},l) R_{5} v_{1} (t- \sigma_{1},l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{}- ( 1- \lambda_{4}) \int_{\Sigma} v_{1}^{\mathrm{T}} \bigl( t-\sigma(t),l \bigr) R_{5} v_{1} \bigl( t-\sigma(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} + ( 1- \lambda_{3}) \int_{\Sigma} v_{1}^{\mathrm{T}} \bigl( t-\sigma(t),l \bigr) R_{6} v_{1} \bigl( t-\sigma(t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{2} (t,l)}{\partial t} =}{} - \int_{\Sigma} v_{1}^{\mathrm{T}} ( t- \sigma_{2},l) R_{6} v_{1} ( t- \sigma_{2},l) \,\mathrm{d} l, \end{aligned}$$

(41)

$$\begin{aligned}& \frac{\partial V_{3} (t,l)}{\partial t} = \tau^{2} (t) \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) Q_{1} u_{1} (t,l) \,\mathrm{d} l-\tau(t) \int_{\Sigma} \int_{t-\tau(t)}^{t} u_{1}^{\mathrm{T}} ( s,l) Q_{1} u_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{3} (t,l)}{\partial t} =}{} + \sigma^{2} (t) \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) Q_{2} v_{1} (t,l) \,\mathrm{d} l-\sigma(t) \int_{\Sigma} \int_{t-\sigma(t)}^{t} v_{1}^{\mathrm{T}} ( s,l) Q_{2} v_{1} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{3} (t,l)}{\partial t} }{} \leq \tau_{2}^{2} \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) Q_{1} u_{1} (t,l) \,\mathrm{d} l- \int_{\Sigma} \biggl[ \int_{t-\tau(t)}^{t} u_{1} ( s,l) \,\mathrm{d} s \biggr]^{\mathrm{T}} Q_{1} \biggl[ \int_{t-\tau(t)}^{t} u_{1} ( s,l) \,\mathrm{d} s \biggr] \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{3} (t,l)}{\partial t} =}{} + \sigma_{2}^{2} \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) Q_{2} v_{1} (t,l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{3} (t,l)}{\partial t} =}{}- \int_{\Sigma} \biggl[ \int_{t-\sigma(t)}^{t} v_{1} ( s,l) \,\mathrm{d} s \biggr]^{\mathrm{T}} Q_{2} \biggl[ \int_{t-\sigma(t)}^{t} v_{1} ( s,l) \,\mathrm{d} s \biggr] \,\mathrm{d} l, \end{aligned}$$

(42)

$$\begin{aligned}& \frac{\partial V_{4} (t,l)}{\partial t} = \int_{\Sigma} F^{\mathrm{T}} \bigl( v_{1} (t,l) \bigr) Q_{3} F \bigl( v_{1} (t,l) \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{4} (t,l)}{\partial t} =}{}- \bigl( 1- \dot{\sigma} ( t) \bigr) \int_{\Sigma} F^{\mathrm{T}} \bigl( v_{1} \bigl( t- \sigma(t),l \bigr) \bigr) Q_{3} F \bigl( v_{1} \bigl( t- \sigma(t),l \bigr) \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{4} (t,l)}{\partial t} } \leq \int_{\Sigma} F^{\mathrm{T}} \bigl( v_{1} (t,l) \bigr) Q_{3} F \bigl( v_{1} (t,l) \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{4} (t,l)}{\partial t} =}{}+ ( \lambda_{2} -1) \int_{\Sigma} F^{\mathrm{T}} \bigl( v_{1} \bigl( t- \tau(t),l \bigr) \bigr) Q_{3} F \bigl( v_{1} \bigl( t- \tau(t),l \bigr) \bigr) \,\mathrm{d} l, \end{aligned}$$

(43)

$$\begin{aligned}& \frac{\partial V_{5} (t,l)}{\partial t} =2 \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) P_{3} \frac{\partial u_{2} (t,l)}{ \partial t} \,\mathrm{d} l+2 \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) P_{4} \frac{\partial v_{2} (t,l)}{\partial t} \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{5} (t,l)}{\partial t}}=2 \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) P_{3} \Biggl[ \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} \biggl( D_{k} \frac{\partial u_{2} (t,l)}{\partial l_{k}} \biggr) - A_{2} u_{2} ( t,l) \\& \hphantom{\frac{\partial V_{5} (t,l)}{\partial t} =}{} + W_{2} G \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr) +W_{2}^{*} F \bigl( v_{1} \bigl( t-\sigma ( t),l \bigr) \bigr) \Biggr] \\& \hphantom{\frac{\partial V_{5} (t,l)}{\partial t} =}{} +2 \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) P_{4} \Biggl[ \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} \biggl( D_{k}^{*} \frac{\partial u_{2} (t,l)}{\partial l_{k}} \biggr) \\& \hphantom{\frac{\partial V_{5} (t,l)}{\partial t} =}{} - C_{2} v_{2} (t,l)+ B_{2} u_{2} \bigl( t- \tau ' ( t),l \bigr) \Biggr], \end{aligned}$$

(44)

$$\begin{aligned}& \frac{\partial V_{6} (t,l)}{\partial t} = \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) R_{7} u_{2} (t,l) \,\mathrm{d} l- \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau_{1} ',l \bigr) R_{7} u_{2} \bigl( t- \tau_{1} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{}+ \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau_{1} ',l \bigr) R_{8} u_{2} \bigl( t- \tau_{1} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \bigl( 1- \dot{\tau} ' (t) \bigr) \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau ' (t),l \bigr) R_{8} u_{2} \bigl( t- \tau ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} + \bigl( 1- \dot{\tau} ' (t) \bigr) \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau ' (t),l \bigr) R_{9} u_{2} \bigl( t- \tau ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau_{2} ',l \bigr) R_{9} u_{2} \bigl(t- \tau_{2} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} + \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) R_{10} v_{2} (t,l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau_{2} ',l \bigr) R_{9} u_{2} \bigl(t- \tau_{2} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} + \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) R_{10} v_{2} (t,l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma_{1} ',l \bigr) R_{10} v_{2} \bigl( t- \sigma_{1} ',l \bigr) \,\mathrm{d} l + \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma_{1} ',l \bigr) R_{11} v_{2} \bigl(t- \sigma_{1} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \bigl( 1- \dot{\sigma} ' (t) \bigr) \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma ' (t),l \bigr) R_{11} v_{2} \bigl( t- \sigma ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} + \bigl( 1- \dot{\sigma} ' (t) \bigr) \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma ' (t),l \bigr) R_{12} v_{2} \bigl( t- \sigma ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma_{2} ',l \bigr) R_{12} v_{2} \bigl(t- \sigma_{2} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} } \leq \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) R_{7} u_{2} (t,l) \,\mathrm{d} l- \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau_{1} ',l \bigr) R_{7} u_{2} \bigl( t- \tau_{1} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{}+ \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau_{1} ',l \bigr) R_{8} u_{2} \bigl( t- \tau_{1} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - ( 1- \eta_{2}) \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau ' (t),l \bigr) R_{8} u_{2} \bigl( t- \tau ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} + ( 1- \eta_{1}) \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau ' (t),l \bigr) R_{9} u_{2} \bigl( t- \tau ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \int_{\Sigma} u_{2}^{\mathrm{T}} \bigl( t- \tau_{2} ',l \bigr) R_{9} u_{2} \bigl(t- \tau_{2} ',l \bigr) \,\mathrm{d} l + \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) R_{10} v_{2} (t,l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma_{1} ',l \bigr) R_{10} v_{2} \bigl( t- \sigma_{1} ',l \bigr) \,\mathrm{d} l + \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma_{1} ',l \bigr) R_{11} v_{2} \bigl(t- \sigma_{1} ',l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - ( 1- \eta_{4}) \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma ' (t),l \bigr) R_{11} v_{2} \bigl( t- \sigma ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} + ( 1- \eta_{3}) \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma ' (t),l \bigr) R_{12} v_{2} \bigl( t- \sigma ' (t),l \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{6} (t,l)}{\partial t} =}{} - \int_{\Sigma} v_{2}^{\mathrm{T}} \bigl( t- \sigma_{2} ',l \bigr) R_{12} v_{2} \bigl( t- \sigma_{2} ',l \bigr) \,\mathrm{d} l, \end{aligned}$$

(45)

$$\begin{aligned}& \frac{\partial V_{7} (t,l)}{\partial t} = \tau^{\prime\,2} (t) \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) Q_{4} u_{2} (t,l) \,\mathrm{d} l- \tau ' (t) \int_{\Sigma} \int_{t- \tau '}^{t} u_{2}^{\mathrm{T}} ( s,l) Q_{4} u_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{7} (t,l)}{\partial t} =}{} + \sigma^{\prime\,2} (t) \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) Q_{5} v_{2} (t,l) \,\mathrm{d} l- \sigma ' (t) \int_{\Sigma} \int_{t- \sigma '}^{t} v_{2}^{\mathrm{T}} ( s,l) Q_{5} v_{2} ( s,l) \,\mathrm{d} s \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{7} (t,l)}{\partial t} } \leq \tau_{2}^{\prime\,2} \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) Q_{4} u_{2} (t,l) \,\mathrm{d} l- \int_{\Sigma} \biggl[ \int_{t- \tau '}^{t} u_{2} ( s,l) \,\mathrm{d} s \biggr]^{\mathrm{T}} Q_{4} \biggl[ \int_{t- \tau '}^{t} u_{2} ( s,l) \,\mathrm{d} s \biggr] \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{7} (t,l)}{\partial t} =}{} + \sigma_{2}^{\prime\,2} \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) Q_{5} v_{2} (t,l) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{7} (t,l)}{\partial t} =}{}- \int_{\Sigma} \biggl[ \int_{t- \sigma '}^{t} v_{2} ( s,l) \,\mathrm{d} s \biggr]^{\mathrm{T}} Q_{5} \biggl[ \int_{t- \sigma '}^{t} v_{2} ( s,l) \,\mathrm{d} s \biggr] \,\mathrm{d} l, \end{aligned}$$

(46)

$$\begin{aligned}& \frac{\partial V_{8} (t,l)}{\partial t} = \int_{\Sigma} G^{\mathrm{T}} \bigl( v_{2} (t,l) \bigr) Q_{6} F \bigl( v_{2} (t,l) \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{8} (t,l)}{\partial t} =}{}- \bigl(1- \dot{\sigma} ' \bigr) \int_{\Sigma} G^{\mathrm{T}} \bigl( v_{2} \bigl( t- \sigma ',l \bigr) \bigr) Q_{6} G \bigl( v_{2} \bigl( t- \sigma ',l \bigr) \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{8} (t,l)}{\partial t}} \leq \int_{\Sigma} G^{\mathrm{T}} \bigl( v_{2} (t,l) \bigr) Q_{6} G \bigl( v_{2} (t,l) \bigr) \,\mathrm{d} l \\& \hphantom{\frac{\partial V_{8} (t,l)}{\partial t} =}{}+ ( \eta_{2} -1) \int_{\Sigma} G^{\mathrm{T}} \bigl( v_{2} ( t- \sigma,l) \bigr) Q_{6} G \bigl( v_{2} \bigl( t- \sigma ',l \bigr) \bigr) \,\mathrm{d} l. \end{aligned}$$

(47)

According to Lemma 3, we have

$$\begin{aligned}& 2 \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) P_{1} \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} \biggl( D_{k} \frac{\partial u_{1} (t,l)}{ \partial l_{k}} \biggr) \, \mathrm{d} l \leq- \frac{\pi^{2}}{2} \int_{\Sigma} u_{1}^{\mathrm{T}} ( t,l) P_{1} D_{L} u_{1} ( t,l) \,\mathrm{d} l, \end{aligned}$$

(48)

$$\begin{aligned}& 2 \sum_{k=1}^{l} \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) P_{2} \frac{\partial}{\partial l_{k}} \biggl( D_{k}^{*} \frac{\partial v_{1} (t,l)}{\partial l_{k}} \biggr) \,\mathrm{d} l \leq- \frac{\pi^{2}}{2} \int_{\Sigma} v_{1}^{\mathrm{T}} ( t,l) P_{2} D_{L}^{*} v_{1} ( t,l) \, \mathrm{d} l, \end{aligned}$$

(49)

$$\begin{aligned}& 2 \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) P_{3} \sum_{k=1}^{l} \frac{\partial}{\partial l_{k}} \biggl( d_{k} \frac{\partial u_{2} (t,l)}{ \partial l_{k}} \biggr) \, \mathrm{d} l \leq- \frac{\pi^{2}}{2} \int_{\Sigma} u_{2}^{\mathrm{T}} ( t,l) P_{3} d_{L} u_{2} ( t,l) \,\mathrm{d} l, \end{aligned}$$

(50)

$$\begin{aligned}& 2 \sum_{k=1}^{l} \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) P_{4} \frac{\partial}{\partial l_{k}} \biggl( d_{k}^{*} \frac{\partial v_{2} (t,l)}{\partial l_{k}} \biggr) \,\mathrm{d} l \leq- \frac{\pi^{2}}{2} \int_{\Sigma} v_{2}^{\mathrm{T}} ( t,l) P_{4} d_{L}^{*} v_{2} ( t,l) \, \mathrm{d} l, \end{aligned}$$

(51)

where

$$\begin{aligned}& \biggl( u_{1}^{\mathrm{T}} (t,l) P_{1} D_{k} \frac{\partial u_{1} ( t,l)}{\partial l_{k}} \biggr)_{k=1}^{L} = \biggl( u_{1}^{\mathrm{T}} (t,l) P_{1} D_{1} \frac{\partial u_{1}^{\mathrm{T}} (t,l)}{\partial l_{1}},\ldots, u_{1}^{\mathrm{T}} (t,l) P_{1} D_{L} \frac{\partial u_{1}^{\mathrm{T}} (t,l)}{\partial l_{L}} \biggr), \\& \biggl( v_{1}^{\mathrm{T}} (t,l) P_{2} D_{k}^{*} \frac{\partial v_{1} ( t,l)}{\partial l_{k}} \biggr)_{k=1}^{L} = \biggl( v_{1}^{\mathrm{T}} (t,l) P_{2} D_{k}^{*} \frac{\partial v_{1} ( t,l)}{\partial l_{1}},\ldots, v_{1}^{\mathrm{T}} (t,l) P_{2} D_{L}^{*} \frac{\partial v_{1} ( t,l)}{\partial l_{L}} \biggr), \\& \biggl( u_{2}^{\mathrm{T}} (t,l) P_{3} d_{k} \frac{\partial u_{2} ( t,l)}{\partial l_{k}} \biggr)_{k=1}^{L} = \biggl( u_{2}^{\mathrm{T}} (t,l) P_{3} d_{1} \frac{\partial u_{2}^{\mathrm{T}} (t,l)}{\partial l_{1}},\ldots, u_{2}^{\mathrm{T}} (t,l) P_{3} d_{L} \frac{\partial u_{2}^{\mathrm{T}} (t,l)}{\partial l_{L}} \biggr), \\& \biggl( v_{2}^{\mathrm{T}} (t,l) P_{4} d_{k}^{*} \frac{\partial v_{2} ( t,l)}{\partial l_{k}} \biggr)_{k=1}^{L} = \biggl( v_{2}^{\mathrm{T}} (t,l) P_{4} d_{k}^{*} \frac{\partial v_{2} ( t,l)}{\partial l_{1}},\ldots, v_{2}^{\mathrm{T}} (t,l) P_{4} d_{L}^{*} \frac{\partial v_{2} ( t,l)}{\partial l_{L}} \biggr). \end{aligned}$$

Considering inequalities (17) and (18), for diagonal matrices $\Lambda_{1} >0$, $\Lambda_{2} >0$, $\Lambda_{3} >0$ and $\Lambda_{4} >0$, the following inequalities hold:

$$\begin{aligned}& 2 F^{\mathrm{T}} \bigl( v_{1} (t,l) \bigr) \Lambda_{1} F \bigl( v_{1} (t,l) \bigr) -2 v_{1}^{\mathrm{T}} ( t,l) K_{1} \Lambda_{1} F \bigl( v_{1} ( t,l) \bigr) \leq0, \end{aligned}$$

(52)

$$\begin{aligned}& 2 F^{\mathrm{T}} \bigl( v_{1} \bigl(t-\sigma(t),l \bigr) \bigr) \Lambda_{2} F \bigl( v_{1} \bigl(t-\sigma(t),l \bigr) \bigr) \\& \quad {} -2 v_{1}^{\mathrm{T}} \bigl( t-\sigma ( t),l \bigr) K_{1} \Lambda_{2} F \bigl( v_{1} \bigl( t-\sigma ( t),l \bigr) \bigr) \leq0, \end{aligned}$$

(53)

$$\begin{aligned}& 2 G^{\mathrm{T}} \bigl( v_{2} (t,l) \bigr) \Lambda_{3} G \bigl( v_{2} (t,l) \bigr) -2 v_{2}^{\mathrm{T}} ( t,l) K_{2} \Lambda_{3} G \bigl( v_{2} ( t,l) \bigr) \leq0, \end{aligned}$$

(54)

$$\begin{aligned}& 2 G^{\mathrm{T}} \bigl( v_{2} \bigl(t- \sigma ' (t),l \bigr) \bigr) \Lambda_{4} G \bigl( v_{2} \bigl(t- \sigma ' (t),l \bigr) \bigr) \\& \quad {} -2 v_{2}^{\mathrm{T}} \bigl( t- \sigma ' ( t),l \bigr) K_{2} \Lambda_{4} G \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr) \leq0. \end{aligned}$$

(55)

According to Lemma 5, we have inequalities as follows:

$$\begin{aligned}& 2 v_{1}^{\mathrm{T}} \bigl( t-\sigma ( t),l \bigr) K_{1} \Lambda_{2} F \bigl( v_{1} \bigl( t-\sigma ( t),l \bigr) \bigr) \\& \quad \leq v_{1}^{\mathrm{T}} \bigl( t-\sigma ( t),l \bigr) K_{1} \Lambda_{2} K_{1} v_{1} \bigl( t-\sigma ( t),l \bigr) \\& \qquad {} + F^{\mathrm{T}} \bigl( v_{1} \bigl( t-\sigma ( t),l \bigr) \bigr) \Lambda_{2} F \bigl( v_{1} \bigl( t-\sigma ( t),l \bigr) \bigr), \end{aligned}$$

(56)

$$\begin{aligned}& 2 v_{2}^{\mathrm{T}} \bigl( t- \sigma ' ( t),l \bigr) K_{2} \Lambda_{4} G \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr) \\& \quad \leq v_{2}^{\mathrm{T}} \bigl( t- \sigma ' ( t),l \bigr) K_{2} \Lambda_{4} K_{2} v_{2} \bigl( t- \sigma ' ( t),l \bigr) \\& \qquad {}+ G^{\mathrm{T}} \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr) \Lambda_{4} G \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr), \end{aligned}$$

(57)

$$\begin{aligned}& 2 v_{1}^{\mathrm{T}} ( t,l) P_{2} B_{1} u_{1} \bigl( t-\tau ( t),l \bigr) \\& \quad \leq v_{1}^{\mathrm{T}} ( t,l) P_{2} v_{1} ( t,l) + u_{1} \bigl( t- \tau ( t),l \bigr) B_{1}^{\mathrm{T}} P_{2} B_{1} u_{1} \bigl( t-\tau ( t),l \bigr), \end{aligned}$$

(58)

$$\begin{aligned}& 2 v_{2}^{\mathrm{T}} ( t,l) P_{4} B_{2} u_{2} \bigl( t- \tau ' ( t),l \bigr) \\& \quad \leq v_{2}^{\mathrm{T}} ( t,l) P_{4} v_{2} ( t,l) + u_{2} \bigl( t- \tau ' ( t),l \bigr) B_{2}^{\mathrm{T}} P_{4} B_{2} u_{2} \bigl( t- \tau ' ( t),l \bigr). \end{aligned}$$

(59)

According to Lemma 6, for a positive scalar ε, there exist

$$\begin{aligned}& 2 u_{2} ( t,l) \mathrm{P}_{3} W_{2} G \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr) \\& \quad \leq\varepsilon u_{2}^{\mathrm{T}} ( t,l) \mathrm{P}_{3} W_{2} u_{2} ( t,l) + \frac{1}{\varepsilon} G^{\mathrm{T}} \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr) \mathrm{P}_{3} W_{2} G \bigl( v_{2} \bigl( t- \sigma ' ( t),l \bigr) \bigr). \end{aligned}$$

(60)

Taking equations (40)-(47) into consideration, derivatives of $V ( t,m,p)$ ($i=1,2,3$) can be formed as follows:

$$ \frac{\partial V(t,l)}{\partial t} = \sum_{i=1}^{8} \frac{\partial V_{i} (t,l)}{\partial t}. $$

(61)

Taking inequalities (48)-(60) into consideration, equation (61) is rewritten as

$$\begin{aligned} \frac{\partial V(t,l)}{\partial t} =& \sum_{i=1}^{8} \frac{\partial V_{i} (t,l)}{\partial t} \\ \leq& \int_{\Sigma} \bigl[ \xi_{1}^{\mathrm{T}} ( t,l) \Xi_{1} \xi_{1} ( t,l) + \xi_{2}^{\mathrm{T}} ( t,l) \Xi_{2} \xi_{2} ( t,l) \bigr] \,dl< 0, \end{aligned}$$

(62)

for $\zeta(t,x)\neq0$, where

$$\begin{aligned}& \xi_{1} = \biggl[ u_{1}^{T} ( t,l), u_{1}^{T} ( t- \tau_{1},l), u_{1}^{T} ( t- \tau_{2},l), u_{1}^{T} ( t-\tau,l), \biggl[ \int_{t-\tau}^{t} u_{1} ( s,l) \,ds \biggr]^{\mathrm{T}}, \\& \hphantom{\xi_{1} =} F^{\mathrm{T}} \bigl( v_{1} \bigl( t- \sigma(t),l \bigr) \bigr) , u_{2}^{T} ( t,l), u_{2}^{T} \bigl( t- \tau_{1} ',l \bigr), u_{1}^{T} \bigl( t- \tau_{2} ',l \bigr), u_{2}^{T} \bigl( t- \tau ',l \bigr), \\& \hphantom{\xi_{1} =} \biggl[ \int_{t- \tau '}^{t} u_{2} ( s,l) \,ds \biggr]^{\mathrm{T}}, G^{\mathrm{T}} \bigl( v_{2} \bigl( t- \sigma ' (t),l \bigr) \bigr) \biggr]^{\mathrm{T}}, \\& \xi_{2} = \biggl[ v_{1}^{T} ( t,l), v_{1}^{T} ( t- \sigma_{1},l), v_{1}^{T} ( t- \sigma_{2},l), v_{1}^{T} ( t-\sigma,l), \biggl[ \int_{t-\sigma}^{t} v_{1} ( s,l) \,ds \biggr]^{\mathrm{T}}, F^{\mathrm{T}} \bigl( v_{1} ( t,l) \bigr) , \\& \hphantom{\xi_{2} =} v_{2}^{T} ( t,l), v_{2}^{T} \bigl( t- \sigma_{1} ',l \bigr), v_{1}^{T} \bigl( t- \sigma_{2} ',l \bigr), v_{2}^{T} \bigl( t- \sigma ',l \bigr), \\& \hphantom{\xi_{2} =} \biggl[ \int_{t- \sigma '}^{t} v_{2} ( s,l) \,ds \biggr]^{\mathrm{T}}, G^{\mathrm{T}} \bigl( v_{2} ( t,l) \bigr) \biggr]^{\mathrm{T}}. \end{aligned}$$

 □

4 Numerical simulation

Consider the reaction-diffusion-delayed GRN (25), (26) with the following parameters:

$$\begin{aligned}& k=1,\qquad L=100,\qquad \gamma=1,\qquad \gamma ' =0.3,\qquad K_{1} = K_{2} =0.65I, \\& A_{1} =\operatorname{diag}( 3,3,3,3,3),\qquad B_{1} =\operatorname{diag}( 0.8,0.8,0.8,0.8,0.8), \\& C_{1} =\operatorname{diag}( 2.5,2.5,2.5,2.5,2.5), \\& D_{1} = \operatorname{diag}(0.1,0.1,0.1,0.1,0.1),\qquad D_{1}^{*} = \operatorname{diag}(0.2,0.2,0.2,0.2,0.2), \\& W_{1} = \left [ \begin{matrix}0 &0 &1&1&-1&&\\ -1&1 &0&0&0 \\ 0 &-1&0&1&0 \\ 0 &-1&1&0&0 \\ 0 &0 &1&1&0 \end{matrix} \right ], \qquad W_{1}^{*} = \left [ \begin{matrix} \gamma&0&0\\ \gamma&0&0\\ 0 &0&-\gamma\\ 0 &0&0\\ 0 &0&0 \end{matrix} \right ], \\& A_{2} =\operatorname{diag}( 0.1,0.1,0.1),\qquad B_{2} =\operatorname{diag}( 1.8,1.8,1.8),\qquad C_{2} =\operatorname{diag}( 2.3,2.3,2.3), \\& d_{1} = \operatorname{diag}(0.1,0.1,0.1), \qquad d_{1}^{*} = \operatorname{diag}(0.2,0.2,0.2), \\& W_{2} =\left [ \begin{matrix} 2.4&-2.4&0\\ 2.4&-2.4&0\\ 2.4&0 &-2.4 \end{matrix} \right ], \qquad W_{2}^{*} = \left [ \begin{matrix} -\gamma '&-\gamma '&0 &0 &0\\ 0 &0 &\gamma '&\gamma '&0\\ 0 &\gamma ' &0 &0 &- \gamma ' \end{matrix} \right ]. \\& \tau =1.5+0.5\sin ( t),\qquad \sigma =2.2+0.8\sin ( t), \\& \tau ' =3.6+0.4\sin ( t),\qquad \sigma ' =2.9+0.6\sin ( t), \end{aligned}$$

where

$$f_{i} ( x) = \frac{x^{2}}{1+ x^{2}}, \qquad g_{i} ( y) = \frac{y^{2}}{1+ y^{2}} \quad (i =1,2,\ldots, n). $$

By using the Toolbox YALMIP in MATLAB to solve the LMI (29) and (30) we can obtain feasible solutions, the processes of asymptotic stability of mRNA concentration and protein concentration under Dirichlet boundary conditions are shown by Figures 1-8. It is obvious that the proposed theory is feasible. The topological structure of interactional GRNs is shown by Figure 9.

Figure 1

The trajectory of $\pmb{u_{11} (t,l)}$ and $\pmb{v_{11} (t,l)}$ .

Figure 2

The trajectory of $\pmb{u_{12} (t,l)}$ and $\pmb{v_{12} (t,l)}$ .

Figure 3

The trajectory of $\pmb{u_{13} (t,l)}$ and $\pmb{v_{13} (t,l)}$ .

Figure 4

The trajectory of $\pmb{u_{14} (t,l)}$ and $\pmb{v_{14} (t,l)}$ .

Figure 5

The trajectory of $\pmb{u_{15} (t,l)}$ and $\pmb{v_{15} (t,l)}$ .

Figure 6

The trajectory of $\pmb{u_{21} (t,l)}$ and $\pmb{v_{21} (t,l)}$ .

Figure 7

The trajectory of $\pmb{u_{22} (t,l)}$ and $\pmb{v_{22} (t,l)}$ .

Figure 8

The trajectory of $\pmb{u_{23} (t,l)}$ and $\pmb{v_{23} (t,l)}$ .

Figure 9

The topological structure of interactional GRNs.

We set up the rules to measure the stabilizing time of interactional GRNs as follows:

$$\begin{aligned}& \gamma ( t)= \int_{\Sigma} \bigl( w_{i} \bigl\vert u_{1i} ( t,l) - u_{1i} (t - \delta,l) \bigr\vert + w_{i} ' \bigl\vert v_{1i} (t,l) v_{1i} (t - \delta,l) \bigr\vert + w_{p} \bigl\vert u_{2p} ( t,l) - u_{2p} (t - \delta,l) \bigr\vert \\& \hphantom{\gamma ( t)=}{}+ w_{p} ' \bigl\vert v_{2p} ( t,l) - v_{2p} (t - \delta,l) \bigr\vert \bigr) \,\mathrm{d} l, \end{aligned}$$

(63)

$$\begin{aligned}& \textstyle\begin{cases} \Theta ( t) =1,\qquad \gamma ( t) > \delta_{0}, \\ \Theta ( t) =2,\qquad \gamma ( t) \leq \delta_{0}, \end{cases}\displaystyle \end{aligned}$$

(64)

where $w_{i} >0$, $w_{i} ' >0$ ($i =1,\ldots, n_{1}$); $w_{p} >0$, $w_{p} ' >0$ ($p =1,\ldots, n_{2}$) are weights of $u_{1i}$, $v_{1i}$, $u_{2p}$ and $v_{2p}$, respectively, $\delta_{0}$ is a positive small quantity, $\Theta(t)$ is a quantity for stability identification. When $\Theta(t) =2$, it indicates interactional GRNs are stable; when $\Theta(t) =1$, it indicates interactional GRNs are unstable.

When $\gamma ' \in \{ 0.3, 0.4, 0.5, 0.6,0.7,0.8,0.9,1.0 \}$, $\gamma =0.4$, $w_{i} = w_{i} ' = w_{p} = w_{p} ' =1/16$, $\delta_{0} =0.1$, the evolutions of $\Theta(t)$ with $\gamma '$ are shown by Figure 10, it is found that the increase of coupling strength $\gamma '$ will lengthen the stabilizing time of interactional GRNs. Furthermore, we can find that the iterative number of solution of LMIs and the time of reaching a steady state have the same change with different $\gamma '$ as shown in Figure 11, therefore we utilize the iterative number to characterize the time of reaching steady state, because of the independence of the initial condition and the step size for the iterative number. When $\gamma\in \{ 0, 0,2,0.4,0.6,0.8,1.0 \}$, the performance of number of iterations with $\gamma '$ and γ is shown in Figure 12, where the color represents the time of reaching a steady state.

Figure 10

The evolution of $\pmb{\Theta(t)}$ with $\pmb{\gamma '}$ .

Figure 11

Iterative number and required time for stable with different $\pmb{\gamma '}$ ( $\pmb{\gamma=0.0.4}$ ).

Figure 12

Performance of number of iterations with $\pmb{\gamma '}$ and γ .

5 Discussion

Further numerical simulations show that:

• (I) Interactional GRNs without reaction-diffusion terms cannot reach a steady state, because of the defection of information from reaction-diffusion terms. In other words, reaction-diffusion terms are important and indispensable for interactional GRNs.

• (II) γ and $\gamma '$ have an absolute effect on the time of reaching steady state as shown in Figure 12, the increase of coupling strength $\gamma '$ and γ will collectively lengthen the stabilizing time of interactional GRNs.

• (III) The matrices $W_{2}$ and $W_{2}^{*}$ qualitatively affect the stability of the interactional GRNs, in other words, the topological structure of the unstable GRNs and $W_{2}^{*}$ are very important for the stability of the interactional GRNs. The out-degree and in-degree of stable GRNs as well as the coupling term of $W_{1}^{*}$ can change at will, but the out-degree and in-degree of unstable GRNs and the coupling term of $W_{2}^{*}$ cannot change freely.

6 Conclusion

In this paper, we have constructed a model for interactional GRNs with reaction-diffusion terms under a Dirichlet boundary condition, and we analyzed the robust stability of interactional GRNs. Through constructing appropriate Lyapunov-Krasovskii functions and linear matrix inequalities (LMIs), we have given stability criteria corresponding to interactional GRNs. By numerical simulations, we found three important conclusions: interactional GRNs without reaction-diffusion terms cannot reach a steady state, because of the defection of information from reaction-diffusion terms, in other words, reaction-diffusion terms are important and indispensable for interactional GRNs; due to the smaller coupling strength $\gamma '$ (in a certain range) and γ, interactional GRNs tend to become stable more quickly; the topological structures of the unstable GRNs and coupling term of $W_{2}^{*}$ determine the stability of interactional GRNs.