State estimation for delayed genetic regulatory networks with reaction diffusion terms and Markovian jump

Abstract

Robust state estimation for delayed genetic regulatory networks with reaction–diffusion terms and uncertainties terms under Dirichlet boundary conditions is addressed in this article. The main purpose of the problem investigation is to design a novel state observer for estimate the true concentrations of mRNA and protein by available measurement outputs. Based on Lyapunov–Krasovskii functions and linear matrix inequalities (LMI), sufficient conditions are given to ensure the robust stability of the estimation error networks. Two examples are presented to illustrate the effectiveness of the proposed approach.

Introduction

In the last decade, many mathematical models of genetic regulatory networks (GRNs) are used to depict the dynamic behavior of interactions between the mRNA and proteins [1, 2]. In general, Boolean model [3,4,5,6,7] and differential models [6,7,8,9,10] are two typical models of GRNs. Owing to the dynamical behavior description of mRNA and protein, the differential equation model of GRNs have received an increasing attention to the investigations on GRNs in the fields of biological and biomedical sciences.

In the interactive process of transcription and translation, time delay is an important and unavoidable factor in dynamics of GRNs [8,9,10, 14, 15, 18], time delay may lead to bifurcation, chaos, oscillation in system [13]. In addition, uncertainties often exist in the most of biological and engineering system, and may cause undesirable dynamic behavior [10, 14,15,16,17,18]. Therefor it is necessary to take delay and uncertainties into consideration to predict the dynamic behavior of GRNs. Usually, gene networks may include some kinds of switching mechanisms due to the randomly occurring phenomena. The switching mechanisms can be described by Markovian jumping linear system [19,20,21]. Due to the existence of unknown time delays, uncertainties and state switching, it is difficult to estimate the concentrations of mRNA and protein directly [20,21,22,23,24]. In practice, state estimation is necessary to artificial input control [20]; therefore, considerable study efforts have been made on the state estimation of GRNs. For example, Li et al. have designed ${H}_{\infty }$ estimator parameters for discrete-time stochastic GRNs and adopted event-triggered to save computer resources [20]; Lu et al. have investigated robust state estimation for the Markov jump GRNs by a linear state estimator [21]; Chen et al. established state estimation of GRNs [22], where stochastic sample-date is utilized for various effective optimization algorithms.

In the existing literature, most of models used to describe the GRNs structure assumed that the concentrations of mRNA and protein are homogeneous in space at all time, but the spatial homogeny is not appropriate for nonuniform distributed concentrations of gene products [25]; therefore, the problem of dynamic analysis for GRNs with reaction–diffusion is worth to research. Han et al. [25] have established asymptotic stability criteria for GRNs with reaction–diffusion under Dirichlet and Neumann boundary conditions, respectively, and gained deep insight that, the information of diffusion–reaction can reduce the conservation of system; in [26], Zou et al. have introduced reaction–diffusion to the interactional GRNs, and proved that diffusion terms are significant for the stability of the interactional GRNs; oscillatory behaviors of GRNs with reaction–diffusion terms has been investigated by Zhang et al. in [27]. Song et al. have proposed sampled-data state estimation of genetic regulatory networks with reaction diffusion [28]. Xiao et al. have investigated stability of genetic regulatory networks by linear parameterization [29].

Although the research on state estimation of GRNs has attracted much attention, but the researches on estimation of GRNs with reaction–diffusion terms are found in only a few place in the literature. In this article, we have designed a novel estimator for Markovian jumping GRNs with reaction–diffusion first, then, we investigated the estimation of the concentrations of mRNA and protein by Lyapunov–Krasovskii functions and linear matrix inequalities (LMI) techniques. Finally, two simulations illustrate the effectiveness of the proposed strategy, which has ability to reduce conservative stability conditions.

Problem formulation

Considering following GRNs with reaction diffusion [25]

$$\left\{\begin{array}{l}\frac{\partial {\widetilde{\alpha }}_{i}(t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({d}_{ki}\frac{\partial {\widetilde{\alpha }}_{i}(t,x)}{\partial {x}_{k}}\right)-{a}_{i}{\widetilde{\alpha }}_{i}\left(t,x\right)+\sum\limits_{j=1}^{n}{b}_{ij}g\left({\widetilde{\beta }}_{j}\left(t-\sigma \left(t\right),x\right)\right)+{Q}_{i} \\ \frac{\partial {\widetilde{\beta }}_{i}(t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({d}_{ki}^{*}\frac{\partial {\widetilde{\beta }}_{i}(t,x)}{\partial {x}_{k}}\right)-{c}_{i}{\widetilde{\beta }}_{i}\left(t,x\right)+{d}_{i}{\widetilde{\alpha }}_{i}\left(t-\tau \left(t\right),x\right) \end{array}\right.$$

(1)

${\widetilde{\alpha }}_{i}\left(t,x\right)$ and ${\widetilde{\beta }}_{i}\left(t,x\right)$ are the concentrations of mRNA and protein of the $i$th node, respectively, the parameters ${a}_{i}$ and ${c}_{i}$ are the degradation rates of the mRNA and protein respectively, and ${d}_{i}$ is the translation rate, $\tau (t)$ and $\sigma (t)$ are time-varying delays denoting the translation delay and feedback regulation delay, respectively. $g(x)$ is a Hill form regulatory function, which represents the feedback regulation of the protein on the transcription, its form is described as Eq. (2)

$$g\left(x\right)=\frac{({\frac{x}{v})}^{H}}{1+({\frac{x}{v})}^{H}},$$

(2)

where $H$ is the Hill coefficient, $v$ is a positive constant, and $g(x)$ satisfies the inequality Eqs. (3) and (4), respectively, because $g(\bullet )$ is monotonically increase function with saturation

$$0\le \frac{g\left(x\right)}{x}\le \varsigma ,\forall x\ne 0,$$

(3)

$${g}^{\mathrm{T}}\left(x\right)\left(g\left(x\right)-Kx\right)\le 0,$$

(4)

$B :=({b}_{ij})\in {R}^{n\times n}$ is a coupling matrix, which is described as

$${b}_{ij}=\left\{\begin{array}{c}{\zeta }_{ij }\\ 0 \\ -{\zeta }_{ij}\end{array}\right.,$$

(5)

where ${\zeta }_{ij}$ represent transcription factor $j$ is an activator of gene $i$, 0 represent that there is no link from node $j$ to $i$, ${-\zeta }_{ij}$ represent transcription factor $j$ is a repressor of gene $i$. ${Q}_{i}={\sum }_{j\in {I}_{i}}{\zeta }_{ij}$, ${I}_{i}$ is the set containing all the repressors of gene $i$.

Where $x={\left({x}_{1},{x}_{2},\cdots ,{x}_{l}\right)}^{\mathrm{T}}\in\Omega \subset {R}^{c}$,$\Omega =\left\{x\left|\left|{x}_{k}\right|\le {L}_{k}\right.\right\}$, ${L}_{k}$ is constant, $k=\mathrm{1,2},\cdots l$, ${D}_{k}=\mathrm{diag}\left\{{D}_{1k},\cdots ,{D}_{nk}\right\}$ and ${D}_{k}^{*}=\mathrm{diag}\left\{{D}_{1k}^{*},\cdots ,{D}_{nk}^{*}\right\}$ denote the transmission diffusion operator along the $i$ th gene of mRNA and protein respectively, where ${D}_{ik}>0, {D}_{ik}^{*}>0$.

The initial conditions are given by

$$\left\{\begin{array}{c}{\widetilde{\alpha }}_{i}\left(s,x\right)={\psi }_{i}\left(s,x\right),s\in \left(-\infty ,0\right],i=\mathrm{1,2},\cdots ,n\\ {\widetilde{\beta }}_{i}\left(s,x\right)={\psi }_{i}^{*}\left(s,x\right),s\in \left(-\infty ,0\right],i=\mathrm{1,2},\cdots ,n\end{array},\right.$$

where ${\psi }_{i}\left(s,x\right)$ and ${\psi }_{i}^{*}\left(s,x\right)$ are bounded and continuous on $\left(-\infty ,0\right]\times \Omega$

We assume that ${\alpha }_{i}^{*}(x)$ and ${\beta }_{i}^{*}(x)$ are the unique solutions of GRN (1), it’s obvious that

$$\left\{\begin{array}{c}0=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({d}_{ik}\frac{\partial {\alpha }_{i}^{*}(x)}{\partial {x}_{k}}\right)-{a}_{i}{\alpha }_{i}^{*}\left(x\right)+\sum\limits_{j=1}^{n}{b}_{ij}{g}_{j}\left({\beta }_{j}^{*}\left(x\right)\right)+{Q}_{i} \\ 0=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({d}_{ik}^{*}\frac{\partial {\alpha }_{i}^{*}(x)}{\partial {x}_{k}}\right)-{c}_{i}{\beta }_{i}^{*}\left(x\right)+{d}_{i}{\alpha }_{i}^{*}\left(x\right),i=\mathrm{1,2},\cdots ,n\end{array}\right.$$

Set ${\alpha }_{i}\left(t,x\right)={\widetilde{\alpha }}_{i}\left(t,x\right)-{\alpha }^{*}\left(x\right)$ and ${\beta }_{i}\left(t,x\right)={\widetilde{\beta }}_{i}\left(t,x\right)-{\beta }^{*}\left(x\right)$, $i=\mathrm{1,2},\cdots ,n$, transform Eq. (1) into the following matrix form:

$$\left\{\begin{array}{c}\frac{\partial \alpha (t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{k}\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)-A\alpha \left(t,x\right)+Bg\left(\beta \left(t-\sigma \left(t\right),x\right)\right) \\ \frac{\partial \beta (t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{k}^{*}\frac{\partial \beta (t,x)}{\partial {x}_{k}}\right)-C\beta \left(t,x\right)+\widetilde{D}\alpha \left(t-\tau \left(t\right),x\right) \end{array}\right.$$

(6)

where $A={\text{diag}}({a}_{1},{a}_{2},\cdots ,{a}_{n})$, $C={\text{diag}}({c}_{1},{c}_{2},\cdots ,{c}_{n})$, $\widetilde{D}={\text{diag}}({d}_{1},{d}_{2},\cdots ,{d}_{n})$, ${D}_{k}={\text{diag}}({D}_{1k},{D}_{2k},\cdots ,{D}_{nk})$, ${D}_{k}^{*}={\text{diag}}({D}_{1k}^{*},{D}_{2k}^{*},\cdots ,{D}_{nk}^{*})$, $\alpha(t,x)=({\alpha }_{1}(t,x),{\alpha }_{2}(t,x),\cdots ,{\alpha }_{n}(t,x))\beta (t,x)=({\beta }_{1}(t,x),{\beta }_{2}(t,x),\cdots ,{\beta }_{n}(t,x)),$ $g(\beta (t-\sigma (t),x))=({g}_{1}(\beta (t-\sigma (t),x)),{g}_{2}(\beta (t-\sigma (t),x)), \break \cdots ,{g}_{n}(\beta (t-\sigma (t),x)) )^{\mathrm{T}},$ ${g}_{i}\left(\beta \left(t-\sigma \left(t\right),x\right)\right)={g}_{i}\left({\beta }_{i}\left(t-\sigma \left(t\right),x\right)\right)-{g}_{i}\left({\beta }^{*}\left(x\right)\right)$, $i=\mathrm{1,2},\cdots ,n$.

Taking the Markov jumping and uncertainties into account, Eq. (6) can be rewrite as

$$\left\{\begin{array}{c}\frac{\partial \alpha (t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{k}({r}_{t})\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)-\left(A({r}_{t})+\Delta A\left({r}_{t}\right)\right)\alpha \left(t,x\right)+\left(B\left({r}_{t}\right)+\Delta B\left({r}_{t}\right)\right)g\left(\beta \left(t-\sigma \left(t\right),x\right)\right) \\ \frac{\partial \beta (t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{k}^{*}({r}_{t})\frac{\partial \beta (t,x)}{\partial {x}_{k}}\right)-\left(C\left({r}_{t}\right)+\Delta C\left({r}_{t}\right)\right)\beta \left(t,x\right)+\left(\widetilde{D}\left({r}_{t}\right)+\Delta \widetilde{D}\left({r}_{t}\right)\right)\alpha \left(t-\tau \left(t\right),x\right) \end{array}\right.$$

(7)

where $\Delta A\left({r}_{t}\right)$, $\Delta B\left({r}_{t}\right)$, $\Delta C\left({r}_{t}\right)$ and $\Delta \widetilde{D}\left({r}_{t}\right)$ are uncertainties for parameters $A\left({r}_{t}\right)$, $B\left({r}_{t}\right)$, $C\left({r}_{t}\right)$ and $\widetilde{D}\left({r}_{t}\right),$ respectively.

We use parameter ${r}_{t}(t\ge 0)$ to represents a right-continuous Markov process on a complete probability space $S=\left\{\mathrm{1,2},\cdots ,N\right\}$ with generator $\Pi ={({\pi }_{ij})}_{N\times N}$ given by

$$\begin{aligned} & P\left\{r\left(t+\delta t\right)=q|r\left(t\right)=p\right\}\\ &\quad =\left\{\begin{array}{c}\left({\pi }_{pq}+\Delta {\pi }_{pq}\right)\delta t+o\left(\delta t\right), if p\ne q\\ 1+\left({\pi }_{pq}+\Delta {\pi }_{pq}\right)\delta t+o\left(\delta t\right), if p=q\end{array}\right.,\end{aligned}$$

where $\delta t>0$, ${\pi }_{pq}\ge 0$ is the known transition rate from $p$ to $q$ if $p\ne q$ where ${\pi }_{pq}=-\sum\nolimits_{q\ne p}{\pi }_{pq}$, and the uncertain transition rate $\Delta {\pi }_{pq}$ satisfies $\left|\Delta {\pi }_{pq}\right|\le {\delta }_{pq}$ and $\Delta {\pi }_{pp}=-\sum\nolimits_{q\ne p}{\Delta \pi }_{pq}$, $p,q\in S$. Then, Eq. (7) can be transformed as following forms:

$$\left\{\begin{array}{c}\frac{\partial \alpha (t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)-\left({A}_{p}+\Delta {A}_{p}\right)\alpha \left(t,x\right)+\left({B}_{p}+\Delta {B}_{p}\right)g\left(\beta \left(t-\sigma \left(t\right),x\right)\right) \\ \frac{\partial \beta (t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}^{*}\frac{\partial \beta (t,x)}{\partial {x}_{k}}\right)-\left({C}_{p}+\Delta {C}_{p}\right)\beta \left(t,x\right)+\left({\widetilde{D}}_{p}+\Delta {\widetilde{D}}_{p}\right)\alpha \left(t-\tau \left(t\right),x\right) \end{array}\right.$$

(8)

To estimate the states of Eq. (8), the following state observer is constructed

$$\left\{\begin{array}{c}\frac{\partial \widehat{\alpha }(t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial \widehat{\alpha }(t,x)}{\partial {x}_{k}}\right)+\widehat{A}\widehat{\alpha }\left(t,x\right)+M\alpha (t,x) \\ \frac{\partial \widehat{\beta }(t,x)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}^{*}\frac{\partial \widehat{\beta }(t,x)}{\partial {x}_{k}}\right)+\widehat{C}\widehat{\beta }\left(t,x\right)+N\beta (t,x) \end{array} \right.$$

(9)

where $\widehat{\alpha }(t,x)$ and $\widehat{\beta }(t,x)$ are the estimates of $\alpha (t,x)$ and $\beta (t,x)$ respectively, and $\widehat{A}$, $\widehat{C}$, $M$ and $N$ are the observe gain matrices to be designed later.

In this paper, we consider Eqs. (8) and (9) have the same initial values under Dirichlet boundary condition as follows:

$$\left\{\begin{array}{c}\alpha \left(t,x\right)=\widehat{\alpha }\left(t,x\right)=0, x\in \partial \Omega ,t\in \left[-d,+\infty \right)\\ \beta \left(t,x\right)=\widehat{\beta }\left(t,x\right)=0, x\in \partial \Omega ,t\in \left[-d,+\infty \right)\end{array}.\right.$$

(10)

We define the error vectors through ${e}_{\alpha }\left(x,t\right)=\alpha \left(t,x\right)-\widehat{\alpha }\left(t,x\right)$ and ${e}_{\beta }\left(x,t\right)=\beta \left(t,x\right)-\widehat{\beta }\left(t,x\right)$. Then the derivative form of ${e}_{\alpha }\left(x,t\right)$ and ${e}_{\beta }\left(x,t\right)$ can be described as

$$\left\{\begin{array}{l}\frac{\partial {e}_{\alpha }\left(t,x\right)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial {e}_{\alpha }\left(t,x\right)}{\partial {x}_{k}}\right)+\widehat{A}{e}_{\alpha }\left(t,x\right)-\widehat{A}\alpha \left(t,x\right)-\left(A+\Delta A\right)\alpha \left(t,x\right)+\left(B+\Delta B\right)g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)-M\alpha \left(t,x\right) \\ \frac{\partial {e}_{\beta }\left(t,x\right)}{\partial t}=\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial {e}_{\beta }\left(t,x\right)}{\partial {x}_{k}}\right)+\widehat{C}{e}_{\beta }\left(t,x\right)-\widehat{C}\beta \left(t,x\right)-\left(C+\Delta C\right)\beta \left(t,x\right)+\left(\widetilde{D}+\Delta \widetilde{D}\right)\alpha \left(t-\tau \left(t\right),x\right) -N\beta \left(t,x\right) \\ {e}_{\alpha }\left(t,x\right)=0, {e}_{\beta }\left(t,x\right)=0, x\in \Omega \\ {e}_{\alpha }\left(t,x\right)=0, {e}_{\beta }\left(t,x\right)=0, x\in \partial \Omega \end{array}\right.$$

(11)

Assumption 1

The time-varying delays satisfying

$$ 0 \le \tau \left( t \right) \le \tau_{{\text{m}}} ,\;0 \le \sigma \left( t \right) \le \sigma_{{\text{m}}} ,\;\mu_{1} \le \dot{\tau }\left( t \right) \le \mu_{2} ,\;\eta_{1} \le \dot{\sigma }\left( t \right) \le \eta_{2} ,\forall t, $$

(12)

where the upper bounds ${\tau }_{\mathrm{m}}$, ${\sigma }_{\mathrm{m}}$, ${\mu }_{2}$ and ${\eta }_{2}$ are nonnegative real numbers, lower bounds ${\mu }_{1}$ and ${\eta }_{1}$ are real numbers.

Assumption 2

Parameter uncertainties $\Delta A\left({r}_{t}\right)$, $\Delta B\left({r}_{t}\right)$, $\Delta C\left({r}_{t}\right)$ and $\Delta \widetilde{D}\left({r}_{t}\right)$ satisfying:

$$\begin{aligned} & \left[\Delta A\left({r}_{t}\right), \Delta B\left({r}_{t}\right), \Delta C\left({r}_{t}\right), \Delta \widetilde{D}\left({r}_{t}\right)\right] \\ & \quad ={M}_{p}{E}_{p}\left(t\right)\left[{N}_{{A}_{p}}, {N}_{{B}_{p}}, {N}_{{C}_{p}},{N}_{{D}_{p}}\right],\end{aligned}$$

(13)

${M}_{p}$, ${N}_{{A}_{p}}, {N}_{{B}_{p}}, {N}_{{C}_{p}}$ and ${N}_{{D}_{p}}$ are given constant matrices, ${E}_{p}$ is a known real time varying function with the bound as follows:

$${{E}_{p}^{\mathrm{T}}\left(t\right)E}_{p}\left(t\right)\le I,$$

(14)

where $i\in \mathrm{Z}$ and $t\ge 0$.

Lemma 1

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

$${\int }_{a}^{b}{f}^{2}(v)\mathrm{d}v\le \frac{{(b-a)}^{2}}{{\pi }^{2}}{\int }_{a}^{b}{[{f}^{^{\prime}}(v)]}^{2}\mathrm{d}v.$$

(15)

Lemma 2

(Green’s Second Identity) If $\Omega$ is a bounded ${C}^{1}$ open set in ${R}^{n}$ and $\mu$, $v\in {C}^{2}(\overline{\Omega })$, then

$${\int }_{\Omega }\mu\Delta v\mathrm{d}x={\int }_{\Omega }v\Delta \mu \mathrm{d}x+{\int }_{\partial \Omega }\left(\mu \frac{\partial v}{\partial \overline{n} }-v\frac{\partial \mu }{\partial \overline{n} }\right)\mathrm{d}S,$$

(16)

where $\frac{\partial v}{\partial \overline{n} }$ and $\frac{\partial \mu }{\partial \overline{n} }$ are the directional derivatives of $v$ and $\mu$ in the direction of the outward pointing normal $\overline{n }$ to the surface element $\mathrm{d}S$, respectively. $\sum\nolimits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}({D}_{k}\frac{\partial }{\partial {x}_{k}})$ can be regarded as Laplacian operator which is formally self-adjoint and differential in lemma 2 inner product for function with Dirichlet boundary.

Lemma 3

[25] From Green formula, under Dirichlet boundary conditions, and using Lemmas 1and 2, we can obtain

$$2{\int }_{\Sigma }{\mu }^{\mathrm{T}}\sum\limits_{k=1}^{l}\frac{\partial }{\partial {l}_{k}}\left(\frac{\partial \mu }{\partial {l}_{k}}\right)\mathrm{d}l\le -\frac{{\pi }^{2}}{2}{\int }_{\Sigma }{\mu }^{\mathrm{T}}\mu \mathrm{d}l. $$

(17)

Lemma 4

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

$${\left({\int }_{0}^{\vartheta }x(s)\mathrm{d}s\right)}^{\mathrm{T}}M\left({\int }_{0}^{\vartheta }x(s)\mathrm{d}s\right)\le \vartheta \left({\int }_{0}^{\vartheta }x(s)Mx(s)\mathrm{d}s\right)\text{.} \, $$

(18)

Lemma 5

For any vectors $X, Y\in {R}^{n}$, and any scalar $\varepsilon >0$, exist following inequality

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

(19)

Observer design

In this section, we will investigate state estimation for Eq. (8) with reaction–diffusion terms via Lyapunov functional approach.

Theorem 1

For given scalars ${\tau }_{\mathrm{m}}$, ${\sigma }_{\mathrm{m}}$, ${\mu }_{1}$, ${\mu }_{2}$, ${\eta }_{1}$, ${\eta }_{2}$ and given estimator parameters $\widehat{A}$, $\widehat{C}$, $M$ and $N$, Eq. (11) under Dirichlet boundary conditions is stable without uncertainties if there exist matrices ${P}_{1p}^{\mathrm{T}}={P}_{1p}>0$, ${P}_{2p}^{\mathrm{T}}={P}_{2p}>0$, ${J}_{1p}^{\mathrm{T}}={J}_{1p}>0$, ${J}_{2p}^{\mathrm{T}}={J}_{2p}>0$, ${R}_{\iota }^{\mathrm{T}}={R}_{\iota }>0$, $(\iota =3,\cdots , 6)$, ${{\Lambda }_{1}^{\mathrm{T}}=\Lambda }_{1}>0$, ${{\Lambda }_{2}^{\mathrm{T}}=\Lambda }_{2}>0$, ${Q}_{\delta }^{\mathrm{T}}={Q}_{\delta }>0$, $(\delta =1,\cdots , 6)$; ${\tau }_{\mathrm{m}}$, ${\sigma }_{\mathrm{m}}\to \infty$, such that the following linear matrix in equality (LMIs) hold:

$${\Xi }_{p}=\left[\begin{array}{c}{\Sigma }_{11p}\\ *\end{array} \quad \begin{array}{c}{\Sigma }_{12p}\\ {\Sigma }_{22p}\end{array}\right]<0,$$

(20)

where

$${\Sigma }_{11p}=\left[\begin{array}{c}{\Pi }_{1p}\\ *\\ *\\ *\\ *\\ *\end{array} \begin{array}{c}0\\ {\Phi }_{1}\\ *\\ *\\ *\\ *\end{array} \begin{array}{c}{P}_{1p}{B}_{p}\\ K{\Lambda }_{2}\\ -{2\Lambda }_{2}\\ *\\ *\\ *\end{array} \begin{array}{c}0\\ 0\\ 0\\ {\Phi }_{2}\\ *\\ *\end{array} \begin{array}{c}0\\ 0\\ 0\\ {\widetilde{D}}_{p}^{\mathrm{T}}{P}_{2p}\\ {\Pi }_{2p}\\ *\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\\ K{\Lambda }_{1}\\ -2{\Lambda }_{1}\end{array}\right],$$

$${\Sigma }_{12p}=\left[\begin{array}{c}{\Pi }_{3p}\\ 0\\ {B}_{p}^{\mathrm{T}}{J}_{1p}\\ 0\\ 0\\ 0\end{array} \quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ {\widetilde{D}}_{p}^{\mathrm{T}}{J}_{2p}\\ {\Pi }_{4p}\\ 0\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],$$

$${\Sigma }_{22p}=\left[\begin{array}{c}{\Pi }_{5p}\\ *\\ *\\ *\\ *\\ *\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ {\Phi }_{3}\\ *\\ *\\ *\\ *\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ 0\\ {\Pi }_{6p}\\ *\\ *\\ *\\ *\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ {\Phi }_{4}\\ *\\ *\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ {-Q}_{5}\\ *\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ {-Q}_{6}\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-Q}_{2}\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-Q}_{4}\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-R}_{4}\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-R}_{6}\end{array}\right]$$

with

$${\Pi }_{1p}=-{\frac{{\pi }^{2}}{2}P}_{1p}{D}_{kp}-2{P}_{1p}{A}_{p}+{Q}_{1}+{\tau }_{\mathrm{m}}^{2}{Q}_{5}+\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{1q},$$

$${\Pi }_{2p}=-\frac{{\pi }^{2}}{2}{P}_{2p}{D}_{kp}^{*}-2{P}_{2p}{C}_{p}+{Q}_{3}+{\sigma }_{\mathrm{m}}^{2}{Q}_{6}+\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{2q},$$

$${\Pi }_{3p}=-\left({A}_{q}^{T}-{\widehat{A}}^{\mathrm{T}}-{M}^{\mathrm{T}}\right){R}_{1q},$$

$${\Pi }_{4p}=\left(-{C}_{q}^{T}-{\widehat{C}}^{T}-{N}^{T}\right){R}_{2q},$$

$${\Pi }_{5p}=-\frac{{\pi }^{2}}{2}{J}_{1q}{D}_{kq}-2{J}_{1q}\widehat{A}+{R}_{3}+\sum\limits_{q=1}^{N}{\pi }_{pq}{J}_{1q},$$

$${\Pi }_{6p}=-\frac{{\pi }^{2}}{2}{J}_{2p}{D}_{kp}^{*}-2{R}_{2p}\widehat{C}+{R}_{5}+\sum\limits_{q=1}^{N}{\pi }_{pq}{J}_{2q},$$

$${\Phi }_{1}=\left({\eta }_{2}-1\right){Q}_{3}+\left(1-{\eta }_{1}\right){Q}_{4},$$

$${\Phi }_{2}=\left({\mu }_{2}-1\right){Q}_{1}+\left(1-{\mu }_{1}\right){Q}_{2},$$

$${\Phi }_{3}=\left({\mu }_{2}-1\right){R}_{3}+\left(1-{\mu }_{1}\right){R}_{4},$$

$${\Phi }_{4}=\left({\eta }_{2}-1\right){R}_{5}+(1-{\eta }_{1}){R}_{6}$$

Proof

Define a Lyapunov–Krasovskii functional candidate for state estimation as

$$\widehat{V}(t,x)=\sum\limits_{\varpi =1}^{5}{V}_{\varpi }\left(t,x\right),$$

(21)

where

$$\begin{aligned} {V}_{1}\left(\alpha ,\beta , t,p\right)& ={\int }_{\Omega }{\alpha }^{\mathrm{T}}(t,x){P}_{1p}\alpha (t,x)\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}\beta \left(t,x\right)\mathrm{d}x\end{aligned}$$

(22)

$$\begin{aligned} {V}_{2}\left(\alpha ,\beta , t\right)& ={\int }_{\Omega }{\int }_{t-\tau (t)}^{t}{\alpha }^{\mathrm{T}}\left(s,x\right){Q}_{1}\alpha \left(s,x\right)\mathrm{d}s\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\int }_{t-{\tau }_{\mathrm{m}}}^{t-\tau (t)}{\alpha }^{\mathrm{T}}\left(s,x\right){Q}_{2}\alpha \left(s,x\right)\mathrm{d}s\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\int }_{t-\sigma (t)}^{t}{\beta }^{\mathrm{T}}\left(s,x\right){Q}_{3}\beta \left(s,x\right)\mathrm{d}s\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\int }_{t-{\sigma }_{\mathrm{m}}}^{t-\sigma (t)}{\beta }^{\mathrm{T}}\left(s,x\right){Q}_{4}\beta \left(s,x\right)\mathrm{d}s\mathrm{d}x,\end{aligned}$$

(23)

$$\begin{aligned} {V}_{3}\left({e}_{\alpha },{e}_{\beta }, t,p\right)& ={\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){J}_{1p}{e}_{\alpha }\left(t,x\right)dx\\ & \quad +{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}{e}_{\beta }\left(t,x\right)dx,\end{aligned}$$

(24)

$$\begin{aligned}{V}_{4}\left({e}_{\alpha },{e}_{\beta }, t\right) & ={\int }_{\Omega }{\int }_{t-\tau (t)}^{t}{e}_{\alpha }^{\mathrm{T}}\left(s,x\right){R}_{3}{e}_{\alpha }\left(s,x\right)\mathrm{d}s\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\int }_{t-{\tau }_{\mathrm{m}}}^{t-\tau (t)}{e}_{\alpha }^{\mathrm{T}}\left(s,x\right){R}_{4}{e}_{\alpha }\left(s,x\right)\mathrm{d}s\mathrm{d}x \\& \quad +{\int }_{\Omega }{\int }_{t-\sigma (t)}^{t}{e}_{\beta }^{\mathrm{T}}\left(s,x\right){R}_{5}{e}_{\beta }\left(s,x\right)\mathrm{d}s\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\int }_{t-{\sigma }_{\mathrm{m}}}^{t-\sigma (t)}{e}_{\beta }^{\mathrm{T}}\left(s,x\right){R}_{6}{e}_{\beta }\left(s,x\right)\mathrm{d}s\mathrm{d}x, \end{aligned}$$

(25)

$$ \begin{aligned} {V}_{5}\left(\alpha ,\beta , t\right)& ={\tau }_{\mathrm{m}}{\int }_{\Omega }{\int }_{-{\tau }_{\mathrm{m}}}^{0}{\int }_{t+\theta }^{t}{\alpha }^{\mathrm{T}}(s,x){Q}_{5}\alpha (s,x)\mathrm{d}s\mathrm{d}\theta \mathrm{d}x\\ & \quad +{\sigma }_{\mathrm{m}}{\int }_{\Omega }{\int }_{-{\sigma }_{\mathrm{m}}}^{0}{\int }_{t+\theta }^{t}{\beta }^{\mathrm{T}}(s,x){Q}_{6}\beta \left(s,x\right)\mathrm{d}s\mathrm{d}\theta \mathrm{d}x\end{aligned}$$

(26)

then, computing the derivatives of ${V}_{\varpi }$, we can get

$$\begin{aligned} \frac{\partial {V}_{1}}{\partial t} & =2{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1i}\frac{\partial \alpha \left(t,x\right)}{\partial t}\mathrm{d}x\\ & \quad +2{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2i}\frac{\partial \beta \left(t,x\right)}{\partial t}\mathrm{d}x\\& =2{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1i}\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)\right.\\ & \quad \left.-{A}_{p}\alpha \left(t,x\right)+{B}_{p}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\vphantom{\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)\right.}\right]\mathrm{d}x\\& \quad +2{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}^{*}\frac{\partial \beta \left(t,x\right)}{\partial {x}_{k}}\right)\right.\\ & \quad \left. -{C}_{p}\beta \left(t,x\right)+{\widetilde{D}}_{p}\alpha \left(t-\tau \left(t\right),x\right)\vphantom{\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)\right.}\right]\mathrm{d}x\\& \quad +{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{1q}\alpha \left(t,x\right)\mathrm{d}x\\ & \quad +\sum\limits_{j=1}^{N}{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{2q}\beta \left(t,x\right)\mathrm{d}x\end{aligned}$$

(27)

$$\begin{aligned} \frac{\partial {V}_{2}}{\partial t}& ={\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){Q}_{1}\alpha \left(t,x\right)\mathrm{d}x \\ & \quad -(1-\dot{\tau }){\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){Q}_{1}\alpha \left(t-\tau ,x\right)\mathrm{d}x\\ & \quad +(1-\dot{\tau }){\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){Q}_{2}\alpha \left(t-\tau ,x\right)\mathrm{d}x\\ & \quad -{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\mathrm{m}},x\right){Q}_{2}\alpha \left(t-{\tau }_{\mathrm{m}},x\right)\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){Q}_{3}\beta \left(t,x\right)\mathrm{d}x\\ & \quad -(1-\dot{\sigma }){\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){Q}_{3}\beta \left(t-\sigma ,x\right)\mathrm{d}x\\ &\quad +\left(1-\dot{\sigma }\right){\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){Q}_{4}\beta \left(t-\sigma ,x\right)\mathrm{d}x\\ & \quad -{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t-{\sigma }_{m},x\right){Q}_{4}\beta \left(t-{\sigma }_{m},x\right)\mathrm{d}x\\ & \le {\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){Q}_{1}\alpha \left(t,x\right)\mathrm{d}x\\ & \quad -(1-{\mu }_{2}){\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){Q}_{1}\alpha \left(t-\tau ,x\right)\mathrm{d}x\\ &\quad +(1-{\mu }_{1}){\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){Q}_{2}\alpha \left(t-\tau ,x\right)\mathrm{d}x\\ & \quad -{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\mathrm{m}},x\right){Q}_{2}m\left(t-{\tau }_{\mathrm{m}},x\right)\mathrm{d}x\\ & \quad +{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){Q}_{3}\beta \left(t,x\right)\mathrm{d}x\\ & \quad -\left(1-{\eta }_{2}\right){\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){Q}_{3}\beta \left(t-\sigma ,x\right)\mathrm{d}x\\ &\quad +(1-{\eta }_{1}){\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){Q}_{4}\beta \left(t-\sigma ,x\right)\mathrm{d}x\\ & \quad -{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t-{\sigma }_{\mathrm{m}},x\right){Q}_{4}\beta \left(t-{\sigma }_{\mathrm{m}},x\right)\mathrm{d}x\end{aligned}$$

(28)

$$\begin{aligned} \frac{\partial {V}_{3}}{\partial t}& =2{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){J}_{1p}\frac{\partial {e}_{\alpha }(t,x)}{\partial t}\mathrm{d}x\\ & \quad +2{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}\frac{\partial {e}_{\beta }(t,x)}{\partial t}\mathrm{d}x\\ &=2{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){J}_{1p}\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial {e}_{\alpha }(t,x)}{\partial {x}_{k}}\right)\right.\\ & \quad +\widehat{A}{e}_{\alpha }\left(t,x\right) -\widehat{A}\alpha \left(t,x\right)-{A}_{p}\alpha \left(t,x\right)\\ &\quad +{B}_{p}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\left.-M\alpha (t,x)\vphantom{\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial {e}_{\alpha }(t,x)}{\partial {x}_{k}}\right)\right.}\right]\mathrm{d}x\\ &\quad +2{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}^{*}\frac{\partial \beta \left(t,x\right)}{\partial {x}_{k}}\right)\right.\\ &\quad +\widehat{C}{e}_{\beta }\left(t,x\right)-\widehat{C}\beta \left(t,x\right)-{C}_{p}\beta \left(t,x\right)\\ &\quad +{\widetilde{D}}_{p}\alpha \left(t-\tau \left(t\right),x\right)-\left.N\beta \left(t,x\right)\vphantom{\left[\sum\limits_{k=1}^{l}\frac{\partial }{\partial {x}_{k}}\left({D}_{kp}\frac{\partial {e}_{\alpha }(t,x)}{\partial {x}_{k}}\right)\right.}\right]\mathrm{d}x\\ &\quad +{\int }_{\Omega }{{e}_{\alpha }}^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{1q}{e}_{\alpha }\left(t,x\right)\mathrm{d}x\\ &\quad +{\int }_{\Omega }{{e}_{\beta }}^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{2q}{e}_{\beta }\left(t,x\right)\mathrm{d}x,\end{aligned}$$

(29)

$$\begin{aligned} \frac{\partial {V}_{4}}{\partial t}& ={\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){R}_{3}{e}_{\alpha }\left(t,x\right)\mathrm{d}x\\ & \quad -(1-\dot{\tau }){\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){R}_{3}{e}_{\alpha }\left(t-\tau ,x\right)\mathrm{d}x\\ & \quad +(1-\dot{\tau }){\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){R}_{4}{e}_{\alpha }\left(t-\tau ,x\right)\mathrm{d}x\\ & \quad -{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\alpha },x\right){R}_{4}{e}_{\alpha }\left(t-{\tau }_{\mathrm{m}},x\right)\mathrm{d}x\\ &\quad +{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){R}_{5}{e}_{\beta }\left(t,x\right)\mathrm{d}x\\ &\quad -(1-\dot{\sigma }){\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){R}_{5}{e}_{\beta }\left(t-\sigma ,x\right)\mathrm{d}x\\ &\quad +\left(1-\dot{\sigma }\right){\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){R}_{6}{e}_{\beta }\left(t-\sigma ,x\right)\mathrm{d}x\\ &\quad -{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t-{\sigma }_{\mathrm{m}},x\right){R}_{6}{e}_{\beta }\left(t-{\sigma }_{\mathrm{m}},x\right)\mathrm{d}x\\ &\le {\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){R}_{3}{e}_{\alpha }\left(t,x\right)\mathrm{d}x\\ & \quad -(1-{\mu }_{1}){\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){R}_{3}{e}_{\alpha }\left(t-\tau ,x\right)\mathrm{d}x \\ & \quad +(1-{\mu }_{1}){\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){R}_{4}{e}_{\alpha }\left(t-\tau ,x\right)\mathrm{d}x\\ &\quad -{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\mathrm{m}},x\right){R}_{4}{e}_{\alpha }\left(t-{\tau }_{\mathrm{m}},x\right)\mathrm{d}x\\ &\quad +{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){R}_{5}{e}_{\beta }\left(t,x\right)\mathrm{d}x\\ &\quad -\left(1-{\eta }_{2}\right){\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){R}_{5}{e}_{\beta }\left(t-\sigma ,x\right)\mathrm{d}x\\ &\quad +\left(1-{\eta }_{1}\right){\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){R}_{6}{e}_{\beta }\left(t-\sigma ,x\right)\mathrm{d}x\\ &\quad -{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t-{\sigma }_{\mathrm{m}},x\right){R}_{6}{e}_{\beta }\left(t-{\sigma }_{\mathrm{m}},x\right)\mathrm{d}x,\end{aligned}$$

(30)

$$\begin{aligned} \frac{\partial {V}_{5}}{\partial t}& ={\tau }_{\mathrm{m}}^{2}{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){Q}_{5}\alpha \left(t,x\right)\mathrm{d}x\\ &\quad -{\tau }_{\mathrm{m}}{\int }_{\Omega }{\int }_{t-{\tau }_{\mathrm{m}}}^{t}{\alpha }^{\mathrm{T}}\left(s,x\right){Q}_{5}\alpha \left(s,x\right)\mathrm{d}s\mathrm{d}x\\ &\quad +{\sigma }_{\mathrm{m}}^{2}{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){Q}_{6}\beta \left(t,x\right)\mathrm{d}x\\ &\quad -{\sigma }_{\mathrm{m}}{\int }_{\Omega }{\int }_{t-{\sigma }_{\mathrm{m}}}^{t}{\beta }^{\mathrm{T}}\left(s,x\right){Q}_{6}\beta \left(s,x\right)\mathrm{d}s\mathrm{d}x\\ & {\le \tau }_{\mathrm{m}}^{2}{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){Q}_{5}\alpha \left(t,x\right)\mathrm{d}x\\ &\quad -{\int }_{\Omega }{\int }_{t-{\tau }_{m}}^{t}{\alpha }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s{Q}_{5}{\int }_{t-{\tau }_{m}}^{t}{\alpha }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s\mathrm{d}x\\ &\quad +{\sigma }_{\alpha }^{2}{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){Q}_{6}\beta \left(t,x\right)\mathrm{d}x\\ &\quad -{\int }_{\Omega }{\int }_{t-{\sigma }_{\mathrm{m}}}^{t}{\beta }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s{Q}_{6}{\int }_{t-{\sigma }_{\mathrm{m}}}^{t}{\beta }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s\mathrm{d}x.\end{aligned}$$

(31)

According to Lemma 3, we have

$$2{\int }_{\Omega }{\alpha }^{\mathrm{T}}(t,x){P}_{1p}\sum\limits_{k=1}^{l}\frac{\partial }{\partial {l}_{k}}\left({D}_{kp}\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)\mathrm{d}x\le -\frac{{\pi }^{2}}{2}{\int }_{\Omega }{\alpha }^{\mathrm{T}}(t,x){P}_{1p}{D}_{kp}\alpha (t,x)\mathrm{d}x \, \text{,} \, $$

(32)

$$2{\int }_{\Omega }{\beta }^{\mathrm{T}}(t,x){P}_{2p}\sum\limits_{k=1}^{l}\frac{\partial }{\partial {l}_{k}}\left({D}_{kp}^{*}\frac{\partial \beta (t,x)}{\partial {l}_{k}}\right)\mathrm{d}x\le -\frac{{\pi }^{2}}{2}{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}{D}_{kp}^{*}\beta \left(t,x\right)\mathrm{d}x , \, $$

(33)

$$2{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}(t,x){J}_{1p}\sum\limits_{k =1}^{l}\frac{\partial }{\partial {l}_{k}}\left({D}_{kp}\frac{\partial {e}_{\alpha }(t,x)}{\partial {l}_{k}}\right)\mathrm{d}x\le -\frac{{\pi }^{2}}{2}{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){{J}_{1p}D}_{kp}{g}_{1}\left(t,x\right)\mathrm{d}l , \, $$

(34)

$$2{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}(t,x){J}_{2p}\sum\limits_{k=1}^{l}\frac{\partial }{\partial {l}_{k}}\left({D}_{kp}^{*}\frac{\partial {e}_{\beta }(t,x)}{\partial {l}_{k}}\right)\mathrm{d}x\le -\frac{{\pi }^{2}}{2}{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}{D}_{kp}^{*}{e}_{\beta }\left(t,x\right)\mathrm{d}l , $$

(35)

where

$$\begin{aligned} &{\left({\alpha }^{\mathrm{T}}(t,x){P}_{1p}{D}_{kp}\frac{\partial \alpha (t,x)}{\partial {x}_{k}}\right)}_{k=1}^{L}\\ & \quad =\left({\alpha }^{\mathrm{T}}(t,x){P}_{1p}{D}_{1p}\frac{\partial \alpha (t,x)}{\partial {x}_{1}},\right.\\ &\quad\qquad \left.\cdots ,{\alpha }^{\mathrm{T}}(t,x){P}_{1p}{D}_{Lp}\frac{\partial \alpha (t,x)}{\partial {x}_{L}}\right)\end{aligned}$$

$$\begin{aligned} &{\left({\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}{D}_{kp}^{*}\frac{\partial \beta \left(t,x\right)}{\partial {x}_{k}}\right)}_{k=1}^{L}\\ & \quad =\left({\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}{D}_{1p}^{*}\frac{\partial \beta \left(t,x\right)}{\partial {x}_{1}},\right.\\ & \quad\qquad \left.\cdots ,{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}{D}_{Lp}^{*}\frac{\partial \beta \left(t,x\right)}{\partial {x}_{L}}\right)\end{aligned}$$

$$\begin{aligned} & {\left({e}_{\alpha }^{\mathrm{T}}(t,x){J}_{1p}{D}_{k}\frac{\partial {e}_{\alpha }(t,x)}{\partial {x}_{k}}\right)}_{k=1}^{L}\\ & \quad=\left({e}_{\alpha }^{\mathrm{T}}(t,x){J}_{1p}{D}_{1p}\frac{\partial {e}_{\alpha }(t,x)}{\partial {x}_{1}},\right.\\ & \qquad \quad \cdots,\left.{e}_{\alpha }^{\mathrm{T}}(t,x){J}_{1p}{D}_{Lp}\frac{\partial {e}_{\alpha }(t,x)}{\partial {x}_{L}}\right)\end{aligned}$$

$$\begin{aligned} & {\left({e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}{D}_{k}^{*}\frac{\partial {e}_{\beta }\left(t,x\right)}{\partial {x}_{k}}\right)}_{k=1}^{L}\\ & \quad =\left({e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}{D}_{1p}^{*}\frac{\partial {e}_{\beta }\left(t,x\right)}{\partial {x}_{1}}\right.,\\ & \quad\qquad\left.\cdots ,{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}{D}_{Lp}^{*}\frac{\partial {e}_{\beta }\left(t,x\right)}{\partial {x}_{L}}\right).\end{aligned}$$

The combination (27), (32) and (33) gives

$$\begin{aligned}\frac{\partial {V}_{1}}{\partial t}& \le 2{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1p}\Bigg[-\frac{{\pi }^{2}}{4}{D}_{kp}-{A}_{p}\alpha \left(t,x\right)\\ & \quad +{B}_{p}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\Bigg]\mathrm{d}x+2{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}\\ &\quad \Bigg[{-\frac{{\pi }^{2}}{4}D}_{kp}^{*}-{C}_{p}\beta \left(t,x\right)+{\widetilde{D}}_{p}\alpha \left(t-\tau \left(t\right),x\right)\Bigg]\mathrm{d}x \\ &\quad +{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{1q}\alpha \left(t,x\right)\mathrm{d}x\\ &\quad +{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{2q}\beta \left(t,x\right)\mathrm{d}x.\end{aligned}$$

(36)

Combine (29), (34) and (35), we can obtain

$$\begin{aligned} \frac{\partial {V}_{3}}{\partial t}& \le 2{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){J}_{1p}\Bigg[-\frac{{\pi }^{2}}{4}{D}_{kp}+\widehat{A}{e}_{\alpha }\left(t,x\right)\\ &\quad -\widehat{A}\alpha \left(t,x\right)-{A}_{p}\alpha \left(t,x\right)\\ &\quad +{B}_{p}g\left(p\left(t-\sigma \left(t\right),x\right)\right)-M\alpha (t,x)\Bigg]\mathrm{d}x\\ & \quad +2{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}\Bigg[{-\frac{{\pi }^{2}}{4}D}_{kp}^{*}+\widehat{C}{e}_{\alpha }\left(t,x\right)\\ &\quad -\widehat{C}\beta \left(t,x\right)-{C}_{p}\beta \left(t,x\right)\\ &\quad +{\widetilde{D}}_{p}\alpha \left(t-\tau \left(t\right),x\right)-N\beta \left(t,x\right)\Bigg]\mathrm{d}x\\ &\quad +{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{J}_{1q}{e}_{\alpha }\left(t,x\right)\mathrm{d}x\\ &\quad +{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{J}_{2q}{e}_{\beta }\left(t,x\right)\mathrm{d}x.\end{aligned}$$

(37)

Considering Eq. (4), for diagonal matrices ${\Lambda }_{1}>0$, ${\Lambda }_{2}>0$, the following inequalities holds:

$$2{g}^{\mathrm{T}}\left(\beta (t,x)\right){\Lambda }_{1}g\left(\beta (t,x)\right)-2{\beta }^{\mathrm{T}}\left(t,x\right){K\Lambda }_{1}g\left(\beta (t,x)\right)\le 0$$

(38)

$$\begin{aligned} & 2{g}^{\mathrm{T}}\left(\beta (t-\sigma (t),x)\right){\Lambda }_{2}g\left(\beta (t-\sigma (t),x)\right)\\ & \quad -2{\beta }^{\mathrm{T}}\left(t-\sigma \left(t\right),x\right){K\Lambda }_{2}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\le 0.\end{aligned}$$

(39)

Derivatives of ${V}_{\varpi }\left(t,x\right)$ can be formed as follows

$$\frac{\partial V(t,x)}{\partial t}=\sum\limits_{\varpi =1}^{5}\frac{\partial {V}_{\varpi }(t,x)}{\partial t}$$

(40)

Taking Eqs. (27)–(31) and Eqs. (36)–(40) yields

$$\begin{aligned} \frac{\partial \widehat{V}(t,x)}{\partial t}&\!\le\! {\int }_{\Omega }\left[\vphantom{\left.\sum\limits_{q=1}^{N}{\pi }_{pq}{J}_{2q}{e}_{\beta }\left(t,x\right)\right]}{\alpha }^{\mathrm{T}}\left(t,x\right)\left(-\frac{{\pi }^{2}}{2}{P}_{1p}{D}_{kp}\!-\!2{P}_{1p}{A}_{p}\!+\!{Q}_{1}\right.\right.\\ & \quad \left.+{\tau }_{\mathrm{m}}^{2}{Q}_{5}\vphantom{\left(-\frac{{\pi }^{2}}{2}{P}_{1p}{D}_{kp}\!-\!2{P}_{1p}{A}_{p}\!+\!{Q}_{1}\right.}\right)\alpha \left(t,x\right) +{\beta }^{\mathrm{T}}\left(t,x\right)\\ & \quad \left(-\frac{{\pi }^{2}}{2}{P}_{2p}{D}_{kp}^{*}\!-\!2{P}_{2p}{C}_{p}+{Q}_{3}+{\sigma }_{\mathrm{m}}^{2}{Q}_{6}\right)\beta (t,x)\\ & \quad +2{\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1p}{B}_{p}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\\ & \quad +2{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}{D}_{p}\alpha \left(t-\tau \left(t\right),x\right)\\ & \quad +\left({\mu }_{2}-1\right){\alpha }^{\mathrm{T}}\left(t-\tau \left(t\right),x\right){Q}_{1}\alpha \left(t-\tau \left(t\right),x\right)\\& \quad +\left(1-{\mu }_{1}\right){\alpha }^{\mathrm{T}}\left(t-\tau \left(t\right),x\right){Q}_{2}\alpha \left(t-\tau \left(t\right),x\right)\\ & \quad -{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\mathrm{m}},x\right){Q}_{2}\alpha \left(t-{\tau }_{\mathrm{m}},x\right)\\& \quad +\left({\eta }_{2}-1\right){\beta }^{\mathrm{T}}\left(t-\sigma \left(t\right),x\right){Q}_{3}\beta \left(t-\sigma \left(t\right),x\right)\\ & \quad +\left({1-\eta }_{1}\right){\beta }^{\mathrm{T}}\left(t-\sigma \left(t\right),x\right){Q}_{4}\beta \left(t-\sigma \left(t\right),x\right)\\& \quad -{\beta }^{\mathrm{T}}\left(t-{\sigma }_{\mathrm{m}},x\right){Q}_{4}\beta \left(t-{\sigma }_{\mathrm{m}},x\right)\\ & \quad +{e}_{\alpha }^{\mathrm{T}}\left(t,x\right)\left(-\frac{{\pi }^{2}}{2}{J}_{1p}{D}_{kp}-2{J}_{1p}\widehat{A}+{R}_{3}\right){e}_{\alpha }\left(t,x\right)\\ & \quad +{e}_{\beta }^{\mathrm{T}}\left(t,x\right)\left(-\frac{{\pi }^{2}}{2}{J}_{2p}{D}_{kp}^{*}-2{R}_{2p}\widehat{C}+{R}_{5}\right){e}_{\beta }\left(t,x\right)\\ & \quad +2{\alpha }^{\mathrm{T}}\left(t,x\right)\left(-{A}_{p}-\widehat{A}-M\right){J}_{1p}{e}_{\alpha }(t,x)\\& \quad +2{\beta }^{\mathrm{T}}\left(t,x\right)\left(-{C}_{p}-\widehat{C}-N\right){J}_{2p}{e}_{\beta }(t,x)\\ & \quad +2{g}^{\mathrm{T}}\left(\beta \left(t-\sigma \left(t\right),x\right)\right){B}_{p}^{\mathrm{T}}{J}_{1p}{e}_{\alpha }(t,x)\\& \quad +2{\alpha }^{\mathrm{T}}\left(t-\tau \left(t\right),x\right){\widetilde{D}}_{p}^{\mathrm{T}}{J}_{2p}{e}_{\beta }(t,x)\\ & \quad -\left(1-{\mu }_{2}\right){e}_{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){R}_{3}{e}_{\alpha }\left(t-\tau ,x\right)\\& \quad +(1-{\mu }_{1}){e}_{\alpha }^{\mathrm{T}}\left(t-\tau ,x\right){R}_{4}{e}_{\alpha }\left(t-\tau ,x\right)\\ & \quad -{e}_{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\mathrm{m}},x\right){R}_{4}{e}_{\alpha }\left(t-{\tau }_{\mathrm{m}},x\right)\\ & \quad -\left(1-{\eta }_{2}\right){e}_{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){R}_{5}{e}_{\beta }\left(t-\sigma ,x\right)\\ & \quad +\left(1-{\eta }_{1}\right){e}_{\beta }^{\mathrm{T}}\left(t-\sigma ,x\right){R}_{6}{e}_{\beta }\left(t-\sigma ,x\right)\\ & \quad -{e}_{\beta }^{\mathrm{T}}\left(t-{\sigma }_{\mathrm{m}},x\right){R}_{6}{e}_{\beta }\left(t-{\sigma }_{\mathrm{m}},x\right)\\ & \quad -{\int }_{t-{\tau }_{\mathrm{m}}}^{t}{\alpha }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s{Q}_{5}{\int }_{t-{\tau }_{\mathrm{m}}}^{t}{\alpha }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s\\ & \quad -{\int }_{t-{\sigma }_{\mathrm{m}}}^{t}{\beta }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s{Q}_{6}{\int }_{t-{\sigma }_{\mathrm{m}}}^{t}{\beta }^{\mathrm{T}}\left(s,x\right)\mathrm{d}s\\ & \quad +2{\beta }^{\mathrm{T}}\left(t,x\right){K\Lambda }_{1}g\left(\beta \left(t,x\right)\right)\\ & \quad -2{g}^{\mathrm{T}}\left(\beta \left(t,x\right)\right){\Lambda }_{1}g\left(\beta \left(t,x\right)\right)\\ & \quad +2{\beta }^{\mathrm{T}}\left(t-\sigma \left(t\right),x\right){K\Lambda }_{2}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\\ & \quad -2{g}^{\mathrm{T}}\left(\beta (t-\sigma (t),x)\right){\Lambda }_{2}g\left(\beta (t-\sigma (t),x)\right)\\ & \quad +{\alpha }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{1q}\alpha \left(t,x\right)\\ & \quad {+\beta }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{P}_{2q}\beta \left(t,x\right) \\ & \quad {+e}_{\alpha }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}{\pi }_{pq}{J}_{1q}{e}_{\alpha }\left(t,x\right)\\ & \quad {+e}_{\beta }^{\mathrm{T}}\left(t,x\right)\left.\sum\limits_{q=1}^{N}{\pi }_{pq}{J}_{2q}{e}_{\beta }\left(t,x\right)\right]\mathrm{d}x\\& ={\int }_{\Omega }{\theta }^{\mathrm{T}}\left(t,x\right){\Xi }_{p}\theta \left(t,x\right)\mathrm{d}x<0\end{aligned}$$

(41)

for $\theta (t,x)\ne 0$, where

$$\begin{aligned} \theta \left(t,x\right)& =\Bigg[{\alpha }^{\mathrm{T}}(t,x),{\beta }^{\mathrm{T}}\left(t-\sigma \left(t\right),x\right),{g}^{\mathrm{T}}\left(\beta (t-\sigma \left(t\right),x\right)),\\ & \quad {\alpha }^{\mathrm{T}}\left(t-\tau \left(t\right),x\right),{\beta }^{\mathrm{T}}\left(t,x\right),{g}^{\mathrm{T}}\left(\beta (t,x\right)),\\ & \quad {e}_{\alpha }^{\mathrm{T}}\left(t,x\right),{{e}_{\alpha }^{\mathrm{T}}\left(t-\tau \left(t\right),x\right), {e}_{\beta }^{\mathrm{T}}\left(t,x\right),e}_{\beta }^{\mathrm{T}}\left(t-\sigma \left(t\right),x\right),\\ & \quad \underset{t-{\tau }_{\mathrm{m}}}{\overset{t}{\int }}{\alpha }^{\mathrm{T}}(s,x)\mathrm{d}s,\underset{t-{\sigma }_{\mathrm{m}}}{\overset{t}{\int }}{\beta }^{\mathrm{T}}(s,x)\mathrm{d}s,{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\mathrm{m}},x\right),\\ & \quad \beta \left(t-{\sigma }_{\mathrm{m}},x\right),\\ & \quad {{e}_{\alpha }^{\mathrm{T}}\left(t-{\tau }_{\mathrm{m}},x\right), {e}_{\beta }^{\mathrm{T}}\left(t-{\sigma }_{\mathrm{m}},x\right)\Bigg]}^{\mathrm{T}}.\end{aligned}$$

Theorem 2

For given scalars ${\tau }_{\mathrm{m}}$, ${\sigma }_{\mathrm{m}}$, ${\mu }_{1}$, ${\mu }_{2}$ ${\eta }_{1}$, ${\eta }_{2}$ ${\varepsilon }_{1}>0$, ${\varepsilon }_{2}>0$, ${\varepsilon }_{3}>0$, ${\varepsilon }_{4}>0$ and given estimator parameters $\widehat{A}$, $\widehat{C}$, $M$ and $N$, Eq. (11) under Dirichlet boundary conditions is stable with uncertainties if there exist matrices ${P}_{1p}^{\mathrm{T}}={P}_{1p}>0$, ${P}_{2p}^{\mathrm{T}}={P}_{2p}>0$, ${J}_{1p}^{\mathrm{T}}={J}_{1p}>0$, ${J}_{2p}^{\mathrm{T}}={J}_{2p}>0$, ${R}_{\iota }^{\mathrm{T}}={R}_{\iota }>0$, $(\iota =3,\cdots , 6)$, ${{\Lambda }_{1}^{\mathrm{T}}=\Lambda }_{1}>0$, ${{\Lambda }_{2}^{\mathrm{T}}=\Lambda }_{2}>0$, ${Q}_{\delta }^{\mathrm{T}}={Q}_{\delta }>0$, $(\delta =1,\cdots , 6)$, such that the following linear matrix in equality (LMIs) hold:

$${\Xi }_{p}=\left[\begin{array}{c}{\Sigma }_{11p}\\ *\end{array}\quad \begin{array}{c}{\Sigma }_{12p}\\ {\Sigma }_{22p}\end{array}\right]<0,$$

(42)

where${\Sigma }_{11p}=\left[\begin{array}{c}{\Pi }_{1p}\\ *\\ *\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ {\Phi }_{1}\\ *\\ *\\ *\\ *\end{array}\quad \begin{array}{c}{P}_{1p}{B}_{p}\\ K{\Lambda }_{2}\\ {\Gamma }_{p}\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ {\Phi }_{2}\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ {\widetilde{D}}_{p}^{T}{P}_{2p}\\ {\Pi }_{2p}\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ K{\Lambda }_{1}\\ -2{\Lambda }_{1}\end{array}\right],$

$${\Sigma }_{12p}=\left[\begin{array}{c}{\Pi }_{3p}\\ 0\\ {B}_{p}^{\mathrm{T}}{J}_{1p}\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ {\widetilde{D}}_{p}^{T}{J}_{2p}\\ {\Pi }_{4p}\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],$$

$${\Sigma }_{22p}=\left[\begin{array}{c}{\Pi }_{5p}\\ *\\ *\\ *\\ *\\ *\\ *\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ {\Phi }_{3}\\ *\\ *\\ *\\ *\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ 0\\ {\Pi }_{6p}\\ *\\ *\\ *\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ {\Phi }_{4}\\ *\\ *\\ *\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ {-Q}_{5}\\ *\\ *\\ *\\ *\\ *\end{array} \quad\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ {-Q}_{6}\\ *\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-Q}_{2}\\ *\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-Q}_{4}\\ *\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-R}_{4}\\ *\end{array}\quad \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ {-R}_{6}\end{array}\right]$$

$${\Pi }_{1p}=-{\frac{{\pi }^{2}}{2}P}_{1p}{D}_{kp}-2{P}_{1p}{A}_{p}+{Q}_{1}+{\tau }_{\mathrm{m}}^{2}{Q}_{5}+{2\varepsilon }_{1}^{-1}{P}_{1p}{M}_{p}{M}_{p}^{\mathrm{T}}+{(\varepsilon }_{1}{P}_{1p}{+{\varepsilon }_{3}{J}_{1p})N}_{Ap}^{\mathrm{T}}{N}_{Ap}+\sum\limits_{q=1}^{N}{(\pi }_{pq}+\Delta {\pi }_{pq}){P}_{1q},$$

$${\Pi }_{2p}=-\frac{{\pi }^{2}}{2}{P}_{2p}{D}_{kp}^{*}-2{P}_{2p}{C}_{p}+{Q}_{3}+{\sigma }_{\mathrm{m}}^{2}{Q}_{6}+{2\varepsilon }_{2}^{-1}{P}_{2p}{M}_{p}{M}_{p}^{\mathrm{T}}{+({\varepsilon }_{2}{P}_{2p}+{\varepsilon }_{4}{J}_{2p})N}_{Cp}^{\mathrm{T}}{N}_{Cp}+\sum\limits_{q=1}^{N}{(\pi }_{pq}+\Delta {\pi }_{pq}){P}_{2q},$$

$${\Pi }_{3p}=-\left({A}_{p}^{T}-{\widehat{A}}^{\mathrm{T}}-{M}^{\mathrm{T}}\right){J}_{1p},$$

$${\Pi }_{4p}=\left(-{C}_{p}^{T}-{\widehat{C}}^{T}-{N}^{T}\right){J}_{2p},$$

$${\Pi }_{5p}=-\frac{{\pi }^{2}}{2}{J}_{1p}{D}_{kp}-2{J}_{1p}\widehat{A}+{R}_{3}+\sum\limits_{q=1}^{N}{(\pi }_{pq}+\Delta {\pi }_{pq}){J}_{1q}+ \text{ } {2\varepsilon }_{3}^{-1}{J}_{1p}{M}_{p}{M}_{p}^{\mathrm{T}} \, ,$$

$${\Pi }_{6p}=-\frac{{\pi }^{2}}{2}{J}_{2p}{D}_{kp}^{*}-2{J}_{2p}\widehat{C}+{R}_{5}\sum\limits_{q=1}^{N}{(\pi }_{pq}+\Delta {\pi }_{pq}){J}_{2q}{+2\varepsilon }_{4}^{-1}{J}_{2p}{M}_{p}{M}_{p}^{\mathrm{T}},$$

${\Gamma }_{p}={{(\varepsilon }_{1}{P}_{1p}+{\varepsilon }_{3}{J}_{1p}){N}_{Bp}^{\mathrm{T}}{N}_{Bp}-2\Lambda }_{2},$

$${\Phi }_{1}=\left({\eta }_{2}-1\right){Q}_{3}+(1-{\eta }_{1}){Q}_{4},$$

$${\Phi }_{2}=\left({\mu }_{2}-1\right){Q}_{1}+(1-{\mu }_{1}){Q}_{2}+({\varepsilon }_{2}{P}_{2p}+{\varepsilon }_{4}{R}_{2p}){N}_{Dp}^{\mathrm{T}}{N}_{Dp},$$

$${\Phi }_{3}=\left({\mu }_{2}-1\right){R}_{3}+(1-{\mu }_{1}){R}_{4},$$

$${\Phi }_{4}=\left({\eta }_{2}-1\right){R}_{5}+\left(1-{\eta }_{1}\right){R}_{6}.$$

Proof

Consider the same Lyapunov–Krasovskii functional as Theorem 1.

The derivative of ${V}_{1}\left(t,x\right)$ can be computed as

$$\begin{aligned} \frac{\partial {V}_{1}}{\partial t}&\le 2{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1p}\Bigg[-\frac{{\pi }^{2}}{4}{D}_{kp}-({A}_{p}+\Delta {A}_{p})\alpha \left(t,x\right)\\ & \quad +({B}_{p}+\Delta {B}_{p})g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\Bigg]\mathrm{d}x\\& \quad +2{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}\Bigg[{-\frac{{\pi }^{2}}{4}}D_{kp}^{*}-{(C}_{p}+{\Delta C}_{p})\beta \left(t,x\right)\\ & \quad +{(\widetilde{D}}_{p}+\Delta {\widetilde{D}}_{i})\alpha \left(t-\tau \left(t\right),x\right)\Bigg]\mathrm{d}x\\& \quad +{\int }_{\Omega }{\alpha }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}({\pi }_{pq}+\Delta {\pi }_{pq}){P}_{1q}\alpha \left(t,x\right)\mathrm{d}x\\ & \quad +\sum\limits_{j=1}^{N}{\int }_{\Omega }{\beta }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}({\pi }_{pq}+\Delta {\pi }_{pq}){P}_{2q}\beta \left(t,x\right)\mathrm{d}x,\end{aligned}$$

(43)

Based on Assumption 2 and Lemma 5, we have

$$\begin{aligned}& {\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1p}\left[-\Delta {A}_{p}\alpha \left(t\right)+\Delta {B}_{p}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\right]\\ &\quad ={\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1p}{M}_{p}{E}_{p}\left[-{N}_{Ap}\alpha \left(t,x\right)+{N}_{Bp}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\right]\\ & \quad \le {2\varepsilon }_{1}^{-1}{\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1p}{M}_{p}{M}_{p}^{\mathrm{T}}\alpha \left(t,x\right)\\ & \qquad +{\varepsilon }_{1}{\alpha }^{\mathrm{T}}\left(t,x\right){P}_{1p}{N}_{Ap}^{\mathrm{T}}{N}_{Ap}\alpha \left(t,x\right)\\ & \qquad +{\varepsilon }_{1}{g}^{\mathrm{T}}\left(\beta \left(t-\sigma \left(t\right),x\right)\right){P}_{1p}{N}_{Bp}^{\mathrm{T}}{N}_{Bp}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\end{aligned}$$

(44)

and

$$\begin{aligned} & {\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}\left[-\Delta {C}_{p}\alpha \left(t\right)+\Delta {\widetilde{D}}_{p}\alpha \left(t-\tau \left(t\right),x\right)\right]\\ &\quad ={\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}{M}_{p}{E}_{p}\left[-{N}_{Cp}\beta \left(t,x\right)+{N}_{Dp}\alpha \left(t-\tau \left(t\right),x\right)\right]\\ &\quad \le {2\varepsilon }_{2}^{-1}{\beta }^{\mathrm{T}}\left(t,x\right){P}_{2p}{M}_{p}{M}_{p}^{\mathrm{T}}\beta \left(t,x\right)\\ &\qquad +{\varepsilon }_{2}{\beta }^{\mathrm{T}}\left(t,x\right){{P}_{2p}N}_{Cp}^{\mathrm{T}}{N}_{Cp}\beta \left(t,x\right)\\ &\qquad +{\varepsilon }_{2}{\alpha }^{\mathrm{T}}\left(t-\tau \left(t\right),x\right){P}_{2p}{N}_{Dp}^{\mathrm{T}}{N}_{Dp}\alpha \left(t-\tau \left(t\right),x\right),\end{aligned}$$

(45)

where ${\varepsilon }_{1}$, ${\varepsilon }_{2}>0.$

The derivative of ${V}_{3}\left(t,x\right)$ can be computed as

$$\begin{aligned}\frac{\partial {V}_{3}}{\partial t}& \le 2{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){J}_{1p}\Bigg[-\frac{{\pi }^{2}}{4}{D}_{kp}+\widehat{A}{e}_{\alpha }\left(t,x\right)\\ &\quad -\widehat{A}\alpha \left(t,x\right)-({A}_{p}+\Delta {A}_{p})\alpha \left(t,x\right)\\ &\quad +({B}_{p}+\Delta {B}_{p})g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\\ &\quad -M\alpha (t,x)\Bigg]\mathrm{d}x+2{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}\\ &\quad \Bigg[-{\frac{{\pi }^{2}}{4}}D_{ki}^{*}+\widehat{C}{e}_{\beta }\left(t,x\right)-\widehat{C}\beta \left(t,x\right)-{(C}_{p}+{\Delta C}_{p})\beta \left(t,x\right)\\ &\quad +{(\widetilde{D}}_{p}+\Delta {\widetilde{D}}_{p})\alpha \left(t-\tau \left(t\right),x\right)-N\beta \left(t,x\right)\Bigg]\mathrm{d}x\\ &\quad +{\int }_{\Omega }{e}_{\alpha }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}\left({\pi }_{pq}+\Delta {\pi }_{pq}\right){J}_{1q}{e}_{\alpha }\left(t,x\right)\mathrm{d}x\\ & \quad +{\int }_{\Omega }{e}_{\beta }^{\mathrm{T}}\left(t,x\right)\sum\limits_{q=1}^{N}\left({\pi }_{pq}+\Delta {\pi }_{pq}\right){J}_{2q}{e}_{\beta }\left(t,x\right)\mathrm{d}x.\end{aligned}$$

(46)

Using Assumption 2 and Lemma 5, we have

$$\begin{aligned} & {e}_{\alpha }^{\mathrm{T}}\left(t,x\right){R}_{1p}\left[\Delta {A}_{p}\alpha \left(t\right)+\Delta {B}_{p}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\right]\\ &\quad ={e}_{\alpha }^{\mathrm{T}}\left(t,x\right){R}_{1p}{M}_{p}{E}_{p}\left[-{N}_{Ap}\alpha \left(t,x\right)+{N}_{Bp}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\right]\\ &\quad \le {2\varepsilon }_{3}^{-1}{e}_{\alpha }^{\mathrm{T}}\left(t,x\right){J}_{1p}{M}_{p}{M}_{p}^{\mathrm{T}}{e}_{\alpha }\left(t,x\right)\\ &\qquad +{\varepsilon }_{3}{\alpha }^{\mathrm{T}}\left(t,x\right){J}_{1p}{N}_{Ap}^{\mathrm{T}}{N}_{Ap}\alpha \left(t,x\right)\\ &\qquad +{\varepsilon }_{3}{g}^{\mathrm{T}}\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\\ &\qquad {J}_{1p}{N}_{Bp}^{\mathrm{T}}{N}_{Bp}g\left(\beta \left(t-\sigma \left(t\right),x\right)\right)\end{aligned}$$

(47)

and

$$\begin{aligned} {e}_{\beta }^{\mathrm{T}} & \left(t,x\right){R}_{2p}\left[-{\Delta C}_{p}\beta \left(t\right)+\Delta {\widetilde{D}}_{p}\alpha \left(t-\tau \left(t\right),x\right)\right]\\ &\quad ={e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}{M}_{p}{E}_{p}\left[-{N}_{Cp}\beta \left(t,x\right)+{N}_{Dp}\alpha \left(t-\tau \left(t\right),x\right)\right]\\ &\quad \le {2\varepsilon }_{4}^{-1}{e}_{\beta }^{\mathrm{T}}\left(t,x\right){J}_{2p}{M}_{p}{M}_{p}^{\mathrm{T}}{e}_{\beta }\left(t,x\right)\\ &\qquad +{\varepsilon }_{4}{\beta }^{\mathrm{T}}\left(t,x\right) {J}_{2p}{N}_{Cp}^{\mathrm{T}}{N}_{Cp}\beta \left(t,x\right)\\ &\qquad +{\varepsilon }_{4}{\alpha }^{\mathrm{T}}\left(t-\tau \left(t\right),x\right){{J}_{2p}N}_{Dp}^{\mathrm{T}}{N}_{Dp}\alpha \left(t-\tau \left(t\right),x\right),\end{aligned}$$

(48)

where ${\varepsilon }_{3}$, ${\varepsilon }_{4}>0$.

Combining (28), (30), (31), (40) and (43)–(48), we can deduce that

$$\frac{\partial \widehat{V}(t,x)}{\partial t}={\int }_{\Omega }{\theta }^{T}\left(t,x\right){\Xi }_{p}\theta \left(t,x\right)\mathrm{d}x<0.$$

(49)

Numerical simulation

In this section, we provide two numerical examples to demonstrate effectiveness and applicability of the proposed state observer.

Example 1

Equation (8) is assumed to have two genes with the following parameters.

$${A}_{1}=\left[\begin{array}{c@{\quad}c}1& 0\\ 0& 1\end{array}\right]{A}_{2}=\left[\begin{array}{c@{\quad}c}2& 0\\ 0& 2\end{array}\right]{B}_{1}=\left[\begin{array}{c@{\quad}c}1& 2\\ 0.8& 0\end{array}\right]{B}_{2}=\left[\begin{array}{c@{\quad}c}-1& 0\\ 1& 2\end{array}\right]$$

$${C}_{1}=\left[\begin{array}{c@{\quad}c}2& 0\\ 0& 2\end{array}\right]{C}_{2}=\left[\begin{array}{c@{\quad}c}3& 0\\ 0& 3\end{array}\right]{\widetilde{D}}_{1}=\left[\begin{array}{c@{\quad}c}1& 0\\ 0& 1\end{array}\right]{\widetilde{D}}_{2}=\left[\begin{array}{c@{\quad}c}0.8& 0\\ 0& 0.8\end{array}\right]$$

$$\begin{aligned} & {L}_{1}={L}_{2}=1{D}_{k1}=\left[\begin{array}{c@{\quad}c}0.1& 0\\ 0& 0.1\end{array}\right]{D}_{k2}=\left[\begin{array}{c@{\quad}c}0.15& 0\\ 0& 0.15\end{array}\right] \\ & {D}_{k1}^{*}=\left[\begin{array}{c@{\quad}c}0.2& 0\\ 0& 0.2\end{array}\right]{D}_{k2}^{*}=\left[\begin{array}{c@{\quad}c}0.1& 0\\ 0& 0.1\end{array}\right](k=\mathrm{1,2})\end{aligned}$$

$$\begin{aligned} &{N}_{A1}= \left[\begin{array}{c@{\quad}c}0.1& 0.2\end{array}\right]{N}_{A2}=\left[\begin{array}{c@{\quad}c}0.1& -0.2\end{array}\right]\\ & {N}_{B1}=\left[\begin{array}{c@{\quad}c}0.2& -0.3\end{array}\right]{N}_{B2}=\left[\begin{array}{c@{\quad}c}0.1& 0.1\end{array}\right]\\ & {N}_{C1}=\left[\begin{array}{c@{\quad}c}0.3& -0.3\end{array}\right]{N}_{C2}=\left[-\begin{array}{c@{\quad}c}0.4& 0.1\end{array}\right];\end{aligned}$$

$$\begin{aligned} & {N}_{D1}=\left[-\begin{array}{c@{\quad}c}0.2& -0.1\end{array}\right]{N}_{D2}=\left[\begin{array}{c@{\quad}c}0.2& -0.1\end{array}\right]\\ & {M}_{1}={\left[\begin{array}{c@{\quad}c}0.2& 0.1\end{array}\right]}^{\mathrm{T}}{M}_{2}={\left[\begin{array}{c@{\quad}c}0.12& 0.15\end{array}\right]}^{\mathrm{T}}\\ & {E}_{1}=\mathrm{sin}\left(t\right){E}_{2}=\mathrm{cos}\left(t\right),\end{aligned}$$

$$\tau =0.3+0.6\times \mathrm{sin}(t)\sigma =0.7+0.4\times \mathrm{sin}(t).$$

The mode evolution of Eq. (8) is shown in Fig. 1, it is obvious that there are two switching modes, and the transmission probability and the uncertain probabilities are assumed as $\pi =\left[\begin{array}{c@{\quad}c}-3& 3\\ 7& -7\end{array}\right]$, and the uncertain probabilities is $\Delta \pi =\left[\begin{array}{c@{\quad}c}0.2& -0.2\\ 0.3& -0.3\end{array}\right]$, the form of Hill function is assumed as ${f}_{i}\left(x\right)=\frac{{x}^{2}}{1+{x}^{2}}$, $(i=\mathrm{1,2},\cdots ,n)$.

Fig. 1

Mode evolution

When ${\tau }_{\mathrm{m}}=0.9$, ${\sigma }_{\mathrm{m}}=1.1$, ${\mu }_{1}=-0.6$, ${\mu }_{2}=0.6$, ${\eta }_{1}=-0.4$, ${\eta }_{2}=0.4$ and ${\varepsilon }_{1}={\varepsilon }_{2}={\varepsilon }_{3}={\varepsilon }_{4}=1$, desired estimator gain matrices can be designed by Matlab LMI Toolbox as follows:

$$\begin{aligned} & \widehat{A}=\left[\begin{array}{c@{\quad}c}0.0224& 0\\ 0& 0.0403\end{array}\right]\widehat{C}=\left[\begin{array}{c@{\quad}c}0.0231 & 0\\ 0& 0.0239\end{array}\right]\\ & M=\left[\begin{array}{c@{\quad}c}-0.0615& 0\\ 0& -0.0388\end{array}\right]N=\left[\begin{array}{c@{\quad}c}-0.0451& 0\\ 0& -0.0417\end{array}\right].\end{aligned}$$

Figures 2 and 3 depict the true concentration of mRNA concentration ${\alpha }_{i}\left(t,x\right) (i=\mathrm{1,2})$, and protein concentration ${\beta }_{i}(t,x)$ under Dirichlet boundary conditions. Figures 4 and 5 exhibit the concentration of estimation of mRNA concentration ${\alpha }_{i}(t,x)$, and concentration of estimation of protein concentration ${\beta }_{i}(t,x)$. The estimation errors of the concentrations of mRNA and protein under Dirichlet boundary conditions are described in Figs. 6, 7. It is noticed that the estimation errors between Eqs. (8) and (9) approaches to zero asymptotically.

Fig. 2

The trajectory of ${\alpha }_{1}(t,x)$ and ${\alpha }_{2}(t,x)$ under Dirichlet boundary conditions

Fig. 3

The trajectory of ${\beta }_{1}(t,x)$ and ${\beta }_{2}(t,x)$ under Dirichlet boundary conditions

Fig. 4

The trajectory of ${\widehat{\alpha }}_{1}(t,x)$ and ${\widehat{\alpha }}_{2}(t,x)$ under Dirichlet boundary conditions

Fig.5

The trajectory of ${\widehat{\beta }}_{1}(t,x)$ and ${\widehat{\beta }}_{2}(t,x)$ under Dirichlet boundary condition

Fig. 6

The trajectory of ${e}_{\alpha 1}(t,x)$ and ${e}_{\alpha 2}(t,x)$ under Dirichlet boundary condition

Fig. 7

The trajectory of ${e}_{\beta 1}(t,x)$ and ${e}_{\beta 2}(t,x)$ under Dirichlet boundary condition

Example 2

If ${A}_{i}=A,$ ${B}_{i}=B$, ${C}_{i}=C,$ ${D}_{i}=D,$ ${D}_{ki}={D}_{k}$ and ${D}_{ki}^{*}={D}_{k}^{*}$, Eq. (8) will degenerate to GRN without Markov jumping parameters.

To compare the conservation of our approach with the other articles, $A,$ $B,$ $C$, $\widetilde{D}$, ${D}_{k}$ and ${D}_{k}^{*}$ are given as follows: $A=\left[\begin{array}{c@{\quad}c@{\quad}c}0.2& 0& 0\\ 0& 1.1& 0\\ 0& 0& 1.2\end{array}\right], B=\left[\begin{array}{c@{\quad}c@{\quad}c}0& 0& -0.5\\ -0.5& 0& 0\\ 0& -0.5& 0\end{array}\right], C=\left[\begin{array}{c@{\quad}c@{\quad}c}0.3& 0& 0\\ 0& 0.7& 0\\ 0& 0& 1.3\end{array}\right], \widetilde{D}\!=\!\left[\begin{array}{c@{\quad}c@{\quad}c}1& 0& 0\\ 0& 0.4& 0\\ 0& 0& 0.7\end{array}\right]{L}_{1}\!=\!{L}_{2}\!=\!{L}_{3}\!=\!1;$

$${D}_{k}=\left[\begin{array}{c@{\quad}c@{\quad}c}0.1& 0& 0\\ 0& 0.1& 0\\ 0& 0& 0.1\end{array}\right], {D}_{k}^{*} =\left[\begin{array}{c@{\quad}c@{\quad}c}0.2& 0& 0\\ 0& 0.2& 0\\ 0& 0& 0.2\end{array}\right]\quad \left(k=\mathrm{1,2},3\right).$$

If ${\mu }_{2}={\eta }_{2}=\mu \in \left\{0.7, 0.83, 0.93, 0.94, 0.99, 1\right\}$, the maximum delay (${\tau }_{\mathrm{m}}={\sigma }_{\mathrm{m}}$) is obtained by utilizing Theorem 1. The following illustrations are made for Table 1:

• (I) When $\mu \le 0.83$, all LMI conditions discussed here are feasible.

• (II) When $0.93\ge \mu \ge 0.99$, the LMI conditions in [30, Theorem 1] is unfeasible, but the LMI conditions in [31, Theorem 1], [25, Theorem 1] and [13, Remark 2] are feasible, delay upper bounds from of proposed Theorem 1 are $\infty$.

• (III) When $\mu =1$, all LMI conditions discussed here are unfeasible except [25, Theorem 1] and [13, Remark 2].

Table 1 Upper bounds on ${\tau }_{\mathrm{m}}={\sigma }_{\mathrm{m}}$ with different ${\mu }_{2}={\eta }_{2}=\mu$

Taking above illustrations into consideration, we can obtain the conclusion such that, in the range of $\mu <1,$ Theorem 1 of this paper is less conservative than [30, Theorem 1], [31, Theorem 1], [25, Theorem 1] and [13, Remark 2].

Conclusion

In this paper, state estimation is investigated for GRNs with both Markovian jumping parameters and reaction diffusion terms, where the jumping parameters have been governed by a homogeneous Markovian chain. A linear parameter estimator is designed for estimation of concentrations of mRNA and protein at first, and the estimation errors are stable by utilization of Lyapunov–Krasovskii functions and sufficient conditions. Two simulations as examples demonstrate the applicability and effectiveness of the obtained result.