# Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

- Based on proposition 1 in [55] and since $f,g:{\left[0,\infty \right)}^{3}\to \mathbb{R}$ are continuous and differentiable functions in which $f(t,0,\eta )\ge 0$ and $g(t,\xi ,0)\ge 0$, for all $t,\xi ,\eta \ge 0$, we can deduce that system (1) has a local unique solution $({u}_{1},{u}_{2})$ on $\Omega \times \left[0,{T}^{\ast}\right)$, and furthermore there are two continuous functions ${N}_{1},{N}_{2}:\left[0,{T}^{\ast}\right)\to \left[0,\infty \right)$ such that:$$\begin{array}{cc}0\le {u}_{1}(x,t)\le {N}_{1}\left(t\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\le {u}_{2}(x,t)\le {N}_{2}\left(t\right),& \phantom{\rule{4.pt}{0ex}}\mathrm{where}\phantom{\rule{4.pt}{0ex}}(x,t)\in \Omega \times \left[0,{T}^{\ast}\right).\end{array}$$
- There is a constant $\gamma \ge 1$ and a continuous function ${L}_{0}:{\left[0,\infty \right)}^{2}\to \left[0,\infty \right)$ such that $\left|g(t,\xi ,\eta )\right|\le {L}_{0}(t,r){(1+\eta )}^{\gamma}$, for all $t,\xi ,\eta \ge 0$ with $\xi \le r$. This, consequently, implies:$$\left|g(t,\xi ,\eta )\right|\le a+b\eta +{\xi}^{2}\eta \le (a+b+\xi ){(1+\eta )}^{2}.$$
- There is a continuous function ${\mu}_{0}:{\left[0,\infty \right)}^{2}\to \left[0,\infty \right)$ so that $f(t,\xi ,\eta )+g(t,\xi ,\eta )\le {\mu}_{0}(t,r)$, $\forall t,\xi ,\eta \ge 0$ with $\xi \le r$. This, consequently, implies:$$f(t,\xi ,\eta )+g(t,\xi ,\eta )=a-\xi \le a.$$
- The solution ${u}_{1}(x,t)$ is still uniformly bounded as a function of t in each bounded interval. To see this, one can refer to Lemma 2.2 in [54].

**Lemma**

**1.**

## 3. Synchronization

**Definition**

**1.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Numerical Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Dynamic behavior of the drive system (1) with ${d}_{1}=0.01$, ${d}_{2}=1$, $a=3.5$, and $b=0.25$ in accordance with the initial conditions given in (10).

**Figure 2.**Dynamic behavior of the response system (4) with ${d}_{1}=0.01$, ${d}_{2}=1$, $a=3.5$, and $b=0.25$ in accordance with the initial conditions given in (11).

**Figure 3.**The solution of the drive system (1) in $2D$ space at (

**a**) $t=0$, (

**b**) $t=1$, and (

**c**) $t=3$.

**Figure 4.**The solution of the response system (4) in $2D$ space at (

**a**) $t=0$, (

**b**) $t=1$, and (

**c**) $t=3$.

**Figure 5.**Dynamic behavior of the solutions of the spatiotemporal synchronization error system (5) with ${d}_{1}=0.01$, ${d}_{2}=1$, and $K=0.2$.

**Figure 6.**The solution of the spatiotemporal synchronization error system (5) in $2D$-space at (

**a**) $t=0$, (

**b**) $t=1$, and (

**c**) $t=3$.

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**MDPI and ACS Style**

Ouannas, A.; Batiha, I.M.; Bekiros, S.; Liu, J.; Jahanshahi, H.; Aly, A.A.; Alghtani, A.H.
Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law. *Entropy* **2021**, *23*, 1516.
https://doi.org/10.3390/e23111516

**AMA Style**

Ouannas A, Batiha IM, Bekiros S, Liu J, Jahanshahi H, Aly AA, Alghtani AH.
Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law. *Entropy*. 2021; 23(11):1516.
https://doi.org/10.3390/e23111516

**Chicago/Turabian Style**

Ouannas, Adel, Iqbal M. Batiha, Stelios Bekiros, Jinping Liu, Hadi Jahanshahi, Ayman A. Aly, and Abdulaziz H. Alghtani.
2021. "Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law" *Entropy* 23, no. 11: 1516.
https://doi.org/10.3390/e23111516