Optimal Control of an Eco-Epidemiological Reaction-Diffusion Model

Abstract

In this paper, a prey–predator diffusion model with isolation and drug treatment control measures for prey infection is studied. The main objective is to find an optimal control that minimizes the population density of infected prey and the costs of isolation and drug treatment for infected prey. Through analysis, the existence and uniqueness of weak solution, as well as the existence and local uniqueness of optimal controls are proven. The first-order necessary condition is derived, and the feasibility of the theoretical proof is verified through numerical simulations.

1. Introduction

In recent years, remarkable progress has been made in interdisciplinary research between ecology and mathematics. Through the construction of differential equation systems, researchers have conducted in-depth investigations into complex behaviors such as species interactions, disease transmission, and resource competition. With the increasingly widespread application of optimal control theory in biomathematics, scholars have gradually focused on how to design optimal strategies within the framework of mathematical models to achieve the collaborative optimization of ecological balance and resource utilization. Against this research background, the close interaction between prey and predator has become a core research issue.

In natural ecosystems, the interactions between prey and predator are often influenced by diseases, especially since the population dynamics of a prey population after infection may differ significantly from those in a healthy state. Therefore, considering the factor of disease infection in prey populations can significantly increase the complexity of prey–predator model and make the research more realistic. In recent years, the application of epidemiology in population biology has received extensive attention and made important progress.

In the study of the dynamic behavior of prey–predator ordinary differential equation model with infected prey, the main focus is on the equilibrium points of the model, their stability, and the system’s response to external disturbances. By analyzing the equilibrium points (such as steady states) and their stability (such as local or global stability), the long-term behavioral patterns of prey–predator models with infected prey are revealed. For example, in 1999, Chattopadhyay, Joydev, and Ovide Arino [1] assumed that the disease only spreads within the prey population, is non-hereditary, and infected prey cannot recover or develop immunity, and the predator population primarily preys on infected prey. They mainly investigated theoretical issues such as the existence and boundedness of solutions, as well as the stability of equilibrium points. Additionally, they simplified the three-dimensional system to a two-dimensional system for analysis and generalized the results back to the original system. In 2001, Xiao, Y. and Chen, L [2] studied a class of prey–predator model where prey is infected and predators prey on infected prey, introducing time-delay factors, particularly the gestational delay of predator. Using delay differential equations, they analyzed non-negativity, the global stability of solutions, permanence, and the impact of time delays on system stability. Notably, they also explored Hopf bifurcation phenomena caused by time delays and demonstrated the dual role of time delays in system stability (both destabilizing and stabilizing effects). In 2011, Niu X, Zhang T, Teng Z [3] investigated a non-autonomous eco-epidemiological model involving infected prey, where the parameters of the ecosystem change over time. The research focused on the asymptotic behavior of the system, especially the stability and periodicity of long-term behavior, and explored the long-term dynamic effects of disease transmission on prey and predator populations.

However, ordinary differential equation (ODE) models overlook the impact of spatial heterogeneity on research, which has motivated researchers to introduce spatial distribution into models and construct more complex partial differential equation (PDE) models. For example, in 2010, Li, Jianjun, and Wenjie Gao [4] studied a class of prey–predator reaction-diffusion model with infected prey. In 2018, Pan M, Yang J, Lin Z [5] analyzed a non-autonomous eco-epidemiological diffusion model involving infected prey, and the model is shown as follows:

$$\begin{array}{cccc}\hfill & S_t-d_I▵S=\Lambda (t)-\beta (t)SI-d(t)S,\hfill & \hfill & x\in \Omega ,t>0,\hfill \\ \hfill & I_t-d_I▵I=\beta (t)SI-c(t)I-\alpha (t)PI,\hfill & \hfill & x\in \Omega ,t>0,\hfill \\ \hfill & P_t-d_P▵P=P(r(t)-b(t)P)+k(t)\alpha (t)PI,\hfill & \hfill & x\in \Omega ,t>0,\hfill \\ \hfill & {\partial }_{\eta }S={\partial }_{\eta }I={\partial }_{\eta }P=0,\hfill & \hfill & x\in \partial \Omega ,t>0,\hfill \\ \hfill & S(x,0)=S_0(x)\ge 0,I(x,0)=I_0(x)\ge 0,P(x,0)=P_0(x)\ge 0,\hfill & \hfill & x\in \overline{\Omega }.\hfill \end{array}$$

Here, S is the population density of susceptible prey, I is the population density of infected prey, P is the population density of predator, $d_I$ are the diffusion coefficients of susceptible prey and infected prey populations, respectively, $d_P$ is the diffusion coefficient of predator population, $\Lambda (t)$ is the recruitment rate of the prey population at time t (including birth rate and migration rate), $\beta (t)$ is the conversion rate of susceptible prey to infected prey at time t, $d(t)$ is the mortality rate of susceptible prey at time t, $c(t)$ is the mortality rate of infected prey at time t, $\alpha (t)$ is the predation rate of predator preying on infected prey at time t, $r(t)$ is the intrinsic growth rate of the predator population at time t, ${\displaystyle \frac{1}{b(t)}}$ is the environmental maximum carrying capacity of the predator population at time t, $k(t)$ is the conversion rate of predator preying on infected prey into predators at time t, x is the spatial location, $\Omega$ is a fixed and bounded domain in ${\mathbb{R}}^N$ with smooth boundary $\partial \Omega$, and $S_0(x),I_0(x),P_0(x)$ are the initial values of the population densities of susceptible prey, infected prey, and predator, respectively. This study mainly analyzes the impact of disease transmission in the prey population on the long-term dynamic behavior of the system. The results show that disease transmission is dominated by $\beta (t)$ when the conversion rate $\beta (t)$ is large enough, and the disease spreads; conversely, it dies out.

With the deepening understanding of disease transmission mechanisms in ecosystems, control measures such as isolation and treatment have increasingly attracted the attention of scholars as prevention and control measures. These control strategies can not only reduce the spread rate of diseases in prey populations but also indirectly influence the dynamic changes of predator populations by optimizing the health status of prey. Therefore, how to design and optimize these control strategies through optimal control methods has become an important topic in current interdisciplinary research between ecology and mathematics. For example, in 2018, RUD Amalia and DK Arif [6] proposed a prey–predator mathematical model with infection and harvesting, where infection and harvesting only occur in the prey population, assuming that prey infections do not spread to predator. Using Pontryagin’s maximum principle, they explored the optimal control problem aimed at maximizing the density of susceptible prey populations while minimizing control costs. In the same year, JSH Simon and JFT Rabago [7] studied the optimal control problem of a prey-predator model with infected prey. Considering disease transmission in the prey population, they investigated the optimal control problem targeting the minimization of both the density of infected prey and control costs, and conducted numerical simulations using numerical methods. In 2019, A Hugo and E Simanjilo [8] studied an eco-epidemiological model with isolation control strategies, as shown in the following model:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial S}{\partial t}}=rS(1-{\displaystyle \frac{S+I}{k}})-(1-u_1)\beta SI-{\displaystyle \frac{p_{1}SY}{m+S}},\hfill \\ \hfill & {\displaystyle \frac{\partial I}{\partial t}}=(1-u_1)\beta SI-{\displaystyle \frac{(1-u_2)p_{2}IY}{m+I}}-(a+e_1)I,\hfill \\ \hfill & {\displaystyle \frac{\partial Y}{\partial t}}=q{\displaystyle \frac{p_{1}SY}{m+S}}+{\displaystyle \frac{(1-u_2)p_{2}IY}{m+I}}-(a+e_1)I-e_{2}Y.\hfill \end{array}$$

Among them, S is the population density of susceptible prey, I is the population density of infected prey, and Y is the population density of the predator, respectively. r is the intrinsic growth rate, k is the environmental carrying capacity, $\beta$ is the rate of transmission, m is half-saturation constant, and $p_1,p_2$ are the predation coefficients with susceptible prey and infected prey, respectively. a is the death rate caused by disease or by predation, $e_1,e_2$ are the natural death rate and the predator’s death rate, respectively. q is the efficiency with which prey is converted into predator. $u_1(x,t)$ is the isolation control rate for susceptible prey and infected prey, and $u_2(x,t)$ is the isolation control rate for infected prey and predator, respectively. t represents time and x represents the spatial location. The main research objective is to find the control functions $u_1(x,t)$ and $u_2(x,t)$ that minimize both the density of the infected prey population and the control costs. The objective functional is defined in the following form:

$$\begin{array}{c}\hfill J={\int }_0^{t_f}(BI+{\displaystyle \frac{1}{2}}{A}_1{u}_1^2+{\displaystyle \frac{1}{2}}{A}_2{u}_2^2)dt.\end{array}$$

Among them, $t_f$ is the final time. B is the cost required for infected prey, and $A_1,A_2$ are the relative cost weights corresponding to the two control functions, respectively. The study results indicate that the isolation of infected individuals plays a crucial role in disease elimination. In 2024, Mekonen Kassahun Getnet and Abayneh Fentie Bezabih [9] studied a scenario where both prey and predator populations are infected, with the specific model described as follows:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial s}{\partial t}}=rs(1-{\displaystyle \frac{s+I}{k}})-(1-u_1)\beta sI-p_{1}sy+u_{2}I,\hfill \\ \hfill & {\displaystyle \frac{\partial I}{\partial t}}=(1-u_1)\beta sI-p_{2}Iy-(u_2+{\delta }_1)I,\hfill \\ \hfill & {\displaystyle \frac{\partial y}{\partial t}}=q{p}_{1}SY+q{p}_{2}Iy-(1-u_1)\alpha yz-\mu y+u_{2}z,\hfill \\ \hfill & {\displaystyle \frac{\partial z}{\partial t}}=(1-u_1)\alpha yz-(u_2+{\delta }_2)z.\hfill \end{array}$$

Among them, s is the population density of susceptible prey, I is the population density of infected prey, y is the population density of susceptible predator, and z is the population density of infected predator, respectively. r is the intrinsic growth rate, k is the environmental carrying capacity, $\beta$ is the rate of transmission, $p_1,p_2$ are the predation coefficients with susceptible prey and infected prey, respectively. ${\delta }_1,{\delta }_2$ are the fatality rates of infected prey and infected predator, respectively. q is the transformation of eaten prey to predation, $\alpha$ is the convolution rate, and $\mu$ is the natural death rate of the susceptible predator, respectively. $u_1(x,t)$ represents preventive measures such as isolation, patient tracking, and reducing interaction rates with infected individuals, $u_2(x,t)$ denotes providing medical care for all infected cases, respectively. t represents time, x represents the spatial location. The main research objective is to find the control functions $u_1(x,t)$ and $u_2(x,t)$ that minimize the population densities of infected prey and predator, as well as the control costs. The objective functional is defined in the following form:

$$\begin{array}{c}\hfill J(u_1,u_2)={\int }_0^{t_f}(c_{1}I(t)+c_{2}z(t)+{\displaystyle \frac{1}{2}}{\sum }_{i=1}^2{b}_i{u}_i^2)dt.\end{array}$$

Among them, $t_f$ is the final time. $c_1,c_2$ correspond to the weights of infected prey and infected predator, respectively, $b_1,b_2$ correspond to the relative costs necessary for control, respectively. Numerical simulations using numerical methods shown that the control effect of $u_1$ is superior to that of $u_2$.

Considering the impact of spatial heterogeneity on the research, we introduce spatial distribution into the model and construct a more complex partial differential model. In 2019, TY Miyaoka, S Lenhart, and JFCA Meyer [10] investigated the optimization of vaccination strategies in a vector-borne reaction-diffusion model to address Zika virus transmission. In 2020, F Dai and B Liu [11] studied the optimal control problem in a general reaction-diffusion eco-epidemiological model where prey may be infected. By integrating prey-predator dynamics with disease transmission in prey, reaction-diffusion equations were used to describe the spatial and temporal evolution of populations and diseases. Through optimal control theory, the results shown that adopting optimal control strategies can effectively reduce disease transmission in prey populations and improve the overall stability of the prey–predator system. In 2020, PT Sowndarrajan, N Nyamoradi, and L Shangerganesh [12] conducted a mathematical analysis of optimal control problems in a prey–predator model with infected prey, considering disease transmission in prey populations and examining how different optimal control strategies affect the dynamics of predator and prey populations. In 2024, ES Baranovskii, RV Brizitskii, and ZY Saritskaia [13] studied the optimal control problem for a more general reaction-diffusion model. However, their paper focused on a single time-independent equation, while this paper investigates a class of coupled systems comprising three time-dependent convection-diffusion equations.

In this paper, we study the optimal control problem of a class of prey–predator dispersal model with isolation and treatment of predator infection, and the problem equation is as follows:

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial S}{\partial t}}-d_S▵S=AS-(1-u_1)\beta SI-dS-{\eta }_{1}SY+u_{2}I,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial I}{\partial t}}-d_I▵I=(1-u_1)\beta SI-cI-{\eta }_{2}IY-u_{2}I,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(2)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial Y}{\partial t}}-d_Y▵Y=rY(1-{\displaystyle \frac{Y}{k}})+p_1{\eta }_{1}SY+p_2{\eta }_{2}IY,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial S}{\partial n}}={\displaystyle \frac{\partial I}{\partial n}}={\displaystyle \frac{\partial Y}{\partial n}}=0,\hfill & \hfill & (x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill & S(x,0)=S_0(x),I(x,0)=I_0(x),Y(x,0)=Y_0(x),\hfill & \hfill & x\in \Omega ,\hfill \end{array}$$

(5)

where $\Omega$ is a bounded smooth domain of ${\mathbb{R}}^N$, $Q_T=\Omega \times (0,T)$, $\mathsf{\Gamma }=\partial \Omega \times (0,T)$. Furthermore, $d_S,d_I,d_Y,A,\beta ,d,{\eta }_1,c,{\eta }_2,r,p_1,p_2,u_1,u_2\in L^{\infty }(Q_T)$, $S_0(x),I_0(x),Y_0(x)\in L^{\infty }(\Omega )$, $d_S,d_I,d_Y,A$ $\beta ,d,{\eta }_1,c,{\eta }_2,r,k,p_1,p_2,S_0(x),I_0(x),Y_0(x)$ are all nonnegative functions and k is a function which is greater than zero.

The variable t represents time, x represents the spatial location. The functions S, I, and Y represent the densities of the susceptible prey, the infected prey, and the predator, respectively. The functions $d_S$, $d_I$, and $d_Y$ represent the diffusion coefficient of susceptible prey, the infected prey, and the predator, respectively. The function A represents the birth rate of susceptible prey. The function $\beta$ represents the infection rate of susceptible prey. The function d represents the death rate for susceptible prey. The function ${\eta }_1$, ${\eta }_2$ represent the prey rate of susceptible prey and infected prey, respectively. The function c represents the natural and disease-induced death rate for infected prey. The function r represents the growth rate of the predator. The function k represents the carrying capacity for the predator. The functions $p_1$, $p_2$ represent the conversion rate of a predator which is susceptible prey to another predator and infected prey to predator, respectively. The function $S_0(x)$, $I_0(x)$, $Y_0(x)$ represent the initial value of the population density of susceptible prey, infected prey, and predator, respectively.

The function $u_1(x,t)$ represents the isolation rate of infected prey, and the function $u_2(x,t)$ represents the cure rate of drug treatment for infected prey, respectively.

Suppose $\mathit{u}=(u_1,u_2)$ is the control, and the admissible control set $U_{ad}$ is defined by

$$\begin{array}{c}\hfill U_{ad}=\{\mathit{u}=(u_1,u_2)\in L^2(Q_T),0\le u_i\le M_i,i=1,2\}.\end{array}$$

(6)

where $M_i$ represents the upper bound of control $u_i,i=1,2.$

The objective functional is defined by

$$\begin{array}{c}\hfill J(u_1,u_2)={\int }_0^T{\int }_{\Omega }\left(wI(x,t)+{\displaystyle \frac{1}{2}}{a}_1{u}_1^2(x,t)+{\displaystyle \frac{1}{2}}{a}_2{u}_2^2(x,t)\right)dxdt,\end{array}$$

(7)

where $\mathit{u}\in U_{ad}$, $w,a_1,a_2\in {\mathbb{R}}^{+}$, w represents the weight factor, $a_1$ is the price for isolating infected prey, and $a_2$ is the price for drug treatment for susceptible and infected prey, respectively. The functional J represents the population density and isolation of infected forage populations, as well as the cost of treatment. Our goal is to seek ${\mathit{u}}^{\ast }=(u_1^{\ast },u_2^{\ast })$ such that the objective function J reaches its minimum.

To simplify the following proof process, the system (1)–(5) is denoted in the following form:

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial z_1}{\partial t}}-d_1▵z_1=b_1{z}_1-(1-u_1)b_2{z}_1{z}_2-b_3{z}_1{z}_3+u_2{z}_2,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial z_2}{\partial t}}-d_2▵z_2=-c_1{z}_2+(1-u_1)b_2{z}_1{z}_2-c_2{z}_2{z}_3-u_2{z}_2,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(9)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial z_3}{\partial t}}-d_3▵z_3=e_1{z}_3-e_2{z}_3^2+e_3{z}_1{z}_3+e_4{z}_2{z}_3,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(10)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial z_1}{\partial n}}={\displaystyle \frac{\partial z_2}{\partial n}}={\displaystyle \frac{\partial z_3}{\partial n}}=0,\hfill & \hfill & (x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(11)

$$\begin{array}{cccc}\hfill & z_1(x,0)=z_{1,0}(x),z_2(x,0)=z_{2,0}(x),z_3(x,0)=z_{3,0}(x),\hfill & \hfill & x\in \Omega ,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & J(u_1,u_2)={\int }_0^T{\int }_{\Omega }\left(w{z}_2(x,t)+{\displaystyle \frac{1}{2}}{a}_1{u}_1^2(x,t)+{\displaystyle \frac{1}{2}}{a}_2{u}_2^2(x,t)\right)dxdt,\hfill \end{array}$$

(13)

Among them, the function $z_i,i=1,2,3$ represents the functions $S,I$, and Y, respectively. The function $d_i,i=1,2,3$ represents the function $d_S,d_I$, and $d_Y$, respectively. The function $b_i,i=1,2,3$ represents the functions $A-d,\beta$, and ${\eta }_1$, respectively. The function $c_i,i=1,2$ represents the functions c and ${\eta }_2$, respectively. The function $e_i,i=1,2,3,4$ represents the functions $r,{\displaystyle \frac{r}{k}},p_1{\eta }_1$, and $p_2{\eta }_2$, respectively. Due to the strong coupling nature of the system of equations, studying the well-posedness, optimal control problems, and numerical calculations of this system presents certain difficulties and complexities.

The paper is organized as follows. In Section 2, we establish the well-posedness of the system (8)–(12). In Section 3, we prove the existence of the optimal controls, derive the first order necessary condition and obtain the formula of the optimal control by optimality system. In Section 4, we prove the local uniqueness of the optimal control. In Section 5, we present numerical experiments for Neumann boundary condition. In Section 6, we provide a summary of the key findings from this study and outline potential directions for future research.

2. The Well-Posedness of the System

In this section, we give the definition of weak solutions of the system (8)–(12).

Definition 1.

We call $\mathit{z}=(z_1,z_2,z_3)\in {(C([0,T];L^2(\Omega ))\cap L^2((0,T);H^1(\Omega )))}^3$ a weak solution of the problem (8)–(12), if for any $\mathit{\varphi }=({\varphi }_1,{\varphi }_2,{\varphi }_3)\in {(L^2((0,T);H^1(\Omega )))}^3$ with ${\displaystyle \frac{\partial {\varphi }_i}{\partial t}}\in L^2(Q_T)$, ${\displaystyle \frac{\partial {\varphi }_i}{\partial n}}=0$ and ${\varphi }_i(x,T)=0,i=1,2,3$, the following equalities hold

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_1{\displaystyle \frac{\partial {\varphi }_1}{\partial t}}dxdt-{\int }_{\Omega }{z}_{1,0}(x){\varphi }_1(x,0)dx+d_1{\displaystyle {\int \int }_{Q_T}}\nabla z_1·\nabla {\varphi }_{1}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(b_1{z}_1-(1-u_1)b_2{z}_1{z}_2-b_3{z}_1{z}_3+u_2{z}_2\right){\varphi }_{1}dxdt,\hfill \\ \hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_2{\displaystyle \frac{\partial {\varphi }_2}{\partial t}}dxdt-{\int }_{\Omega }{z}_{2,0}(x){\varphi }_2(x,0)dx+d_2{\displaystyle {\int \int }_{Q_T}}\nabla z_2·\nabla {\varphi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(-c_1{z}_2+(1-u_1)b_2{z}_1{z}_2-c_2{z}_2{z}_3-u_2{z}_2\right){\varphi }_{2}dxdt,\hfill \\ \hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_3{\displaystyle \frac{\partial {\varphi }_3}{\partial t}}dxdt-{\int }_{\Omega }{z}_{3,0}(x){\varphi }_3(x,0)dx+d_3{\displaystyle {\int \int }_{Q_T}}\nabla z_3·\nabla {\varphi }_{3}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}(e_1{z}_3-e_2{z}_3^2+e_3{z}_1{z}_3+e_4{z}_2{z}_3){\varphi }_{3}dxdt.\hfill \end{array}$$

Lemma 1

([14]). Let $B_1\subset B_2\subset B_3$ be three Banach spaces. Assume that $B_1\hookrightarrow \hookrightarrow B_2\hookrightarrow B_3$. Then,

$$\begin{array}{c}\hfill \{u\in L^p(0,T;B_1):u_t\in L^q(0,T;B_3)\}\hookrightarrow \hookrightarrow L^p(0,T;B_2),1<p,q<\infty .\end{array}$$

Lemma 2.

For $u\in L^{\infty }(\Omega )$, there exists a sequence ${\left\{u_n\right\}}_{n=1}^{\infty }\in C_0^{\infty }(\Omega )$, such that

$$\begin{array}{c}\hfill \parallel u_n{\parallel }_{L^{\infty }(\Omega )}\le {\parallel u\parallel }_{L^{\infty }(\Omega )},\end{array}$$

and

$$\begin{array}{c}\hfill u_n\stackrel{\ast }{\rightharpoonup }u,in{L}^{\infty }(\Omega ).\end{array}$$

Proof. 

For $ϵ>0$, let $J_{ϵ}(x)$ be a mollifier such that

$$\begin{array}{c}\hfill J_{ϵ}(x)=0,|x|\ge ϵ,\mathrm{and}{\int }_{{\mathbb{R}}^n}{J}_{ϵ}(x)dx=1.\end{array}$$

(14)

Take $\delta >0$ such that

$$\begin{array}{c}\hfill {\Omega }^{\delta }:=\{x\in \Omega ;\mathrm{dist}(x,\partial \Omega )<\delta \}\ne \varphi .\end{array}$$

Define

$$\begin{array}{c}\hfill u^{\delta }(x)=\left\{\begin{array}{cc}u(x),\hfill & x\in {\Omega }^{\delta },\hfill \\ 0,\hfill & x\in {\mathbb{R}}^n\setminus {\Omega }^{\delta }.\hfill \end{array}\right.\end{array}$$

(15)

For $0<ϵ<\delta$, define

$$\begin{array}{c}\hfill u_{ϵ}^{\delta }(x)={\int }_{{\mathbb{R}}^n}{J}_{ϵ}(x-y)u^{\delta }(y)dy={\int }_{{\mathbb{R}}^n}{J}_{ϵ}(y)u^{\delta }(x-y)dy,x\in {\mathbb{R}}^n.\end{array}$$

(16)

From (14)–(16), we have

$$\begin{array}{c}\hfill \parallel u_{ϵ}^{\delta }{\parallel }_{L^{\infty }({\mathbb{R}}^n)} \le \parallel u^{\delta }{\parallel }_{L^{\infty }({\mathbb{R}}^n)} \le {\parallel u\parallel }_{L^{\infty }(\Omega )}.\end{array}$$

(17)

By Theorem 2.29 in [15], we have

$$\begin{array}{c}\hfill u_{ϵ}^{\delta }(x)\in C_0^{\infty }(\Omega )\mathrm{and}\underset{ϵ\to 0^{+}}{lim}{\parallel u_{ϵ}^{\delta }-u^{\delta }\parallel }_{L^2(\Omega )}=0.\end{array}$$

(18)

For any $\phi \in L^1(\Omega )$, there exists a sequence ${\phi }_k\in C_0^{\infty }(\Omega )$, such that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\parallel {\phi }_k-\phi \parallel }_{L^1(\Omega )}=0.\end{array}$$

(19)

From (15), (17), and (18), we have

$$\begin{array}{cc}\hfill & \underset{\delta \to 0^{+}}{\overline{lim}}\underset{ϵ\to 0^{+}}{\overline{lim}}\left|{\int }_{\Omega }(u_{ϵ}^{\delta }-u)\phi dx\right|\hfill \\ \hfill \le & \underset{\delta \to 0^{+}}{\overline{lim}}\underset{ϵ\to 0^{+}}{\overline{lim}}\left(\left|{\int }_{\Omega }(u_{ϵ}^{\delta }-u^{\delta }){\phi }_{k}dx\right|+\left|{\int }_{\Omega }(u_{ϵ}^{\delta }-u)(\phi -{\phi }_k)dx\right|+\left|{\int }_{\Omega }(u^{\delta }-u)\phi dx\right|\right)\hfill \\ \hfill \le & {2\parallel u\parallel }_{L^{\infty }(\Omega )}{\int }_{\Omega }\left|\phi -{\phi }_k\right|dx.\hfill \end{array}$$

(20)

It follows from (19) and (20) that

$$\begin{array}{c}\hfill \underset{\delta \to 0^{+}}{lim}\underset{ϵ\to 0^{+}}{lim}{\int }_{\Omega }(u_{ϵ}^{\delta }-u)\phi dx=0.\end{array}$$

For $i,j=1,2,\cdots$, take ${\delta }_i={\displaystyle \frac{\delta }{i}}$, ${ϵ}_j^i={\displaystyle \frac{{\delta }_i}{j+1}}$. Then,

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}\underset{j\to \infty }{lim}{\int }_{\Omega }(u_{{ϵ}_j^i}^{{\delta }_i}-u)\phi dx=0.\end{array}$$

By using a diagonal selection, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{\Omega }(u_n-u)\phi dx=0.\end{array}$$

(21)

where $u_n=u_{{ϵ}_n^n}^{{\delta }_n},n=1,2,\cdots$. The proof is completed from (17), (18), and (21).  □

Next, we give the well-posedness of the problem (8)–(12).

Theorem 1.

For small enough $T>0$, there exists a uniquely weak positive solution $\mathit{z}$ to the problem (8)–(12), such that for $i=1,2,3$, ${\displaystyle \frac{\partial z_i}{\partial t}}\in L^2((0,T);H^{-1}(\Omega ))$ and

$$\begin{array}{c}\hfill \parallel z_i{\parallel }_{L^2((0,T);H^1(\Omega ))}+\parallel {\displaystyle \frac{\partial z_i}{\partial t}}{\parallel }_{L^2((0,T);H^{-1}(\Omega ))}\le C,\end{array}$$

(22)

where C is depending on T, $\parallel z_{i,0}{(x)\parallel }_{L^{\infty }(\Omega )}$, $\parallel b_2{\parallel }_{L^{\infty }(Q_T)}$, $\parallel e_1{\parallel }_{L^{\infty }(Q_T)}$, $\parallel e_3{\parallel }_{L^{\infty }(Q_T)}$, $\parallel e_4{\parallel }_{L^{\infty }(Q_T)}$ with $i=1,2,3$. Moreover, it still holds that

$$\begin{array}{c}\hfill 0\le z_i\le H_i{e}^{{\tau }_{i}T},i=1,2,3,\end{array}$$

(23)

where

$$\begin{array}{c}\hfill H_1=\hspace{0.17em}\parallel z_{1,0}{(x)\parallel }_{L^{\infty }(\Omega )},H_2=\hspace{0.17em}\parallel z_{2,0}{(x)\parallel }_{L^{\infty }(\Omega )},H_3=\hspace{0.17em}{\parallel z_{3,0}(x)\parallel }_{L^{\infty }(\Omega )},\end{array}$$

(24)

$$\begin{array}{c}\hfill {\tau }_1=\parallel e{b}_2{H}_1{\parallel }_{L^{\infty }(Q_T)},{\tau }_2=\parallel e{b}_2{H}_1{\parallel }_{L^{\infty }(Q_T)},{\tau }_3={\parallel e_1+e_3{H}_{1}e+e_4{H}_{2}e\parallel }_{L^{\infty }(Q_T)}.\end{array}$$

(25)

Proof. 

For $i=1,2,3$, $\alpha =1,2$, $k=1,2,3,4$, and any $(x,t)\in Q_T$, it follows from Lemma 2 that there exist nonnegative sequences ${\left\{d_i^{(n)}\right\}}_{n=1}^{\infty },{\left\{b_i^{(n)}\right\}}_{n=1}^{\infty },{\left\{c_{\alpha }^{(n)}\right\}}_{n=1}^{\infty },{\left\{e_k^{(n)}\right\}}_{n=1}^{\infty }\in C_0^{\infty }(Q_T)$, ${\left\{z_{i,0}^{(n)}\right\}}_{n=1}^{\infty }\in C_0^{\infty }(\Omega ),$ such that

$$\begin{array}{c}\parallel d_i^{(n)}{\parallel }_{L^{\infty }(Q_T)}\le \hspace{0.17em}\parallel d_i{\parallel }_{L^{\infty }(Q_T)},\parallel b_i^{(n)}{\parallel }_{L^{\infty }(Q_T)}\le \hspace{0.17em}\parallel b_i{\parallel }_{L^{\infty }(Q_T)},\parallel c_{\alpha }^{(n)}{\parallel }_{L^{\infty }(Q_T)}\le {\parallel c_{\alpha }\parallel }_{L^{\infty }(Q_T)},\end{array}$$

(26)

$$\begin{array}{c}\parallel e_k^{(n)}{\parallel }_{L^{\infty }(Q_T)}\le \hspace{0.17em}\parallel e_k{\parallel }_{L^{\infty }(Q_T)},\parallel z_{i,0}^{(n)}{(x)\parallel }_{L^{\infty }(\Omega )}\le {\parallel z_{i,0}(x)\parallel }_{L^{\infty }(\Omega )}\end{array}$$

(27)

and

$$\begin{array}{c}{d}_i^{(n)}\stackrel{\ast }{\rightharpoonup }{d}_i,b_i^{(n)}\stackrel{\ast }{\rightharpoonup }{b}_i,c_{\alpha }^{(n)}\stackrel{\ast }{\rightharpoonup }{c}_{\alpha },e_k^{(n)}\stackrel{\ast }{\rightharpoonup }{e}_k,\mathrm{in}{L}^{\infty }(Q_T),\hspace{0.17em}\mathrm{as}\hspace{0.17em}n\to \infty ,\end{array}$$

(28)

$$\begin{array}{c}{z}_{i,0}^{(n)}(x)\to z_{i,0}(x),\mathrm{in}{L}^2(\Omega ),\hspace{0.17em}\mathrm{as}\hspace{0.17em}n\to \infty .\end{array}$$

(29)

For $i=1,2,3$, consider the following system

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial z_i^{(n)}}{\partial t}}-d_i^{(n)}▵z_i^{(n)}=g_i(x,t,{\mathit{z}}^{(n)}),\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(30)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial z_i^{(n)}}{\partial n}}=0,\hfill & \hfill & (x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(31)

$$\begin{array}{cccc}\hfill & z_i^{(n)}(x,0)=z_{i,0}^{(n)}(x),\hfill & \hfill & x\in \Omega ,\hfill \end{array}$$

(32)

where

$$\begin{array}{cc}\hfill & g_1(x,t,\gamma )=b_1^{(n)}{\gamma }_1-(1-u_1^{(n)})b_2^{(n)}{\gamma }_1{\gamma }_2-b_3^{(n)}{\gamma }_1{\gamma }_3+u_2^{(n)}{\gamma }_2,\hfill \\ \hfill & g_2(x,t,\gamma )=-c_1^{(n)}{\gamma }_2+(1-u_1^{(n)})b_2^{(n)}{\gamma }_1{\gamma }_2-c_2^{(n)}{\gamma }_2{\gamma }_3-u_2^{(n)}{\gamma }_2,\hfill \\ \hfill & g_3(x,t,\gamma )=e_1^{(n)}{\gamma }_3-e_2^{(n)}{\gamma }_3^2+e_3^{(n)}{\gamma }_1{\gamma }_3+e_4^{(n)}{\gamma }_2{\gamma }_3.\hfill \end{array}$$

Define

$$\begin{array}{c}\hfill \overline{\mathit{z}}=(H_1{e}^{{\tau }_{1}t},H_2{e}^{{\tau }_{2}t},H_3{e}^{{\tau }_{3}t}),\underline{\mathit{z}}=(0,0,0).\end{array}$$

(33)

where $H_1,H_2,H_3,{\tau }_1,{\tau }_2,{\tau }_3$ are defined as (24) and (25).

Take $T\le {\displaystyle \frac{1}{e{b}_2{H}_1}}$. By computation, one can see that for any $\gamma$ with ${\underline{z}}_i\le {\gamma }_i\le {\overline{z}}_i$ and (33), we have

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial {\overline{z}}_i}{\partial t}}-d_i^{(n)}▵{\overline{z}}_i\ge g_i(t,x,\gamma ),{\displaystyle \frac{\partial {\underline{z}}_i}{\partial t}}-d_i^{(n)}▵{\underline{z}}_i\le g_i(t,x,\gamma ),\hfill & \hfill & (x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial {\overline{z}}_i}{\partial n}}\ge 0,{\displaystyle \frac{\partial {\underline{z}}_i}{\partial n}}\le 0,\hfill & \hfill & (x,t)\in \mathsf{\Gamma },\hfill \\ \hfill & {\overline{z}}_{i,0}(x)=H_i\ge z_{i,0}^{(n)}(x),{\underline{z}}_{i,0}(x)\le z_{i,0}^{(n)}(x),\hfill & \hfill & x\in \Omega .\hfill \end{array}$$

Hence, $\overline{\mathit{z}}$ and $\underline{\mathit{z}}$ are the upper solution and the lower solution of the system (30)–(32), respectively. Hence, it follows from [16] that there exists a unique solution ${\mathit{z}}^{(\mathit{n})}=(z_1^{(n)},z_2^{(n)},z_3^{(n)})\in {(C({\overline{Q}}_T)\cap C^2(Q_T))}^3$ such that

$$\underline{z_i}\le z_i^{(n)}\le \overline{z_i},i=1,2,3.$$

That is to say,

$$0\le z_i^{(n)}\le H_i{e}^{{\tau }_{i}T},i=1,2,3.$$

(34)

Multiplying the Equation (30) by $z_1^{(n)}$ and integrating by parts over $\Omega \times (0,t)$ with $0<t<T$, we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\int }_{\Omega }{\left(z_1^{(n)}(x,t)\right)}^{2}dx+d_1{\int }_0^T{\int }_{\Omega }{|\nabla z_1^{(n)}(x,t)|}^{2}dxdt\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\int }_{\Omega }{(z_{1,0}^{(n)}(x))}^{2}dx+{\int }_0^T{\int }_{\Omega }\left(b_1^{(n)}{z}_1^{(n)}-(1-u_1^{(n)})b_2^{(n)}{z}_1^{(n)}{z}_2^{(n)}-b_3^{(n)}{z}_1^{(n)}{z}_3^{(n)}+u_2^{(n)}{z}_2^{(n)}\right)z_1^{(n)}dxdt.\hfill \end{array}$$

It follows from (26)–(27) and (34) that

$$\underset{0\le t\le T}{sup}{\int }_{\Omega }{\left(z_1^{(n)}(x,t)\right)}^{2}dx+d_1{\int }_0^T{\int }_{\Omega }{|\nabla z_1^{(n)}(x,t)|}^{2}dxdt\le C.$$

(35)

For any $\phi \in L^2((0,T);H_0^1(\Omega ))$, we have

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }{\displaystyle \frac{\partial z_1^{(n)}}{\partial t}}\phi dxdt\hfill \\ \hfill =& d_1{\int }_0^T{\int }_{\Omega }▵z_1^{(n)}\phi dxdt+{\int }_0^T{\int }_{\Omega }\left(b_1^{(n)}{z}_1^{(n)}-(1-u_1^{(n)})b_2^{(n)}{z}_1^{(n)}{z}_2^{(n)}-b_3^{(n)}{z}_1^{(n)}{z}_3^{(n)}+u_2^{(n)}{z}_2^{(n)}\right)\phi dxdt\hfill \\ \hfill \le & d_1{\int }_0^T{\int }_{\Omega }\nabla z_1^{(n)}·\nabla \phi dxdt+C{\parallel \phi \parallel }_{L^2(Q_T)}\hfill \\ \hfill \le & C(\parallel \nabla \phi {\parallel }_{L^2(Q_T)}+\parallel \phi {\parallel }_{L^2(Q_T)})\hfill \\ \hfill \le & C(\parallel \phi {\parallel }_{L^2((0,T);H_0^1(\Omega ))})\hfill \end{array}$$

due to (26)–(27), (34), and (35). Hence, we have

$$\parallel {\displaystyle \frac{\partial z_1^{(n)}}{\partial t}}{\parallel }_{L^2((0,T);H^{-1}(\Omega ))}\le C.$$

(36)

Similarly, one can get

$$\underset{0\le t\le T}{sup}{\int }_{\Omega }{\left(z_2^{(n)}(x,t)\right)}^{2}dx+d_2{\int }_0^T{\int }_{\Omega }{|\nabla z_2^{(n)}(x,t)|}^{2}dxdt+\parallel {\displaystyle \frac{\partial z_2^{(n)}}{\partial t}}{\parallel }_{L^2((0,t);H^{-1}(\Omega ))}\le C,$$

(37)

and

$$\underset{0\le t\le T}{sup}{\int }_{\Omega }{\left(z_3^{(n)}(x,t)\right)}^{2}dx+d_3{\int }_0^T{\int }_{\Omega }{|\nabla z_3^{(n)}(x,t)|}^{2}dxdt+\parallel {\displaystyle \frac{\partial z_3^{(n)}}{\partial t}}{\parallel }_{L^2((0,t);H^{-1}(\Omega ))}\le C.$$

(38)

Hence, it follows from Lemma 1, Lemma C.1 in [17] and (34)–(38), that there exist subsequences of ${\left\{n\right\}}_{n=1}^{\infty }$, denoted by ${\left\{n_j\right\}}_{j=1}^{\infty }$, and a function $\mathit{z}=(z_1,z_2,z_3)\in {(C([0,T];L^2(\Omega ))\cap L^2((0,T);H^1(\Omega )))}^3$ with $\mathit{z}\in {(L^{\infty }(Q_T))}^3$ and ${\displaystyle \frac{\partial \mathit{z}}{\partial t}}\in (L^2{((0,T);H^{-1}(\Omega ))}^3$, such that for $i=1,2,3$,

$$\begin{array}{c}\nabla z_i^{(n_j)}\rightharpoonup \nabla z_i\mathrm{in}{L}^2(Q_T;{\mathbb{R}}^n),{\displaystyle \frac{\partial z_i^{(n_j)}}{\partial t}}\rightharpoonup {\displaystyle \frac{\partial z_i}{\partial t}}\mathrm{in}{L}^2((0,T);H^{-1}(\Omega )),asj\to \infty ,\end{array}$$

(39)

$$\begin{array}{c}{z}_i^{(n_j)}\to z_i\mathrm{in}{L}^2(Q_T),z_i^{(n_j)}\stackrel{\ast }{\rightharpoonup }{z}_i\mathrm{in}{L}^{\infty }(Q_T),asj\to \infty .\end{array}$$

(40)

Since ${\mathit{z}}^{(n_j)}$ is the solution to the system (30)–(32), for any ${\varphi }_i\in C^{\infty }(\overline{Q_T})$ with ${\displaystyle \frac{\partial {\varphi }_i}{\partial n}}=0$ and ${\varphi }_i(x,T)=0(i=1,2,3)$, we have the following equalities hold

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_1^{(n_j)}{\displaystyle \frac{\partial {\varphi }_1}{\partial t}}dxdt-{\int }_{\Omega }{z}_{1,0}^{(n_j)}(x){\varphi }_1(x,0)dx+d_1{\displaystyle {\int \int }_{Q_T}}\nabla z_1^{(n_j)}·\nabla {\varphi }_{1}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(b_1^{(n_j)}{z}_1^{(n_j)}-(1-u_1^{(n_j)})b_2^{(n_j)}{z}_1^{(n_j)}{z}_2^{(n_j)}-b_3^{(n_j)}{z}_1^{(n_j)}{z}_3^{(n_j)}+u_2^{(n_j)}{z}_2^{(n_j)}\right){\varphi }_{1}dxdt.\hfill \end{array}$$

(41)

Similarly, we have

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_2^{(n_j)}{\displaystyle \frac{\partial {\varphi }_2}{\partial t}}dxdt-{\int }_{\Omega }{z}_{2,0}^{(n_j)}(x){\varphi }_2(x,0)dx+d_2{\displaystyle {\int \int }_{Q_T}}\nabla z_2^{(n_j)}·\nabla {\varphi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(-c_1^{(n_j)}{z}_2^{(n_j)}+(1-u_1^{(n_j)})b_2^{(n_j)}{z}_1^{(n_j)}{z}_2^{(n_j)}-c_2^{(n_j)}{z}_2^{(n_j)}{z}_3^{(n_j)}-u_2^{(n_j)}{z}_2^{(n_j)}\right){\varphi }_{2}dxdt\hfill \end{array}$$

(42)

and

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_3^{(n_j)}{\displaystyle \frac{\partial {\varphi }_3}{\partial t}}dxdt-{\int }_{\Omega }{z}_{3,0}^{(n_j)}(x){\varphi }_3(x,0)dx+d_3{\displaystyle {\int \int }_{Q_T}}\nabla z_3^{(n_j)}·\nabla {\varphi }_{3}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(e_1^{(n_j)}{z}_3^{(n_j)}-e_2^{(n_j)}{(z_3^{(n_j)})}^2+e_3^{(n_j)}{z}_1^{(n_j)}{z}_3^{(n_j)}+e_4^{(n_j)}{z}_2^{(n_j)}{z}_3^{(n_j)}\right){\varphi }_{3}dxdt.\hfill \end{array}$$

(43)

Letting $n\to \infty$ in (41)–(43), we can get from (28)–(29), (39), and (40) that

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_1{\displaystyle \frac{\partial {\varphi }_1}{\partial t}}dxdt-{\int }_{\Omega }{z}_{1,0}(x){\varphi }_1(x,0)dx+d_1{\displaystyle {\int \int }_{Q_T}}\nabla z_1·\nabla {\varphi }_{1}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(b_1{z}_1-(1-u_1)b_2{z}_1{z}_2-b_3{z}_1{z}_3+u_2{z}_2\right){\varphi }_{1}dxdt,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_2{\displaystyle \frac{\partial {\varphi }_2}{\partial t}}dxdt-{\int }_{\Omega }{z}_{2,0}(x){\varphi }_2(x,0)dx+d_2{\displaystyle {\int \int }_{Q_T}}\nabla z_2·\nabla {\varphi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(-c_1{z}_2+(1-u_1)b_2{z}_1{z}_2-c_2{z}_2{z}_3-u_2{z}_2\right){\varphi }_{2}dxdt,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_3{\displaystyle \frac{\partial {\varphi }_3}{\partial t}}dxdt-{\int }_{\Omega }{z}_{3,0}(x){\varphi }_3(x,0)dx+d_3{\displaystyle {\int \int }_{Q_T}}\nabla z_3·\nabla {\varphi }_{3}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(e_1{z}_3-e_2{z}_3^2+e_3{z}_1{z}_3+e_4{z}_2{z}_3\right){\varphi }_{3}dxdt.\hfill \end{array}$$

Since $\mathit{z}\in {(C([0,T];L^2(\Omega ))\cap L^2((0,T);H^1(\Omega )))}^3$, we can extend the space of the test function from ${(C^{\infty }(\overline{Q_T}))}^3$ to ${(L^2((0,T);H^1(\Omega )))}^3$ with ${\displaystyle \frac{\partial \varphi }{\partial t}}\in {(L^2(Q_T))}^3$.

Hence, $\mathit{z}$ is the solution of the system (8)–(12). Furthermore, we can obtain the estimates (22) and (23) from (35)–(40).

Next, we prove the uniqueness.

Let ${\tilde{\mathit{z}}}^{(n)}=({\tilde{z}}_1^{(n)},{\tilde{z}}_2^{(n)},{\tilde{z}}_3^{(n)})$ and ${\widehat{\mathit{z}}}^{(n)}=({\widehat{z}}_1^{(n)},{\widehat{z}}_2^{(n)},{\widehat{z}}_3^{(n)})$ be two solutions of the system (30)–(32). Denote $\mathit{v}={\tilde{z}}^{(n)}-{\widehat{z}}^{(n)}$. Then, $\mathit{v}=(v_1,v_2,v_3)$ satisfies the following system:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial v_1}{\partial t}}-d_1^{(n)}\Delta v_1=b_1^{(n)}{v}_1-(1-u_1^{(n)})b_2^{(n)}{v}_1{\tilde{z}}_2^{(n)}-(1-u_1^{(n)})b_2^{(n)}{\widehat{z_1}}^{(n)}{v}_2-b_3^{(n)}{v}_1{\widetilde{z_3}}^{(n)}-b_3^{(n)}{\widehat{z_1}}^{(n)}{v}_3\hfill \\ \hfill & +u_2^{(n)}{v}_2,(x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial v_2}{\partial t}}-d_2^{(n)}\Delta v_2=-c_1^{(n)}{v}_2+(1-u_1^{(n)})b_2^{(n)}{v}_2{\widetilde{z_1}}^{(n)}-(1-u_1^{(n)})b_2^{(n)}{v}_1{\widehat{z_2}}^{(n)}-c_2^{(n)}{v}_2{\widetilde{z_3}}^{(n)}-c_2^{(n)}{\widehat{z_2}}^{(n)}{v}_3\hfill \\ \hfill & -u_2^{(n)}{v}_2,(x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial v_3}{\partial t}}-d_3^{(n)}\Delta v_3=e_1^{(n)}{v}_3-e_2^{(n)}{v}_3({\widetilde{z_3}}^{(n)}+{\widehat{z_3}}^{(n)})+e_3^{(n)}{v}_3{\widetilde{z_1}}^{(n)}+e_3^{(n)}{v}_1{\widehat{z_3}}^{(n)}+e_4^{(n)}{v}_3{\widetilde{z_2}}^{(n)}+e_4^{(n)}{v}_2{\widehat{z_3}}^{(n)},\hfill \\ \hfill & (x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial v_1}{\partial n}}={\displaystyle \frac{\partial v_2}{\partial n}}={\displaystyle \frac{\partial v_3}{\partial n}}=0,(x,t)\in \mathsf{\Gamma },\hfill \\ \hfill & v_1(x,0)=v_2(x,0)=v_3(x,0)=0,x\in \Omega .\hfill \end{array}$$

(44)

Multiply (44) by $v_1$ to get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\int }_{\Omega }{v}_1^2(x,t)dx+d_1{\int }_0^T{\int }_{\Omega }{\left|\nabla v_1\right|}^{2}dxds\le C{\int }_0^T{\int }_{\Omega }\left(v_1^2+v_2^2+v_3^2\right)dxds\hfill \end{array}$$

(45)

due to (23). Similarly, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\int }_{\Omega }{v}_2^2(x,t)dx+d_2{\int }_0^T{\int }_{\Omega }{|\nabla v_2|}^{2}dxds\le C{\int }_0^T{\int }_{\Omega }(v_1^2+v_2^2+v_3^2)dxds,\end{array}$$

(46)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\int }_{\Omega }{v}_3^2(x,t)dx+d_3{\int }_0^T{\int }_{\Omega }{|\nabla v_3|}^{2}dxds\le C{\int }_0^T{\int }_{\Omega }(v_1^2+v_2^2+v_3^2)dxds.\end{array}$$

(47)

From (45)–(47), we can get

$${\int }_{\Omega }(v_1^2+v_2^2+v_3^2)(x,t)dx\le C{\int }_0^t{\int }_{\Omega }(v_1^2+v_2^2+v_3^2)(x,s)ds.$$

By the Grownwall inequality, we have

$${\int }_{\Omega }(v_1^2+v_2^2+v_3^2)(x,t)dx\le 0.$$

Hence, it can yield

$$v_1=v_2=v_3=0.$$

The proof is completed. □

3. The Necessary Condition of the Optimal Control Problem

In this section, we study the optimal control problem of the system (8)–(12). Firstly, we prove the existence of the optimal control. Next, we derive the necessary condition of the optimal control problem and obtain the formula of the optimal control.

3.1. The Existence of the Optimal Control Problem

In this subsection, we prove the existence of the optimal control.

Theorem 2.

Suppose $u=(u_1,u_2)$, then there exists an optimal control $u^{\ast }=(u_1^{\ast },u_2^{\ast })$ such that

$$J(u^{\ast })=J(u_1^{\ast },u_2^{\ast })=\underset{(u_1,u_2)\in U_{ad}}{inf}J(u_1,u_2),$$

(48)

where $U_{ad}$ is defined as (6).

Proof. 

Since the control $u_1,u_2$ and the state $z_1,z_2,z_3$ are uniformly bounded, there exists a sequence $\{{(u_1^{(n)},u_2^{(n)}\}}_{n=1}^{\infty }\in U_{ad}$ such that

$$\underset{n\to \infty }{lim}J(u_1^{(n)},u_2^{(n)})=\underset{(u_1,u_2)\in U_{ad}}{inf}J(u_1,u_2).$$

(49)

Denote by $(z_1^{(n)},z_2^{(n)},z_3^{(n)})$ the solution of the problem (8)–(12) with $u_1=u_1^{(n)}$, $u_2=u_2^{(n)}$.

By Theorem 1, we obtain

$$||z_i^{(n)}{||}_{L^{\infty }(Q_T)}+||z_i^{(n)}{||}_{L^2((0,T);H^1(\Omega ))}+\parallel {\displaystyle \frac{\partial z_i^{(n)}}{\partial t}}{\parallel }_{L^2((0,T);H^{-1}(\Omega ))}\le C,i=1,2,3,$$

where C is a positive constant independent of n. Thus, there exists a subsequence of ${\left\{n\right\}}_{n=1}^{\infty }$, denoted by ${\left\{n_i\right\}}_{i=1}^{\infty }$, and $(u_1^{\ast },u_2^{\ast })\in U_{ad}$, $(z_1^{\ast },z_2^{\ast },z_3^{\ast })\in {\left(L^2((0,T);H^1(\Omega ))\right)}^3$ with $({(z_1^{\ast })}_t,{(z_2^{\ast })}_t,{(z_3^{\ast })}_t)$$\in {\left(L^2((0,T);H^{-1}(\Omega ))\right)}^3$, such that

$$(u_1^{(n_i)},u_2^{(n_i)})\stackrel{\ast }{\rightharpoonup }(u_1^{\ast },u_2^{\ast })\mathrm{in}{(L^{\infty }(Q_T))}^2\mathrm{as}i\to \infty ,$$

(50)

$$(z_1^{(n_i)},z_2^{(n_i)},z_3^{(n_i)})\rightharpoonup (z_1^{\ast },z_2^{\ast },z_3^{\ast })\mathrm{in}{\left(L^2((0,T);H^1(\Omega ))\right)}^3\mathrm{as}i\to \infty ,$$

(51)

$$({(z_1^{(n_i)})}_t,{(z_2^{(n_i)})}_t,{(z_3^{(n_i)})}_t)\rightharpoonup ({(z_1^{\ast })}_t,{(z_2^{\ast })}_t,{(z_3^{\ast })}_t)\mathrm{in}{\left(L^2((0,T);H^{-1}(\Omega ))\right)}^3\mathrm{as}i\to \infty .$$

(52)

From (51) and (52), we get

$$(z_1^{(n_i)},z_2^{(n_i)},z_3^{(n_i)})\to (z_1^{\ast },z_2^{\ast },z_3^{\ast })\mathrm{in}{(L^2(Q_T))}^3\mathrm{as}i\to \infty .$$

(53)

Since $(z_1^{(n_i)},z_2^{(n_i)},z_3^{(n_i)})$ is the weak solution of the problem (8)–(12) with $u_1=u_1^{(n_i)},u_2=u_2^{(n_i)}(i=1,2,3)$, for any $({\varphi }_1,{\varphi }_2,{\varphi }_3)\in {(C^{\infty }(\overline{Q_T}))}^3$ with ${\displaystyle \frac{\partial {\varphi }_1}{\partial n}}={\displaystyle \frac{\partial {\varphi }_2}{\partial n}}={\displaystyle \frac{\partial {\varphi }_3}{\partial n}}=0$ and ${\varphi }_1(x,T)={\varphi }_2(x,T)={\varphi }_3(x,T)=0$, we have

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_1^{(n_i)}{\displaystyle \frac{\partial {\varphi }_1}{\partial t}}dxdt-{\int }_{\Omega }{z}_{1,0}(x){\varphi }_1(x,0)dx+d_1{\displaystyle {\int \int }_{Q_T}}\nabla z_1^{(n_i)}·\nabla {\varphi }_{1}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(b_1{z}_1^{(n_i)}-(1-u_1^{(n_i)})b_2{z}_1^{(n_i)}{z}_2^{(n_i)}-b_3{z}_1^{(n_i)}{z}_3^{(n_i)}+u_2^{(n_i)}{z}_2^{(n_i)}\right){\varphi }_{1}dxdt.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_2^{(n_i)}{\displaystyle \frac{\partial {\varphi }_2}{\partial t}}dxdt-{\int }_{\Omega }{z}_{2,0}(x){\varphi }_2(x,0)dx+d_2{\displaystyle {\int \int }_{Q_T}}\nabla z_2^{(n_i)}·\nabla {\varphi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(-c_1{z}_2^{(n_i)}+(1-u_1^{(n_i)})b_2{z}_1^{(n_i)}{z}_2^{(n_i)}-c_2{z}_2^{(n_i)}{z}_3^{(n_i)}-u_2^{(n_i)}{z}_2^{(n_i)}\right){\varphi }_{2}dxdt,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_3^{(n_i)}{\displaystyle \frac{\partial {\varphi }_3}{\partial t}}dxdt-{\int }_{\Omega }{z}_{3,0}(x){\varphi }_3(x,0)dx+d_3{\displaystyle {\int \int }_{Q_T}}\nabla z_3^{(n_i)}·\nabla {\varphi }_{3}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(e_1{z}_3^{(n_i)}-e_2{(z_3^{(n_i)})}^2+e_3{z}_1^{(n_i)}{z}_3^{(n_i)}+e_4{z}_2^{(n_i)}{z}_3^{(n_i)}\right){\varphi }_{3}dxdt.\hfill \end{array}$$

Letting $i\to \infty$, we get

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_1^{\ast }{\displaystyle \frac{\partial {\varphi }_1}{\partial t}}dxdt-{\int }_{\Omega }{z}_{1,0}(x){\varphi }_1(x,0)dx+d_1{\displaystyle {\int \int }_{Q_T}}\nabla z_1^{\ast }·\nabla {\varphi }_{1}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(b_1{z}_1^{\ast }-(1-u_1^{\ast })b_2{z}_1^{\ast }{z}_2^{\ast }-b_3{z}_1^{\ast }{z}_3^{\ast }+u_2^{\ast }{z}_2^{\ast }\right){\varphi }_{1}dxdt,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_2^{\ast }{\displaystyle \frac{\partial {\varphi }_2}{\partial t}}dxdt-{\int }_{\Omega }{z}_{2,0}(x){\varphi }_2(x,0)dx+d_2{\displaystyle {\int \int }_{Q_T}}\nabla z_2^{\ast }·\nabla {\varphi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(-c_1{z}_2^{\ast }+(1-u_1^{\ast })b_2{z}_1^{\ast }{z}_2^{\ast }-c_2{z}_2^{\ast }{z}_3^{\ast }-u_2^{\ast }{z}_2^{\ast }\right){\varphi }_{2}dxdt,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{z}_3^{\ast }{\displaystyle \frac{\partial {\varphi }_3}{\partial t}}dxdt-{\int }_{\Omega }{z}_{3,0}(x){\varphi }_3(x,0)dx+d_3{\displaystyle {\int \int }_{Q_T}}\nabla z_3^{\ast }·\nabla {\varphi }_{3}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(e_1{z}_3^{\ast }-e_2{(z_3^{\ast })}^2+e_3{z}_1^{\ast }{z}_3^{\ast }+e_4{z}_2^{\ast }{z}_3^{\ast }\right){\varphi }_{3}dxdt.\hfill \end{array}$$

from (50) and (53).

Hence, ${\mathit{z}}^{\ast }=(z_1^{\ast },z_2^{\ast },z_3^{\ast })$ is the weak solution the problem (8)–(12) with $u_1=u_1^{\ast },$$u_2=u_2^{\ast }.$

By the weak semi-continuity of $L^2$ norm, we can obtain

$$\begin{array}{cc}\hfill & \underset{i\to \infty }{\underline{lim}}J\left(u_1^{\left(n_i\right)},u_2^{\left(n_i\right)}\right)\hfill \\ \hfill =& \underset{i\to \infty }{\underline{lim}}{\int }_0^T{\int }_{\Omega }\left(w{z}_2^{(n_i)}+{\displaystyle \frac{1}{2}}{a}_1{(u_1^{(n_i)})}^2+{\displaystyle \frac{1}{2}}{a}_2{(u_2^{(n_i)})}^2\right)dxdt\hfill \\ \hfill \ge & {\int }_0^T{\int }_{\Omega }\left(w{z}_2^{\ast }+{\displaystyle \frac{1}{2}}{a}_1{(u_1^{\ast })}^2+{\displaystyle \frac{1}{2}}{a}_2{(u_2^{\ast })}^2\right)dxdt\hfill \\ \hfill =& J(u_1^{\ast },u_2^{\ast }).\hfill \end{array}$$

It follows from (49) that

$$\begin{array}{c}\hfill J(u_1^{\ast },u_2^{\ast })\le \underset{i\to \infty }{\underline{lim}}J(u_1^{(n_i)},u_2^{(n_i)})=\underset{i\to \infty }{lim}J(u_1^{(n_i)},u_2^{(n_i)})=\underset{(u_1,u_2)\in U_{ad}}{inf}J(u_1,u_2).\end{array}$$

The proof is completed. □

3.2. The First Order Necessary Condition

In this subsection, we study the first order necessary condition.

First of all, we consider the following problem.

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial w_1}{\partial t}}-d_1\Delta w_1=a_{11}{w}_1+a_{12}{w}_2+a_{13}{w}_3+f_1,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(54)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial w_2}{\partial t}}-d_2\Delta w_2=a_{21}{w}_1+a_{22}{w}_2+a_{23}{w}_3+f_2,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(55)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial w_3}{\partial t}}-d_3\Delta w_3=a_{31}{w}_1+a_{32}{w}_2+a_{33}{w}_3+f_3,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(56)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial w_1}{\partial n}}={\displaystyle \frac{\partial w_2}{\partial n}}={\displaystyle \frac{\partial w_3}{\partial n}}=0,\hfill & \hfill & (x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(57)

$$\begin{array}{cccc}\hfill & w_1(x,0)=w_{0,1},w_2(x,0)=w_{0,2},w_3(x,0)=w_{0,3},\hfill & \hfill & x\in \Omega ,\hfill \end{array}$$

(58)

where $a_{ij}\in L^{\infty }(Q_T),f_i\in L^2(Q_T),w_{0,i}\in H^1(\Omega )\mathrm{for}i,j=1,2,3.$

Next, we give the following lemma.

Lemma 3.

There exists a uniquely solution $(w_1,w_2,w_3)\in {(W_2^{2,1}(Q_T))}^3$ to the problem (54)–(58) such that

$$\begin{array}{cc}\hfill & ||w_1{||}_{W_2^{2,1}(Q_T)}+||w_2{||}_{W_2^{2,1}(Q_T)}+||w_3{||}_{W_2^{2,1}(Q_T)}\hfill \\ \hfill \le & C\left(||f_1{||}_{L^2(Q_T)}+||f_2{||}_{L^2(Q_T)}+||f_3{||}_{L^2(Q_T)}+||w_{0,1}{||}_{H^1(\Omega )}+||w_{0,2}{||}_{H^1(\Omega )}+||w_{0,3}{||}_{H^1(\Omega )}\right),\hfill \end{array}$$

where C is depending on $\Omega ,T,||a_{ij}{||}_{L^{\infty }(Q_T)},i,j=1,2,3.$

In addition, if $w_{0,i}\in L^{\infty }(\Omega )$ and $f_i\in L^{\infty }(Q_T)$ for $i=1,2,3$, then we have

$$\begin{array}{c}\hfill ||w_1{||}_{L^{\infty }(Q_T)}+||w_2{||}_{L^{\infty }(Q_T)}+||w_3{||}_{L^{\infty }(Q_T)}\le H{e}^{\tau T},\end{array}$$

(59)

where

$$\begin{array}{c}\hfill H=\underset{i=1,2,3}{max}\{||w_{0,i}{||}_{L^{\infty }(\Omega )},||f_i{||}_{L^{\infty }(Q_T)}\},\tau =\underset{i=1,2,3}{max}\{\sum _{j=1}^3||a_{ij}{||}_{L^{\infty }(Q_T)}+1\}.\end{array}$$

(60)

Proof. 

We only prove (59).

Define

$$\begin{array}{c}\hfill ({\overline{w}}_1,{\overline{w}}_2,{\overline{w}}_3)=(H{e}^{\tau t},H{e}^{\tau t},H{e}^{\tau t}),({\underline{w}}_1,{\underline{w}}_2,{\underline{w}}_3)=(-H{e}^{\tau t},-H{e}^{\tau t},-H{e}^{\tau t}).\end{array}$$

(61)

where $H,\tau$ are defined as (60) and (61). Similar to the proof of (34), we have

$$\begin{array}{c}\hfill {\underline{w}}_i\le w_i\le {\overline{w}}_i,i=1,2,3.\end{array}$$

The proof is completed. □

Theorem 3.

Assume ${\mathit{u}}^{\ast }=(u_1^{\ast },u_2^{\ast })$ is the optimal control, denote by ${\mathit{z}}^{\ast }=(z_1^{\ast },z_2^{\ast },z_3^{\ast })$ the solution to the problem (8)–(12) with $u_1=u_1^{\ast },u_2=u_2^{\ast }.$ Let ${\mathit{P}}^{\ast }=(P_1^{\ast },P_2^{\ast },P_3^{\ast })$ be the solution to the following problem

$$\begin{array}{cccc}\hfill & -{\displaystyle \frac{\partial P_1^{\ast }}{\partial t}}-d_1\Delta P_1^{\ast }=a_{11}{P}_1^{\ast }+a_{21}{P}_2^{\ast }+a_{31}{P}_3^{\ast },\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(62)

$$\begin{array}{cccc}\hfill & -{\displaystyle \frac{\partial P_2^{\ast }}{\partial t}}-d_2\Delta P_2^{\ast }=a_{12}{P}_1^{\ast }+a_{22}{P}_2^{\ast }+a_{32}{P}_3^{\ast }+w,\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(63)

$$\begin{array}{cccc}\hfill & -{\displaystyle \frac{\partial P_3^{\ast }}{\partial t}}-d_3\Delta P_3^{\ast }=a_{13}{P}_1^{\ast }+a_{23}{P}_2^{\ast }+a_{33}{P}_3^{\ast },\hfill & \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(64)

$$\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial P_1^{\ast }}{\partial n}}={\displaystyle \frac{\partial P_2^{\ast }}{\partial n}}={\displaystyle \frac{\partial P_3^{\ast }}{\partial n}}=0,\hfill & \hfill & (x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(65)

$$\begin{array}{cccc}\hfill & P_1^{\ast }(x,T)=P_2^{\ast }(x,T)=P_3^{\ast }(x,T)=0,\hfill & \hfill & x\in \Omega ,\hfill \end{array}$$

(66)

where

$$\begin{array}{c}{a}_{11}=b_1-(1-u_1^{\ast })b_2{z}_2^{\ast }-b_3{z}_3^{\ast },a_{12}=-(1-u_1^{\ast })b_2{z}_1^{\ast }+u_2^{\ast },a_{13}=-b_3{z}_1^{\ast },\\ a_{21}=(1-u_1^{\ast })b_2{z}_2^{\ast },a_{22}=-c_1+(1-u_1^{\ast })b_2{z}_1^{\ast }-c_2{z}_3^{\ast }-u_2^{\ast },a_{23}=-c_2{z}_2^{\ast },\\ a_{31}=e_3{z}_3^{\ast },a_{32}=e_4{z}_3^{\ast },a_{33}=e_1-2e_2{z}_3^{\ast }+e_3{z}_1^{\ast }+e_4{z}_2^{\ast }.\end{array}$$

Then, for any $\mathit{u}=(u_1,u_2)\in U_{ad}$, it holds that

$${\displaystyle {\int \int }_{Q_T}}\left(q_1\left(b_2{z}_1^{\ast }{z}_2^{\ast }(P_1^{\ast }-P_2^{\ast })+a_1{u}_1^{\ast }\right)+q_2\left(z_2^{\ast }(P_1^{\ast }-P_2^{\ast })+a_2{u}_2^{\ast }\right)\right)dxdt\ge 0.$$

Proof. 

For any $\mathit{u}=(u_1,u_2)\in U_{ad}$, denote $\mathit{q}=(q_1,q_2)=(u_1-u_1^{\ast },u_2-u_2^{\ast })$. For $n=1,2,\cdots$, denote by ${\mathit{z}}^{(n)}=(z_1^{(n)},z_2^{(n)},z_3^{(n)})$ the solution of the problem (8)–(12) with $\mathit{u}={\mathit{u}}^{\ast }+{\displaystyle \frac{1}{n}}\mathit{q}$. Denote ${\mathit{v}}^{(n)}=(v_1^{(n)},v_2^{(n)},v_3^{(n)})={\mathit{z}}^{(n)}-{\mathit{z}}^{\ast }$, then ${\mathit{v}}^{(n)}$ satisfies

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial v_1^{(n)}}{\partial t}}-d_1\Delta v_1^{(n)}=b_1{v}_1^{(n)}-(1-u_1^{\ast })b_2{z}_2^{(n)}{v}_1^{(n)}-(1-u_1^{\ast })b_2{z}_1^{\ast }{v}_2^{(n)}+{\displaystyle \frac{1}{n}}{q}_1{b}_2{z}_1^{(n)}{z}_2^{(n)}-b_3{v}_1^{(n)}{z}_3^{(n)}\hfill \\ \hfill & -b_3{v}_3^{(n)}{z}_1^{\ast }+u_2^{\ast }{v}_2^{(n)}+{\displaystyle \frac{1}{n}}{q}_2{z}_2^{(n)},(x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial v_2^{(n)}}{\partial t}}-d_2\Delta v_2^{(n)}=-c_1{v}_2^{(n)}+(1-u_1^{\ast })b_2{z}_2^{(n)}{v}_1^{(n)}+(1-u_1^{\ast })b_2{z}_1^{\ast }{v}_2^{(n)}-{\displaystyle \frac{1}{n}}{q}_1{b}_2{z}_1^{(n)}{z}_2^{(n)}-c_2{v}_2^{(n)}{z}_3^{(n)}\hfill \\ \hfill & -c_2{v}_3^{(n)}{z}_2^{\ast }-u_2^{\ast }{v}_2^{(n)}-{\displaystyle \frac{1}{n}}{q}_2{z}_2^{(n)},(x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial v_3^{(n)}}{\partial t}}-d_3\Delta v_3^{(n)}=e_1{v}_3^{(n)}-e_2{v}_3^{(n)}(z_3^{(n)}+z_3^{\ast })+e_3{v}_1^{(n)}{z}_3^{(n)}+e_3{z}_1^{\ast }{v}_3^{(n)}+e_4{v}_2^{(n)}{z}_3^{(n)}+e_4{z}_2^{\ast }{v}_3^{(n)},\hfill \\ \hfill & (x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial v_1^{(n)}}{\partial n}}={\displaystyle \frac{\partial v_2^{(n)}}{\partial n}}={\displaystyle \frac{\partial v_3^{(n)}}{\partial n}}=0,(x,t)\in \mathsf{\Gamma },\hfill \\ \hfill & v_1^{(n)}(x,0)=v_2^{(n)}(x,0)=v_3^{(n)}(x,0)=0,x\in \Omega .\hfill \end{array}$$

From Lemma 3, ${\mathit{v}}^{(n)}\in {(W_2^{2,1}(Q_T)\cap L^{\infty }(Q_T))}^3$ and

$$\parallel v_1^{(n)}{\parallel }_{L^2(Q_T)}+\hspace{0.17em}\parallel v_2^{(n)}{\parallel }_{L^2(Q_T)}+\hspace{0.17em}{\parallel v_3^{(n)}\parallel }_{L^2(Q_T)}\le {\displaystyle \frac{1}{n}}C,$$

we can obtain

$${\mathit{z}}^{(n)}\to {\mathit{z}}^{\ast }\mathrm{in}{(L^2(Q_T))}^3\mathrm{as}n\to \infty .$$

(67)

Denote ${\Psi }^{(n)}=n{\mathit{v}}^{(n)}$. Then, ${\Psi }^{(n)}\in {(W_2^{2,1}(Q_T)\cap L^{\infty }(Q_T))}^3$ is the solution of the following system

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\Psi }_1^{(n)}}{\partial t}}-d_1\Delta {\Psi }_1^{(n)}=b_1{\Psi }_1^{(n)}-(1-u_1^{\ast })b_2{z}_2^{(n)}{\Psi }_1^{(n)}-(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2^{(n)}+q_1{b}_2{z}_1^{(n)}{z}_2^{(n)}-b_3{\Psi }_1^{(n)}{z}_3^{(n)}\hfill \\ \hfill & -b_3{\Psi }_3^{(n)}{z}_1^{\ast }+u_2^{\ast }{\Psi }_2^{(n)}+q_2{z}_2^{(n)},(x,t)\in Q_T,\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\Psi }_2^{(n)}}{\partial t}}-d_2\Delta {\Psi }_2^{(n)}=-c_1{\Psi }_2^{(n)}+(1-u_1^{\ast })b_2{z}_2^{(n)}{\Psi }_1^{(n)}+(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2^{(n)}-q_1{b}_2{z}_1^{(n)}{z}_2^{(n)}-c_2{\Psi }_2^{(n)}{z}_3^{(n)}\hfill \\ \hfill & -c_2{\Psi }_3^{(n)}{z}_2^{\ast }-u_2^{\ast }{\Psi }_2^{(n)}-q_2{z}_2^{(n)},(x,t)\in Q_T,\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\Psi }_3^{(n)}}{\partial t}}-d_3\Delta {\Psi }_3^{(n)}=e_1{\Psi }_3^{(n)}-e_2{\Psi }_3^{(n)}(z_3^{(n)}+z_3^{\ast })+e_3{\Psi }_1^{(n)}{z}_3^{(n)}+e_3{z}_1^{\ast }{\Psi }_3^{(n)}+e_3{\Psi }_2^{(n)}{z}_3^{(n)}+e_4{z}_2^{\ast }{\Psi }_3^{(n)},\hfill \\ \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\Psi }_1^{(n)}}{\partial n}}={\displaystyle \frac{\partial {\Psi }_2^{(n)}}{\partial n}}={\displaystyle \frac{\partial {\Psi }_3^{(n)}}{\partial n}}=0,(x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill & {\Psi }_1^{(n)}(x,0)={\Psi }_2^{(n)}(x,0)={\Psi }_3^{(n)}(x,0)=0,x\in \Omega .\hfill \end{array}$$

(72)

From Lemma 3, we know that there exists a subsequence of ${\left\{{\Psi }^{(n)}\right\}}_{n=1}^{\infty }$, denoted by itself, and $\Psi \in {(W_2^{2,1}(Q_T)\cap L^{\infty }(Q_T))}^3$ such that

$${\Psi }^{(n)}\to \Psi \mathrm{in}{(L^2(Q_T))}^3,\nabla {\Psi }^{(n)}\rightharpoonup \nabla \Psi \mathrm{in}(L^2{(Q_T;{\mathbb{R}}^N)}^3,asn\to \infty .$$

(73)

Since ${\Psi }^{(n)}$ is the weak solution of the system (68)–(72), we have for any $\mathit{\varphi }=({\varphi }_1,{\varphi }_2,{\varphi }_3)\in {(C^{\infty }({\overline{Q}}_T))}^3$ with ${\displaystyle \frac{\partial {\varphi }_1^{\ast }}{\partial n}}={\displaystyle \frac{\partial {\varphi }_2^{\ast }}{\partial n}}={\displaystyle \frac{\partial {\varphi }_3^{\ast }}{\partial n}}=0$ and ${\varphi }_1(x,T)={\varphi }_2(x,T)={\varphi }_3(x,T)=0$,

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{\Psi }_1^{(n)}{\displaystyle \frac{\partial {\varphi }_1}{\partial t}}dxdt+d_1{\displaystyle {\int \int }_{Q_T}}\nabla {\Psi }_1^{(n)}·\nabla {\varphi }_{1}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}(b_1{\Psi }_1^{(n)}-(1-u_1^{\ast })b_2{z}_2^{(n)}{\Psi }_1^{(n)}-(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2^{(n)}+q_1{b}_2{z}_1^{(n)}{z}_2^{(n)}-b_3{\Psi }_1^{(n)}{z}_3^{(n)}\hfill \\ \hfill & -b_3{\Psi }_3^{(n)}{z}_1^{\ast }+u_2^{\ast }{\Psi }_2^{(n)}+q_2{z}_2^{(n)}){\varphi }_{1}dxdt,\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill & -{\int }_{Q_T}{\Psi }_2^{(n)}{\displaystyle \frac{\partial {\varphi }_2}{\partial t}}dxdt+d_2{\int }_{Q_T}\nabla {\Psi }_2^{(n)}·\nabla {\varphi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}(-c_1{\Psi }_2^{(n)}+(1-u_1^{\ast })b_2{z}_2^{(n)}{\Psi }_1^{(n)}+(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2^{(n)}-q_1{b}_2{z}_1^{(n)}{z}_2^{(n)}-c_2{\Psi }_2^{(n)}{z}_3^{(n)}\hfill \\ \hfill & -c_2{\Psi }_3^{(n)}{z}_2^{\ast }-u_2^{\ast }{\Psi }_2^{(n)}-q_2{z}_2^{(n)}){\varphi }_{2}dxdt,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{\Psi }_3^{(n)}{\displaystyle \frac{\partial {\varphi }_3}{\partial t}}dxdt+d_3{\displaystyle {\int \int }_{Q_T}}\nabla {\Psi }_3^{(n)}·\nabla {\varphi }_{3}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(e_1{\Psi }_3^{(n)}-e_2{\Psi }_3^{(n)}(z_3^{(n)}+z_3^{\ast })+e_3{\Psi }_1^{(n)}{z}_3^{(n)}+e_3{z}_1^{\ast }{\Psi }_3^{(n)}+e_3{\Psi }_2^{(n)}{z}_3^{(n)}+e_4{z}_2^{\ast }{\Psi }_3^{(n)}\right){\varphi }_{3}dxdt.\hfill \end{array}$$

Letting $n\to \infty$ in the above equalities, we can get from (67) and (73) that

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{\Psi }_1{\displaystyle \frac{\partial {\varphi }_1}{\partial t}}dxdt+d_1{\displaystyle {\int \int }_{Q_T}}\nabla {\Psi }_1·\nabla {\varphi }_{1}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}(b_1{\Psi }_1-(1-u_1^{\ast })b_2{z}_2^{\ast }{\Psi }_1-(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2+q_1{b}_2{z}_1^{\ast }{z}_2^{\ast }-b_3{\Psi }_1{z}_3^{\ast }-b_3{\Psi }_3{z}_1^{\ast }\hfill \\ \hfill & +u_2^{\ast }{\Psi }_2+q_2{z}_2^{\ast }){\varphi }_{1}dxdt,\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill & -{\int }_{Q_T}{\Psi }_2{\displaystyle \frac{\partial {\varphi }_2}{\partial t}}dxdt+d_2{\int }_{Q_T}\nabla {\Psi }_2·\nabla {\varphi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}(-c_1{\Psi }_2+(1-u_1^{\ast })b_2{z}_2^{\ast }{\Psi }_1+(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2-q_1{b}_2{z}_1^{\ast }{z}_2^{\ast }-c_2{\Psi }_2{z}_3^{\ast }\hfill \\ \hfill & -c_2{\Psi }_3{z}_2^{\ast }-u_2^{\ast }{\Psi }_2-q_2{z}_2^{\ast }){\varphi }_{2}dxdt,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\displaystyle {\int \int }_{Q_T}}{\Psi }_3{\displaystyle \frac{\partial {\varphi }_3}{\partial t}}dxdt+d_3{\displaystyle {\int \int }_{Q_T}}\nabla {\Psi }_3·\nabla {\varphi }_{3}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left(e_1{\Psi }_3-2e_2{\Psi }_3{z}_3^{\ast }+e_3{\Psi }_1{z}_3^{\ast }+e_3{z}_1^{\ast }{\Psi }_3+e_3{\Psi }_2{z}_3^{\ast }+e_4{z}_2^{\ast }{\Psi }_3\right){\varphi }_{3}dxdt.\hfill \end{array}$$

Hence, $\Psi =({\psi }_1,{\psi }_2,{\psi }_3)$ is the solution of the following system

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\Psi }_1}{\partial t}}-d_1\Delta {\Psi }_1=b_1{\Psi }_1-(1-u_1^{\ast })b_2{z}_2^{\ast }{\Psi }_1-(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2+q_1{b}_2{z}_1^{\ast }{z}_2^{\ast }-b_3{\Psi }_1{z}_3^{\ast }-b_3{\Psi }_3{z}_1^{\ast }\hfill \\ \hfill & +u_2^{\ast }{\Psi }_2+q_2{z}_2^{\ast },(x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial {\Psi }_2}{\partial t}}-d_2\Delta {\Psi }_2=-c_1{\Psi }_2+(1-u_1^{\ast })b_2{z}_2^{\ast }{\Psi }_1+(1-u_1^{\ast })b_2{z}_1^{\ast }{\Psi }_2-q_1{b}_2{z}_1^{\ast }{z}_2^{\ast }-c_2{\Psi }_2{z}_3^{\ast }\hfill \\ \hfill & -c_2{\Psi }_3{z}_2^{\ast }-u_2^{\ast }{\Psi }_2-q_2{z}_2^{\ast },(x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial {\Psi }_3}{\partial t}}-d_3\Delta {\Psi }_3=e_1{\Psi }_3-2e_2{\Psi }_3{z}_3^{\ast }+e_3{\Psi }_1{z}_3^{\ast }+e_3{z}_1^{\ast }{\Psi }_3+e_3{\Psi }_2{z}_3^{\ast }+e_4{z}_2^{\ast }{\Psi }_3,(x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial {\Psi }_1}{\partial n}}={\displaystyle \frac{\partial {\Psi }_2}{\partial n}}={\displaystyle \frac{\partial {\Psi }_3}{\partial n}}=0,(x,t)\in \mathsf{\Gamma },\hfill \\ \hfill & {\Psi }_1(x,0)={\Psi }_2(x,0)={\Psi }_3(x,0)=0,x\in \Omega .\hfill \end{array}$$

By computation, we have

$$\begin{array}{cc}\hfill & J({\mathit{u}}^{\ast }+{\displaystyle \frac{1}{n}}\mathit{q})-J({\mathit{u}}^{\ast })\hfill \\ \hfill =& {\int }_0^T{\int }_{\Omega }\left(w{z}_2^{(n)}-w{z}_2^{\ast }+{\displaystyle \frac{1}{2}}{a}_1\left({(u_1^{\ast }+{\displaystyle \frac{1}{n}}{q}_1)}^2-{(u_1^{\ast })}^2\right)+{\displaystyle \frac{1}{2}}{a}_2\left({(u_2^{\ast }+{\displaystyle \frac{1}{n}}{q}_2)}^2-{(u_2^{\ast })}^2\right)\right)dxdt\hfill \\ \hfill =& {\int }_0^T{\int }_{\Omega }\left(w{v}_2^{(n)}+{\displaystyle \frac{1}{2}}{a}_1({\displaystyle \frac{2}{n}}{u}_1^{\ast }{q}_1+{\displaystyle \frac{1}{n^2}}{q}_1^2)+{\displaystyle \frac{1}{2}}{a}_2({\displaystyle \frac{2}{n}}{u}_2^{\ast }{q}_2+{\displaystyle \frac{1}{n^2}}{q}_2^2)\right)dxdt.\hfill \end{array}$$

It follows from (67) and (73) that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{J({\mathit{u}}^{\ast }+\frac{1}{n}\mathit{q})-J({\mathit{u}}^{\ast })}{1/n}}\hfill \\ \hfill =& {\int }_0^T{\int }_{\Omega }\left(w{\Psi }_2^{(n)}+{\displaystyle \frac{1}{2}}{a}_1(2u_1^{\ast }{q}_1+{\displaystyle \frac{1}{n}}{q}_1^2)+{\displaystyle \frac{1}{2}}{a}_2(2u_2^{\ast }{q}_2+{\displaystyle \frac{1}{n}}{q}_2^2)\right)dxdt.\hfill \end{array}$$

Then, we can get

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\displaystyle \frac{J({\mathit{u}}^{\ast }+\frac{1}{n}\mathit{q})-J({\mathit{u}}^{\ast })}{1/n}}={\int }_0^T{\int }_{\Omega }(w{\Psi }_2+a_1{u}_1^{\ast }{q}_1+a_2{u}_2^{\ast }{q}_2)dxdt.\end{array}$$

(74)

Since $(P_1^{\ast },P_2^{\ast },P_3^{\ast })$ is the solution to the system (62)–(66), we have

$$0=\underset{Q_T}{\int \int }(-{\Psi }_1{\displaystyle \frac{\partial P_1^{\ast }}{\partial t}}+d_1\nabla {\Psi }_1·\nabla P_1^{\ast }-a_{11}{\Psi }_1{P}_1^{\ast }-a_{21}{\Psi }_1{P}_2^{\ast }-a_{31}{P}_3^{\ast }{\Psi }_1)dxdt,$$

(75)

$${\displaystyle {\int \int }_{Q_T}}w{\Psi }_{2}dxdt={\displaystyle {\int \int }_{Q_T}}(-{\Psi }_2{\displaystyle \frac{\partial P_2^{\ast }}{\partial t}}+d_2\nabla {\Psi }_2·\nabla P_2^{\ast }-a_{12}{\Psi }_2{P}_1^{\ast }-a_{22}{\Psi }_2{P}_2^{\ast }-a_{32}{P}_3^{\ast }{\Psi }_2)dxdt,$$

(76)

$$0={\displaystyle {\int \int }_{Q_T}}(-{\Psi }_3{\displaystyle \frac{\partial P_3^{\ast }}{\partial t}}+d_3\nabla {\Psi }_3·\nabla P_3^{\ast }-a_{13}{\Psi }_3{P}_1^{\ast }-a_{23}{\Psi }_3{P}_2^{\ast }-a_{33}{P}_3^{\ast }{\Psi }_3)dxdt.$$

(77)

Summing up the equations (75)–(77), we can obtain

$$\begin{array}{cc}\hfill & {\displaystyle {\int \int }_{Q_T}}w{\Psi }_{2}dxdt\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}((-{\Psi }_1{\displaystyle \frac{\partial P_1^{\ast }}{\partial t}}-d_1{\Psi }_1·▵P_1^{\ast })+(-{\Psi }_2{\displaystyle \frac{\partial P_2^{\ast }}{\partial t}}-d_2{\Psi }_2·▵P_2^{\ast })\hfill \\ \hfill & +(-{\Psi }_3{\displaystyle \frac{\partial P_3^{\ast }}{\partial t}}-d_3{\Psi }_3·▵P_3^{\ast })-\sum _{i,j=1}^3{a}_{ij}{\Psi }_j{P}_i^{\ast })dxdt,\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}(P_1^{\ast }({\displaystyle \frac{\partial {\Psi }_1}{\partial t}}-d_1▵{\Psi }_1)+P_2^{\ast }({\displaystyle \frac{\partial {\Psi }_2}{\partial t}}-d_2▵{\Psi }_2)+P_3^{\ast }({\displaystyle \frac{\partial {\Psi }_3}{\partial t}}-d_3▵{\Psi }_3)\hfill \\ \hfill & -\sum _{i,j=1}^3{a}_{ij}{\Psi }_j{P}_i^{\ast })dxdt,\hfill \\ \hfill =& {\displaystyle {\int \int }_{Q_T}}\left((q_1{b}_2{z}_1^{\ast }{z}_2^{\ast }+q_2{z}_2^{\ast })P_1^{\ast }-(q_1{b}_2{z}_1^{\ast }{z}_2^{\ast }+q_2{z}_2^{\ast })P_2^{\ast }\right)dxdt.\hfill \end{array}$$

(78)

Combining (74) with (78), we have

$$\underset{n\to \infty }{lim}{\displaystyle \frac{J({\mathit{u}}^{\ast }+\frac{1}{n}\mathit{q})-J({\mathit{u}}^{\ast })}{1/n}}=\underset{Q_T}{\int \int }\left(\left(b_2{z}_1^{\ast }{z}_2^{\ast }(P_1^{\ast }-P_2^{\ast })+a_1{u}_1^{\ast }\right)q_1+\left(z_2^{\ast }(P_1^{\ast }-P_2^{\ast })+a_2{u}_2^{\ast }\right)q_2\right)dxdt.$$

Since $J({\mathit{u}}^{\ast })={inf}_{\mathit{u}\in U_{ad}}J(\mathit{u})$, we get

$$\underset{n\to \infty }{lim}{\displaystyle \frac{J({\mathit{u}}^{\ast }+\frac{1}{n}\mathit{q})-J({\mathit{u}}^{\ast })}{1/n}}\ge 0.$$

Hence, it holds that

$${\displaystyle {\int \int }_{Q_T}}\left(\left(b_2{z}_1^{\ast }{z}_2^{\ast }(P_1^{\ast }-P_2^{\ast })+a_1{u}_1^{\ast }\right)q_1+\left(z_2^{\ast }(P_1^{\ast }-P_2^{\ast })+a_2{u}_2^{\ast }\right)q_2\right)dxdt\ge 0.$$

The proof is completed. □

From Theorem 3, we can get the formula of the optimal control by standard argument [18].

Corollary 1.

If $\mathit{u}=(u_1^{\ast },u_2^{\ast })$ is an optimal control, then

$$u_1^{\ast }=min\{M_1,{\displaystyle \frac{b_2}{a_1}}{z}_1{z}_2{(P_2-P_1)}^{+}\},u_2^{\ast }=min\{M_2,{\displaystyle \frac{1}{a_2}}{z}_2{(P_2-P_1)}^{+}\}.$$

(79)

where $(z_1,z_2,z_3,P_1,P_2,P_3)$ is a solution of the optimality system

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial z_1}{\partial t}}-d_1▵z_1=b_1{z}_1-(1-min\{M_1,{\displaystyle \frac{b_2}{a_1}}{z}_1{z}_2{(P_2-P_1)}^{+}\})b_2{z}_1{z}_2-b_3{z}_1{z}_3\hfill \\ \hfill & +min\{M_2,{\displaystyle \frac{1}{a_2}}{z}_2{(P_2-P_1)}^{+}\}{z}_2,(x,t)\in Q_T,\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial z_2}{\partial t}}-d_2▵z_2=-c_1{z}_2+(1-min\{M_1,{\displaystyle \frac{b_2}{a_1}}{z}_1{z}_2{(P_2-P_1)}^{+}\})b_2{z}_1{z}_2-c_2{z}_2{z}_3\hfill \\ \hfill & -min\{M_2,{\displaystyle \frac{1}{a_2}}{z}_2{(P_2-P_1)}^{+}\}{z}_2,(x,t)\in Q_T,\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial z_3}{\partial t}}-d_3▵z_3=e_1{z}_3-e_2{z}_3^2+e_3{z}_1{z}_3+e_4{z}_2{z}_3,(x,t)\in Q_T,\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{\partial P_1}{\partial t}}-d_1\Delta P_1=a_{11}{P}_1+a_{21}{P}_2+a_{31}{P}_3,(x,t)\in Q_T,\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{\partial P_2}{\partial t}}-d_2\Delta P_2=a_{12}{P}_1+a_{22}{P}_2+a_{32}{P}_3+w,(x,t)\in Q_T,\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{\partial P_3}{\partial t}}-d_3\Delta P_3=a_{13}{P}_1+a_{23}{P}_2+a_{33}{P}_3,(x,t)\in Q_T,\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial z_1}{\partial n}}={\displaystyle \frac{\partial z_2}{\partial n}}={\displaystyle \frac{\partial z_3}{\partial n}}=0,(x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial P_1}{\partial n}}={\displaystyle \frac{\partial P_2}{\partial n}}={\displaystyle \frac{\partial P_3}{\partial n}}=0,(x,t)\in \mathsf{\Gamma },\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill & z_1(x,0)=z_{1,0}(x),z_2(x,0)=z_{2,0}(x),z_3(x,0)=z_{3,0}(x),x\in \Omega ,\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill & P_1(x,T)=P_2(x,T)=P_3(x,T)=0,x\in \Omega .\hfill \end{array}$$

(89)

4. The Local Uniqueness of the Optimal Control

In this section, we prove the uniqueness of the solution to the optimality system (80)–(89) when T is sufficiently small.

Theorem 4.

There exists a positive constant Λ, such that if T is sufficiently small, the optimality system (80)–(89) has only one solution.

Proof. 

Take $T<1$. Assume $({\tilde{z}}_1,{\tilde{z}}_2,{\tilde{z}}_3,{\tilde{P}}_1,{\tilde{P}}_2,{\tilde{P}}_3)$ and $({\overline{z}}_1,{\overline{z}}_2,{\overline{z}}_3,{\overline{P}}_1,{\overline{P}}_2,{\overline{P}}_3)$ are two solutions of the optimality system (80)–(89). From Theorem 1, we have

$$\begin{array}{c}\hfill \parallel {\tilde{z}}_i{\parallel }_{L^{\infty }(Q_T)}\le C_1,{\parallel {\overline{z}}_i\parallel }_{L^{\infty }(Q_T)}\le C_1,i=1,2,3,\end{array}$$

(90)

where $C_1$ is a positive constant independent of T. From Theorem 4, we have

$$\begin{array}{c}\hfill \parallel {\tilde{P}}_i{\parallel }_{L^{\infty }(Q_T)}\le C_2,{\parallel {\overline{P}}_i\parallel }_{L^{\infty }(Q_T)}\le C_2,i=1,2,3,\end{array}$$

(91)

where $C_2$ is a positive constant independent of T. Let

$$\begin{array}{c}\hfill {\tilde{z}}_1=e^{\lambda t}{\tilde{\xi }}_1,{\tilde{z}}_2=e^{\lambda t}{\tilde{\xi }}_2,{\tilde{z}}_3=e^{\lambda t}{\tilde{\xi }}_3,{\tilde{P}}_1=e^{-\lambda t}{\tilde{\zeta }}_1,{\tilde{P}}_2=e^{-\lambda t}{\tilde{\zeta }}_2,{\tilde{P}}_3=e^{-\lambda t}{\tilde{\zeta }}_3,\end{array}$$

(92)

$$\begin{array}{c}\hfill {\overline{z}}_1=e^{\lambda t}{\overline{\xi }}_1,{\overline{z}}_2=e^{\lambda t}{\overline{\xi }}_2,{\overline{z}}_3=e^{\lambda t}{\overline{\xi }}_3,{\overline{P}}_1=e^{-\lambda t}{\overline{\zeta }}_1,{\overline{P}}_2=e^{-\lambda t}{\overline{\zeta }}_2,{\overline{P}}_3=e^{-\lambda t}{\overline{\zeta }}_3.\end{array}$$

(93)

where $\lambda >0$ is to be determined.

From (79), define

$$\begin{array}{c}\hfill {\tilde{u}}_1=min\{M_1,{\displaystyle \frac{b_2}{a_1}}{\tilde{z}}_1{\tilde{z}}_2{({\tilde{P}}_2-{\tilde{P}}_1)}^{+}\},{\tilde{u}}_2=min\{M_2,{\displaystyle \frac{1}{a_2}}{\tilde{z}}_2{({\tilde{P}}_2-{\tilde{P}}_1)}^{+}\},\end{array}$$

(94)

$$\begin{array}{c}\hfill {\overline{u}}_1=min\{M_1,{\displaystyle \frac{b_2}{a_1}}{\overline{z}}_1{\overline{z}}_2{({\overline{P}}_2-{\overline{P}}_1)}^{+}\},{\overline{u}}_2=min\{M_2,{\displaystyle \frac{1}{a_2}}{\overline{z}}_2{({\overline{P}}_2-{\overline{P}}_1)}^{+}\}.\end{array}$$

(95)

Take (92)–(95) into the equations (80), we obtain

$$\begin{array}{c}\hfill \lambda {\tilde{\xi }}_1+{\displaystyle \frac{\partial {\tilde{\xi }}_1}{\partial t}}-d_1▵{\tilde{\xi }}_1=b_1{\tilde{\xi }}_1-(1-{\tilde{u}}_1)b_2{e}^{\lambda t}{\tilde{\xi }}_1{\tilde{\xi }}_2-b_3{e}^{\lambda t}{\tilde{\xi }}_1{\tilde{\xi }}_3+{\tilde{u}}_2{\tilde{\xi }}_2,\\ \hfill \lambda {\overline{\xi }}_1+{\displaystyle \frac{\partial {\overline{\xi }}_1}{\partial t}}-d_1▵{\overline{\xi }}_1=b_1{\overline{\xi }}_1-(1-{\overline{u}}_1)b_2{e}^{\lambda t}{\overline{\xi }}_1{\overline{\xi }}_2-b_3{e}^{\lambda t}{\overline{\xi }}_1{\overline{\xi }}_3+{\overline{u}}_2{\overline{\xi }}_2.\end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill & \lambda ({\tilde{\xi }}_1-{\overline{\xi }}_1)+{\displaystyle \frac{\partial ({\tilde{\xi }}_1-{\overline{\xi }}_1)}{\partial t}}-d_1▵({\tilde{\xi }}_1-{\overline{\xi }}_1)\hfill \\ \hfill =& b_1({\tilde{\xi }}_1-{\overline{\xi }}_1)-b_2{e}^{\lambda t}\left((1-{\tilde{u}}_1){\tilde{\xi }}_1{\tilde{\xi }}_2-(1-{\overline{u}}_1){\overline{\xi }}_1{\overline{\xi }}_2\right)-b_3{e}^{\lambda t}({\tilde{\xi }}_1{\tilde{\xi }}_3-{\overline{\xi }}_1{\overline{\xi }}_3)\hfill \\ \hfill & +({\tilde{u}}_2{\tilde{\xi }}_2-{\overline{u}}_2{\overline{\xi }}_2).\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill & \lambda ({\tilde{\xi }}_2-{\overline{\xi }}_2)+{\displaystyle \frac{\partial ({\tilde{\xi }}_2-{\overline{\xi }}_2)}{\partial t}}-d_2▵({\tilde{\xi }}_2-{\overline{\xi }}_2)\hfill \\ \hfill =& -c_1({\tilde{\xi }}_2-{\overline{\xi }}_2)+b_2{e}^{\lambda t}\left((1-{\tilde{u}}_1){\tilde{\xi }}_1{\tilde{\xi }}_2-(1-{\overline{u}}_1){\overline{\xi }}_1{\overline{\xi }}_2\right)-c_2{e}^{\lambda t}({\tilde{\xi }}_2{\tilde{\xi }}_3-{\overline{\xi }}_2{\overline{\xi }}_3)\hfill \\ \hfill & -({\tilde{u}}_2{\tilde{\xi }}_2-{\overline{u}}_2{\overline{\xi }}_2),\hfill \\ \hfill & \lambda ({\tilde{\xi }}_3-{\overline{\xi }}_3)+{\displaystyle \frac{\partial ({\tilde{\xi }}_3-{\overline{\xi }}_3)}{\partial t}}-d_3▵({\tilde{\xi }}_3-{\overline{\xi }}_3)\hfill \\ \hfill =& e_1({\tilde{\xi }}_3-{\overline{\xi }}_3)-e_2{e}^{\lambda t}({\tilde{\xi }}_3^2-{\overline{\xi }}_3^2)+e_3{e}^{\lambda t}({\tilde{\xi }}_1{\tilde{\xi }}_3-{\overline{\xi }}_1{\overline{\xi }}_3)+e_4{e}^{\lambda t}({\tilde{\xi }}_2{\tilde{\xi }}_3-{\overline{\xi }}_2{\overline{\xi }}_3),\hfill \\ \hfill & \lambda ({\tilde{\zeta }}_1-{\overline{\zeta }}_1)-{\displaystyle \frac{\partial ({\tilde{\zeta }}_1-{\overline{\zeta }}_1)}{\partial t}}-d_1▵({\tilde{\zeta }}_1-{\overline{\zeta }}_1)\hfill \\ \hfill =& b_1({\tilde{\zeta }}_1-{\overline{\zeta }}_1)-b_2{e}^{\lambda t}\left((1-{\tilde{u}}_1){\tilde{\xi }}_2{\tilde{\zeta }}_1-(1-{\overline{u}}_1){\overline{\xi }}_2{\overline{\zeta }}_1\right)-b_3{e}^{\lambda t}({\tilde{\xi }}_3{\tilde{\zeta }}_1-{\overline{\xi }}_3{\overline{\zeta }}_1)\hfill \\ \hfill & +b_2{e}^{\lambda t}\left((1-{\tilde{u}}_1){\tilde{\xi }}_2{\tilde{\zeta }}_2-(1-{\overline{u}}_1){\overline{\xi }}_2{\overline{\zeta }}_2\right)+e_3{e}^{\lambda t}({\tilde{\xi }}_3{\tilde{\zeta }}_3-{\overline{\xi }}_3{\overline{\zeta }}_3),\hfill \\ \hfill & \lambda ({\tilde{\zeta }}_2-{\overline{\zeta }}_2)-{\displaystyle \frac{\partial ({\tilde{\zeta }}_2-{\overline{\zeta }}_2)}{\partial t}}-d_2▵({\tilde{\zeta }}_2-{\overline{\zeta }}_2)\hfill \\ \hfill =& -c_1({\tilde{\zeta }}_2-{\overline{\zeta }}_2)-b_2{e}^{\lambda t}\left((1-{\tilde{u}}_1){\tilde{\xi }}_1{\tilde{\zeta }}_1-(1-{\overline{u}}_1){\overline{\xi }}_1{\overline{\zeta }}_1\right)+({\tilde{u}}_2{\tilde{\zeta }}_1-{\overline{u}}_2{\overline{\zeta }}_1)\hfill \\ \hfill & +b_2{e}^{\lambda t}\left((1-{\tilde{u}}_1){\tilde{\xi }}_1{\tilde{\zeta }}_2-(1-{\overline{u}}_1){\overline{\xi }}_1{\overline{\zeta }}_2\right)-c_2{e}^{\lambda t}({\tilde{\xi }}_3{\tilde{\zeta }}_2-{\overline{\xi }}_3{\overline{\zeta }}_2)-({\tilde{u}}_2{\tilde{\zeta }}_2-{\overline{u}}_2{\overline{\zeta }}_2)\hfill \\ \hfill & +e_4{e}^{\lambda t}({\tilde{\xi }}_3{\tilde{\zeta }}_3-{\overline{\xi }}_3{\overline{\zeta }}_3),\hfill \\ \hfill & \lambda ({\tilde{\zeta }}_3-{\overline{\zeta }}_3)-{\displaystyle \frac{\partial ({\tilde{\zeta }}_3-{\overline{\zeta }}_3)}{\partial t}}-d_3▵({\tilde{\zeta }}_3-{\overline{\zeta }}_3)\hfill \\ \hfill =& -b_3{e}^{\lambda t}({\tilde{\xi }}_1{\tilde{\zeta }}_1-{\overline{\xi }}_1{\overline{\zeta }}_1)-c_2{e}^{\lambda t}({\tilde{\xi }}_2{\tilde{\zeta }}_2-{\overline{\xi }}_2{\overline{\zeta }}_2)+e_1({\tilde{\zeta }}_3-{\overline{\zeta }}_3)-2e_2{e}^{\lambda t}({\tilde{\xi }}_3{\tilde{\zeta }}_3-{\overline{\xi }}_3{\overline{\zeta }}_3)\hfill \\ \hfill & +e_3{e}^{\lambda t}({\tilde{\xi }}_1{\tilde{\zeta }}_3-{\overline{\xi }}_1{\overline{\zeta }}_3)+e_4{e}^{\lambda t}({\tilde{\xi }}_2{\tilde{\zeta }}_3-{\overline{\xi }}_2{\overline{\zeta }}_3).\hfill \end{array}$$

Denote $l_1={\tilde{\xi }}_1-{\overline{\xi }}_1$, $l_2={\tilde{\xi }}_2-{\overline{\xi }}_2$, $l_3={\tilde{\xi }}_3-{\overline{\xi }}_3$, $m_1={\tilde{\zeta }}_1-{\overline{\zeta }}_1$, $m_2={\tilde{\zeta }}_2-{\overline{\zeta }}_2$, $m_3={\tilde{\zeta }}_3-{\overline{\zeta }}_3$. Then $(l_1,l_2,l_3,m_1,m_2,m_3)$ satisfies the following system

$$\begin{array}{cc}\hfill & \lambda l_1+{\displaystyle \frac{\partial l_1}{\partial t}}-d_1▵l_1=a_{11}{l}_1+a_{12}{l}_2+a_{13}{l}_3+h_1,(x,t)\in Q_T,\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill & \lambda l_2+{\displaystyle \frac{\partial l_2}{\partial t}}-d_2▵l_2=a_{21}{l}_1+a_{22}{l}_2+a_{23}{l}_3+h_2,(x,t)\in Q_T,\hfill \\ \hfill & \lambda l_3+{\displaystyle \frac{\partial l_3}{\partial t}}-d_3▵l_3=a_{31}{l}_1+a_{32}{l}_2+a_{33}{l}_3+h_3,(x,t)\in Q_T,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \lambda m_1-{\displaystyle \frac{\partial m_1}{\partial t}}-d_1▵m_1=a_{41}{l}_1+a_{42}{l}_2+a_{43}{l}_3+a_{44}{m}_1+a_{45}{m}_2+a_{46}{m}_3+h_4,\hfill \\ \hfill & (x,t)\in Q_T,\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill & \lambda m_2-{\displaystyle \frac{\partial m_2}{\partial t}}-d_2▵m_2=a_{51}{l}_1+a_{52}{l}_2+a_{53}{l}_3+a_{54}{m}_1+a_{55}{m}_2+a_{56}{m}_3+h_5,\hfill \\ \hfill & (x,t)\in Q_T,\hfill \\ \hfill & \lambda m_3-{\displaystyle \frac{\partial m_3}{\partial t}}-d_3▵m_3=a_{61}{l}_1+a_{62}{l}_2+a_{63}{l}_3+a_{64}{m}_1+a_{65}{m}_2+a_{66}{m}_3+h_6,\hfill \\ \hfill & (x,t)\in Q_T,\hfill \\ \hfill & {\displaystyle \frac{\partial l_1}{\partial n}}={\displaystyle \frac{\partial l_2}{\partial n}}={\displaystyle \frac{\partial l_3}{\partial n}}={\displaystyle \frac{\partial m_1}{\partial n}}={\displaystyle \frac{\partial m_2}{\partial n}}={\displaystyle \frac{\partial m_3}{\partial n}}=0,(x,t)\in \mathsf{\Gamma },\hfill \\ \hfill & l_1(x,0)=l_2(x,0)=l_3(x,0)=m_1(x,T)=m_2(x,T)=m_3(x,T)=0,x\in \Omega ,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & a_{11}=b_1-b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\tilde{\xi }}_2-b_3{e}^{\lambda t}{\tilde{\xi }}_3,a_{12}=-b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\overline{\xi }}_1+{\tilde{u}}_2,a_{13}=-b_3{e}^{\lambda t}{\overline{\xi }}_1,\hfill \\ \hfill & h_1=-b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_1{\overline{\xi }}_2+({\tilde{u}}_2-{\overline{u}}_2){\overline{\xi }}_2,\hfill \\ \hfill & a_{21}=b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\tilde{\xi }}_2,a_{22}=-c_1+b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\overline{\xi }}_1-c_2{e}^{\lambda t}{\tilde{\xi }}_3-{\tilde{u}}_2,a_{23}=-c_2{e}^{\lambda t}{\overline{\xi }}_2,\hfill \\ \hfill & h_2=b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_1{\overline{\xi }}_2-({\tilde{u}}_2-{\overline{u}}_2){\overline{\xi }}_2,\hfill \\ \hfill & a_{31}=e_3{e}^{\lambda t}{\tilde{\xi }}_3,a_{32}=e_4{e}^{\lambda t}{\tilde{\xi }}_3,a_{33}=e_1-e_2{e}^{\lambda t}({\tilde{\xi }}_3+{\overline{\xi }}_3)+e_3{e}^{\lambda t}{\overline{\xi }}_1+e_4{e}^{\lambda t}{\overline{\xi }}_2,\hfill \\ \hfill & h_3=0,\hfill \\ \hfill & a_{41}=0,a_{42}=-b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\tilde{\zeta }}_1+b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\tilde{\zeta }}_2,a_{43}=-b_3{e}^{\lambda t}{\tilde{\zeta }}_1+e_3{e}^{\lambda t}{\tilde{\xi }}_3,\hfill \\ \hfill & a_{44}=b_1-b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\overline{\xi }}_2-b_3{e}^{\lambda t}{\overline{\xi }}_3,a_{45}=b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\overline{\xi }}_2,a_{46}=e_3{e}^{\lambda t}{\overline{\xi }}_3,\hfill \\ \hfill & h_4=-b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_2{\overline{\zeta }}_1+b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_2{\overline{\zeta }}_2,\hfill \\ \hfill & a_{51}=-b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\overline{\zeta }}_1+b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\tilde{\zeta }}_2,a_{52}=0,a_{53}=c_2{e}^{\lambda t}{\tilde{\zeta }}_2+e_4{e}^{\lambda t}{\tilde{\zeta }}_3,\hfill \\ \hfill & a_{54}=(1-{\tilde{u}}_1){\overline{\xi }}_1+{\tilde{u}}_2,a_{55}=-c_1+b_2{e}^{\lambda t}(1-{\tilde{u}}_1){\overline{\xi }}_1+c_2{e}^{\lambda t}{\overline{\xi }}_3-{\tilde{u}}_2,a_{56}=e_4{e}^{\lambda t}{\overline{\xi }}_3,\hfill \\ \hfill & h_5=-b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_1{\overline{\zeta }}_1+({\tilde{u}}_2-{\overline{u}}_2){\overline{\zeta }}_1+b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_1{\overline{\zeta }}_2-({\tilde{u}}_2-{\overline{u}}_2){\overline{\zeta }}_2,\hfill \\ \hfill & a_{61}=-b_3{e}^{\lambda t}{\tilde{\zeta }}_1+e_3{e}^{\lambda t}{\tilde{\zeta }}_3,a_{62}=-c_2{e}^{\lambda t}{\tilde{\zeta }}_2+e_4{e}^{\lambda t}{\tilde{\zeta }}_3,a_{63}=-2e_2{e}^{\lambda t}{\tilde{\zeta }}_3,\hfill \\ \hfill & a_{64}=-b_3{e}^{\lambda t}{\overline{\xi }}_1,a_{65}=-c_2{e}^{\lambda t}{\overline{\xi }}_2,a_{66}=e_1-2e_2{e}^{\lambda t}{\overline{\xi }}_3+e_3{e}^{\lambda t}{\overline{\xi }}_1+e_4{e}^{\lambda t}{\overline{\xi }}_2,\hfill \\ \hfill & h_6=0.\hfill \end{array}$$

Note that from (90)–(95) that

$$\begin{array}{cc}\hfill & |{\tilde{u}}_1-{\overline{u}}_1|\hfill \\ \hfill & \le {\displaystyle \frac{b_2}{a_1}}|\widetilde{z_1}\widetilde{z_2}({\tilde{P}}_2-{\tilde{P}}_1)-\overline{z_1}\overline{z_2}({\overline{P}}_2-{\overline{P}}_1)|\hfill \\ \hfill & \le {\displaystyle \frac{b_2}{a_1}}|\widetilde{z_1}\widetilde{z_2}\left(({\tilde{P}}_2-{\tilde{P}}_1)-({\overline{P}}_2-{\overline{P}}_1)\right)+\widetilde{z_1}({\overline{P}}_2-{\overline{P}}_1)(\widetilde{z_2}-\overline{z_2})+\overline{z_2}({\overline{P}}_2-{\overline{P}}_1)(\widetilde{z_1}-\overline{z_1})|\hfill \\ \hfill & \le {\displaystyle \frac{b_2}{a_1}}|\widetilde{z_1}\widetilde{z_2}\left(({\tilde{P}}_2-{\overline{P}}_2)-({\tilde{P}}_1-{\overline{P}}_1)\right)+\widetilde{z_1}({\overline{P}}_2-{\overline{P}}_1)\left(e^{\lambda t}({\tilde{\xi }}_2-{\overline{\xi }}_2)\right)+\overline{z_2}({\overline{P}}_2-{\overline{P}}_1)\left(e^{\lambda t}({\tilde{\xi }}_1-{\overline{\xi }}_1)\right)|\hfill \\ \hfill & \le {\displaystyle \frac{b_2}{a_1}}|\widetilde{z_1}\widetilde{z_2}{e}^{-\lambda t}(m_2-m_1)+\widetilde{z_1}({\overline{P}}_2-{\overline{P}}_1)e^{\lambda t}{l}_2+\overline{z_2}({\overline{P}}_2-{\overline{P}}_1)e^{\lambda t}{l}_1|\hfill \\ \hfill & \le C\left(e^{\lambda t}|l_1|+e^{\lambda t}|l_2|+e^{-\lambda t}(|m_2|+|m_1|)\right).\hfill \end{array}$$

(98)

and

$$\begin{array}{cc}\hfill & |{\tilde{u}}_2-{\overline{u}}_2|\hfill \\ \hfill & \le {\displaystyle \frac{1}{a_2}}|\widetilde{z_2}({\tilde{P}}_2-{\tilde{P}}_1)-\overline{z_2}({\overline{P}}_2-{\overline{P}}_1)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{a_2}}|\widetilde{z_2}\left(({\tilde{P}}_2-{\tilde{P}}_1)-({\overline{P}}_2-{\overline{P}}_1)\right)+({\overline{P}}_2-{\overline{P}}_1)(\widetilde{z_2}-\overline{z_2})|\hfill \\ \hfill & \le {\displaystyle \frac{1}{a_2}}|\widetilde{z_2}\left(({\tilde{P}}_2-{\overline{P}}_2)-({\tilde{P}}_1-{\overline{P}}_1)\right)+({\overline{P}}_2-{\overline{P}}_1)(\widetilde{z_2}-\overline{z_2})|\hfill \\ \hfill & \le {\displaystyle \frac{1}{a_2}}|\widetilde{z_2}{e}^{-\lambda t}(m_2-m_1)+({\overline{P}}_2-{\overline{P}}_1)e^{\lambda t}{l}_2|\hfill \\ \hfill & \le C\left(e^{\lambda t}|l_2|+e^{-\lambda t}(|m_2|+|m_1|)\right).\hfill \end{array}$$

(99)

where C is a constant that does not depend on $\lambda ,t,T$. Multiply (96) by $l_1$ and integrate by parts to get

$$\begin{array}{cc}\hfill & {\displaystyle {\int \int }_{Q_T}}(\lambda l_1^2+{\displaystyle \frac{\partial l_1}{\partial t}}{l}_1-d_1{l}_1▵l_1)dxdt\hfill \\ \hfill & =\lambda {\displaystyle {\int \int }_{Q_T}}{l}_1^{2}dxdt+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{l}_1^2(x,T)dx+d_1{\displaystyle {\int \int }_{Q_T}}{|\nabla l_1|}^{2}dxdt\hfill \\ \hfill & ={\displaystyle {\int \int }_{Q_T}}(a_{11}{l}_1^2+a_{12}{l}_1{l}_2+a_{13}{l}_1{l}_3+h_1{l}_1)dxdt\hfill \\ \hfill & \le C{\displaystyle {\int \int }_{Q_T}}(l_1^2+l_2^2+l_3^2+h_1^2)dxdt.\hfill \end{array}$$

(100)

Note that from (90)–(91) and (98)–(99), we have

$$\begin{array}{cc}\hfill {\displaystyle {\int \int }_{Q_T}}{h}_1^{2}dxdt& ={\displaystyle {\int \int }_{Q_T}}{\left(b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_1{\overline{\xi }}_2-({\tilde{u}}_2-{\overline{u}}_2){\overline{\xi }}_2\right)}^{2}dxdt\hfill \\ \hfill & \le C{\displaystyle {\int \int }_{Q_T}}\left(e^{-\lambda t}{\left(({\overline{u}}_1-{\tilde{u}}_1)-({\tilde{u}}_2-{\overline{u}}_2)\right)}^2\right)dxdt\hfill \\ \hfill & \le C{\displaystyle {\int \int }_{Q_T}}(l_1^2)dxdt.\hfill \end{array}$$

(101)

From (100) and (101), we have

$$\begin{array}{c}\hfill \lambda {\displaystyle {\int \int }_{Q_T}}{l}_1^{2}dxdt\le C{\displaystyle {\int \int }_{Q_T}}(l_1^2+l_2^2+l_3^2)dxdt.\end{array}$$

(102)

Similarly, we can obtain

$$\begin{array}{c}\hfill \lambda {\displaystyle {\int \int }_{Q_T}}{l}_2^{2}dxdt\le C{\displaystyle {\int \int }_{Q_T}}(l_1^2+l_2^2+l_3^2)dxdt.\end{array}$$

(103)

and

$$\begin{array}{c}\hfill \lambda {\displaystyle {\int \int }_{Q_T}}{l}_3^{2}dxdt\le C{\displaystyle {\int \int }_{Q_T}}(l_1^2+l_2^2+l_3^2)dxdt.\end{array}$$

(104)

Multiply (97) by $m_1$ and integrate by parts to get

$$\begin{array}{cc}\hfill & {\displaystyle {\int \int }_{Q_T}}(\lambda {m_1}^2-{\displaystyle \frac{\partial m_1}{\partial t}}{m}_1-d_1{m}_1▵m_1)dxdt\hfill \\ \hfill & =\lambda {\displaystyle {\int \int }_{Q_T}}{m}_1^{2}dxdt+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{m}_1^2(x,0)dx+{\displaystyle {\int \int }_{Q_T}}{|\nabla m_1|}^{2}dxdt\hfill \\ \hfill & ={\displaystyle {\int \int }_{Q_T}}(a_{41}{l}_1{m}_1+a_{42}{l}_2{m}_1+a_{43}{l}_3{m}_1+a_{44}{m}_1^2+a_{45}{m}_2{m}_1+a_{46}{m}_3{m}_1+h_4{m}_1)dxdt\hfill \\ \hfill & \le C{\displaystyle {\int \int }_{Q_T}}\left((|a_{41}|l_1^2+|a_{42}|l_2^2+|a_{43}|l_3^2)+(\sum _{i=1}^6|a_{4i}|+e^{2\lambda t})m_1^2+|a_{45}|m_2^2+|a_{46}|m_3^2+e^{-2\lambda t}{h}_4^2\right)dxdt\hfill \\ \hfill & \le C{\displaystyle {\int \int }_{Q_T}}\left(e^{2\lambda t}(l_1^2+l_2^2+l_3^2+m_1^2)+(m_1^2+m_2^2+m_3^2)+e^{-2\lambda t}{h}_4^2\right)dxdt.\hfill \end{array}$$

(105)

Due to the boundedness of ${\overline{P}}_1$, we have

$$\begin{array}{cc}\hfill {\displaystyle {\int \int }_{Q_T}}{e}^{-2\lambda t}{h}_4^{2}dxdt& ={\displaystyle {\int \int }_{Q_T}}\left(e^{-2\lambda t}{\left(-b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_2{\overline{\zeta }}_1+b_2{e}^{\lambda t}({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_2{\overline{\zeta }}_2\right)}^2\right)dxdt\hfill \\ \hfill & \le C{\displaystyle {\int \int }_{Q_T}}{\left(({\overline{u}}_1-{\tilde{u}}_1){\overline{\xi }}_2({\overline{\zeta }}_2-{\overline{\zeta }}_1)\right)}^{2}dxdt\hfill \\ \hfill & \le C{\displaystyle {\int \int }_{Q_T}}\left(e^{2\lambda t}(l_1^2+l_2^2)+(m_1^2+m_2^2)\right)dxdt.\hfill \end{array}$$

(106)

From (105) and (106), we have

$$\begin{array}{c}\hfill \lambda {\displaystyle {\int \int }_{Q_T}}{m}_1^{2}dxdt\le C{\displaystyle {\int \int }_{Q_T}}{e}^{2\lambda t}(l_1^2+l_2^2+l_3^2+m_1^2)dxdt+C{\displaystyle {\int \int }_{Q_T}}(m_1^2+m_2^2+m_3^2)dxdt.\end{array}$$

(107)

Similarly, we have

$$\begin{array}{cc}\hfill & \lambda {\displaystyle {\int \int }_{Q_T}}{m}_2^{2}dxdt\le C{\displaystyle {\int \int }_{Q_T}}{e}^{2\lambda t}(l_1^2+l_2^2+l_3^2+m_2^2)dxdt+C{\displaystyle {\int \int }_{Q_T}}(m_1^2+m_2^2+m_3^2)dxdt.\hfill \end{array}$$

(108)

and

$$\begin{array}{cc}\hfill & \lambda {\displaystyle {\int \int }_{Q_T}}{m}_3^{2}dxdt\le C{\displaystyle {\int \int }_{Q_T}}{e}^{2\lambda t}(l_1^2+l_2^2+l_3^2+m_3^2)dxdt+C{\displaystyle {\int \int }_{Q_T}}(m_1^2+m_2^2+m_3^2)dxdt.\hfill \end{array}$$

(109)

It follows from (102)–(104) and (107)–(109) that

$$\begin{array}{cc}\hfill & \lambda {\displaystyle {\int \int }_{Q_T}}(l_1^2+l_2^2+l_3^2+m_1^2+m_2^2+m_3^2)dxdt\hfill \\ \hfill & \le (C_1+C_2{e}^{2\lambda t}){\displaystyle {\int \int }_{Q_T}}(l_1^2+l_2^2+l_3^2+m_1^2+m_2^2+m_3^2)dxdt.\hfill \end{array}$$

Take $\lambda >C_1+C_2{e}^{2\lambda t}$ and $\Lambda =\min \{{\displaystyle \frac{1}{2\lambda }}ln{\displaystyle \frac{\lambda -C_1}{C_2}},1\}$, then for $0<T<\Lambda$, we have

$$\begin{array}{c}\hfill {\displaystyle {\int \int }_{Q_T}}(l_1^2+l_2^2+l_3^2+m_1^2+m_2^2+m_3^2)dxdt=0.\end{array}$$

then we have

$$\begin{array}{c}\hfill l_1=l_2=l_3=m_1=m_2=m_3=0.\end{array}$$

which implies

$$\begin{array}{c}\hfill {\tilde{\xi }}_1={\overline{\xi }}_1,{\tilde{\xi }}_2={\overline{\xi }}_2,{\tilde{\xi }}_3={\overline{\xi }}_3,{\tilde{\zeta }}_1={\overline{\zeta }}_1,{\tilde{\zeta }}_2={\overline{\zeta }}_2,{\tilde{\zeta }}_3={\overline{\zeta }}_3.\end{array}$$

Hence,

$$\begin{array}{c}\hfill {\tilde{z}}_1={\overline{z}}_1,{\tilde{z}}_2={\overline{z}}_2,{\tilde{z}}_3={\overline{z}}_3,{\tilde{P}}_1={\overline{P}}_1,{\tilde{P}}_2={\overline{P}}_2,{\tilde{P}}_3={\overline{P}}_3.\end{array}$$

The proof is completed. □

From Corollary 1 and Theorem 4, one can get the uniqueness of the optimal control to the problem ${min}_{(u_1,u_2)\in U_{ad}}J(u_1,u_2)$ if T is sufficiently small.

5. Numerical Experiments

In this section, we use the finite difference method to numerically simulate the optimal control problem for the prey–predator model with disease in prey.

Firstly, we briefly describe the numerical procedure used to solve the optimal control problem in Algorithm 1.

Algorithm 1 Iterative algorithm

Input $d_S,d_I,d_Y,A,\beta ,d,{\eta }_1,c,{\eta }_2,r,k,p_1,p_2,a_1,a_2,w,M_1,M_2,S_0(x),I_0(x),Y_0(x)$. Initialize the control variables $u_i^{(k)}=0,i=1,2,k=0$. With the known controls $u_i^{(k)}$, solve the state variables $S^{(k+1)},I^{(k+1)},Y^{(k+1)}$ by the forward difference method. With the known controls $u_i^{(k)}$ and the state variables $S^{(k+1)},I^{(k+1)},Y^{(k+1)}$, solve the adjoint variables $P_i^{(k+1)},i=1,2,3$ by the backward difference method. Using the state variables $S^{(k+1)},I^{(k+1)},Y^{(k+1)}$ and the adjoint variables $P_i^{(k+1)}$, update the optimal control variables $u_i^{(k+1)}=0,i=1,2$ by using the relation $u_1=min\{M_1,{\displaystyle \frac{\beta SI}{a_1}}{(P_2^{(k+1)}-P_1^{(k+1)})}^{+}\},u_2=min\{M_2,{\displaystyle \frac{I}{a_2}}{(P_2^{(k+1)}-P_1^{(k+1)})}^{+}\}$. If $|u_i^{(k+1})-u_i^{(k)}|<ϵ$, output the results. Otherwise, set $k=k+1$ and repeat from step 3.

Before performing the numerical simulations, it is necessary to determine the values of the parameters that appear in models (1)–(5) and (62)–(66). We make appropriate assumptions about the values of these parameters. For clarity, the values of the parameters are listed in the following table.

The numerical experiments are carried out in the unit square domain $Q_T=(0,1)\times (0,1)$.

By using the above iterative method, we numerically solve the optimal harvesting problem under the Neumann boundary condition with the parameters listed in

Table 1. The values of the model parameters.

Parameter	Value	Source	Parameter	Value	Source
$d_S$	0.0001	[11]	$d_I$	0.0002	[11]
$d_Y$	0.00006	[11]	A	0.5	Assumed
$\beta$	0.7	[12]	d	0.2	Assumed
${\eta }_1$	0.33	[12]	c	0.7	[12]
${\eta }_2$	0.44	[12]	r	0.12	Assumed
k	100	Assumed	$p_1$	0.525	[12]
$p_2$	0.525	[12]	$a_1$	0.5	Assumed
$a_2$	0.2	Assumed	w	0.04	[12]
$M_1$	0.4	Assumed	$M_2$	0.2	Assumed
$S_0(x)$	$10x^2$	Assumed	$I_0(x)$	$x^3$	Assumed
$Y_0(x)$	$1-x$	Assumed	$ϵ$	$0.00001$	Assumed

. The optimal controls are shown in Figure 1, and the optimal states are shown in Figure 2.

In Figure 1a, $u_1(x,t)$ represents optimal control of the isolation treatment of infected prey. In Figure 1b, $u_2(x,t)$ represents optimal control of drug treatment of infected prey.

In Figure 2a, $S(x,t)$ represents the densities of the susceptible prey. In Figure 2b, $I(x,t)$ represents the densities of the infected prey. In Figure 2c, $Y(x,t)$ represents the densities of the predator.

6. Conclusions

In this paper, a prey–predator diffusion model with isolated and drug treatment control measures for prey infection is studied. This coupled parabolic model simulates the interaction between susceptible prey, infected prey, and predator populations. In this model, we aim to find the optimal control pair under boundary conditions so that the population density of the infected population and the cost of isolation and treatment are minimized. Through mathematical analysis, we proved the existence and uniqueness of the weak solution to the model, derived the first-order necessary conditions for the optimal control, further demonstrated the existence and local uniqueness of the optimal control, and verified the feasibility of the model’s optimal control strategy through numerical simulations. In the future, the model can be further expanded by introducing stochastic factors to better reflect the uncertainties in the real world, especially in the context of disease transmission. The results of this study are of great significance to the fields of ecological modeling, epidemiology, and public health. By optimizing control measures, this study provides theoretical references for how to efficiently manage disease outbreaks in animal populations and formulate more effective control policies and intervention measures in different ecological and epidemiological contexts.