Optimal control for the coupled chemotaxis-fluid models in two space dimensions

1.   Introduction

In this paper, we study the coupled chemotaxis-fluid models with the initial-bounary conditions

where $\Omega \subset \mathbb{R}^2$ is a bounded domain with smooth boundary $\partial \Omega$. $\nu$ is the outward normal vector to $\partial \Omega$, and $\gamma$, $\mu$ are positive constants. $n$, $c$ denote the bacterial density, the oxygen concentration, respectively. $u$, $\pi$ are the fluid velocity and the associated pressure. Here, the function $f$ denotes a control that acts on chemical concentration, which lies in a closed convex set $\mathcal{U}$. We observe that in the subdomains where $f \geq 0$ we inject oxygen, and conversely where $f \leq 0$ we extract oxygen.

In order to understand the development of system (1.1), let us mention some previous contributions in this direction. Jin [11] dealed with the time periodic problem of (1.1) in spatial dimension $n = 2,3$. Jin [12] also obtained the existence of large time periodic solution in $\Omega \subset R^3$ without the term $u\cdot\nabla u$.

Espejo and Suzuki [6] discussed the chemotaxis-fluid model

They proved the global existence of weak solution. Tao and Winkler [17] proved the existence of global classical solution and the uniform boundedness. Tao and Winkler [18] also obtained the global classical solution and uniform boundedness under the condition of $\mu > 23$.

The optimal control problems governed by the coupled partial differential equations is important. Colli et al. [4] studied the distributed control problem for a phase-field system of conserved type with a possibly singular potential. Liu and Zhang [14] considered the optimal control of a new mechanochemical model with state constraint. Chen et al. [3] studied the distributed optimal control problem for the coupled Allen-Cahn/Cahn-Hilliard equations. Recently, Guillén-González et al. [9] studied a bilinear optimal control problem for the chemo-repulsion model with the linear production term. The existence, uniqueness and regularity of strong solutions of this model are deduced. They also derived the first-order optimality conditions by using a Lagrange multipliers theorem. Frigeri et al. [8] studied an optimal control problem for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential. Some other results can be found in [2,5,13,15,19].

In this paper, we discuss the optimal control problem for (1.1). We adjust the external source $f$, so that the bacterial density $n$, oxygen concentration $c$ and fluid velocity $u$ are as close as possible to a desired state $n_d$, $c_d$ and $u_d$, and at the final moment $T$ is as close as possible to a desired state $n_{\Omega}$, $c_{\Omega}$ and $u_{\Omega}$. The main difficulties for treating the problem are caused by the nonlinearity of $u\cdot \nabla u$. Our method is based on fixed point method and Simon's compactness results. We overcome the above difficulties and derive first-order optimality conditions by using a Lagrange multipliers theorem.

2.   Basic estimates of linearized problem

In this section, we will construct the existence and some priori estimates of the linearized problem for the chemotaxis-Navier-Stokes system in a bounded domain $\Omega \subset \mathbb{R}^2$. The proofs in this section will be established for a detailed framework.

In the following lemmas we will state the Gagliardo-Nirenberg interpolation inequality [7].

Lemma 2.1. Let $l$ and $k$ be two integers satisfying $0 \leqslant l<k$. Suppose that $1 \leqslant q$, $r \leqslant \infty$, $p>0$ and $\frac{l}{k} \leqslant a \leqslant 1$ such that

Then, for any $u \in W^{k, q}(\Omega) \cap L^{r}(\Omega)$, there exist two positive constants $C_1$ and $C_2$ depending only on $\Omega$, $q$, $k$, $r$ and $N$ such that the following inequality holds

with the following exception: If $1<q<\infty$ and $k-l-\frac{N}{q}$ is a non-negative integer, the (2.1) holds only for $a$ satisfying $\frac{l}{k} \leq a<1$.

The following log-interpolation inequality has been proved by [1].

Lemma 2.2. Let $\Omega \subset \mathbb{R}^{2}$ be a bounded domain with smooth boundary. Then for all non-negative $u\in H^1(\Omega)$, there holds

where $\delta$ is any positive number, and $p(\cdot)$ is an increasing function.

We first consider the existence of solutions to the linear problem of system (1.1). Assume functions $u_0\in H^1(\Omega)$, $\hat{u} \in L^{4}\left(0,T; L^{4}(\Omega)\right), \hat{n} \in L^{2}\left(0,T; L^{2}(\Omega)\right)$, and consider

By using fixed point method, the existence of solutions can be easily obtained. Therefore, we ignore the process of proof and just give the regularity estimate.

Lemma 2.3. Let $u_0\in H^1(\Omega)$, $\hat{u} \in L^{4}\left(0,T; L^{4}(\Omega)\right)$, $\hat{n} \in L^{2}\left(0,T; H^{1}(\Omega)\right), \nabla\varphi\in L^\infty(Q)$, and $u$ be the solution of the problem (2.2), then $u \in L^{\infty}\left(0, T; H^{1}(\Omega)\right) \cap L^{2}\left(0, T; H^{2}(\Omega)\right)$ and $u_t\in L^2(0,T; L^2(\Omega))$.

Proof. Multiplying the first equation of (2.2) by $u$, and integrating it over $\Omega$, we get

By Gronwall's inequality, we have

Operating the Helmholtz projection operator $P$ to the first equation of (2.2), we know

where $A: = -P \Delta$ is called Stokes operator, which is an unbounded self-adjoint positive operator in $L^2$ with compact inverse, for more properties of Stokes operator, we refer to [10]. Note that $\nabla\cdot u = 0$, that is $P u = u$, $P \Delta u = \Delta u$, $P u_{t} = u_{t}$. So, in following calculations, we ignore the projection operator $P$. Multiplying this equation by $\Delta u$, and integrating it over $\Omega$, we get

For the terms on the right, we have

Therefore, we get

By Gronwall's inequality, we derive

Multiplying the first equation of (2.2) by $u_t$, and combining with above inequality, we have

Summing up, we complete the proof.

For the above solution $u$, we consider the following linear problem

Along with fixed point method, the existence of solutions can be easily obtained. Thus we omit the proof and only give the regularity estimate.

Lemma 2.4. Let $c_0\in H^2(\Omega)$, $\hat{n} \in L^{2}\left(0,T; H^{1}(\Omega)\right)$, $f \in L^{2}\left(0,T; H^{1}(\Omega)\right)$, $u$ be the solution of the problem (2.2), and $c$ be the solution of (2.3). Then $c \in L^{\infty}\left((0, T), H^{2}(\Omega)\right) \cap L^{2}\left((0, T), H^{3}(\Omega)\right)$ and $c_t\in L^2(0,T; L^2(\Omega))$.

Proof. Multiplying the first equation of (2.3) by $c$, and integrating it over $\Omega$, we infer from $\int_{\Omega} c(u \cdot \nabla c) = -\frac{1}{2} \int_{\Omega} c^{2} \nabla \cdot u d x = 0$ that

Therefore, we have

Multiplying the first equation of (2.3) by $-\Delta c$, and integrating it over $\Omega$, we get

Using the Young inequality and the Hölder inequality, we obtain

Combining this and above inequalities, we conclude

We therefore verify that

Applying $\nabla$ to the first equation of (2.3), multiplying it by $\nabla \Delta c$, and integrating over $\Omega$ give

For the terms on the right, we obtain

Straightforward calculations yield

Multiplying the first equation of (2.3) by $c_t$, and combining with above inequality, we have

and thereby precisely arrive at the conclusion.

With above solutions $u$ and $c$ in hand, we deal with the following linear problem.

By a similar argument as the above two problems, the existence of solutions can be easily obtained. Therefore, we only give the regularity estimate.

Lemma 2.5. Suppose $0\leq n_0\in H^1(\Omega)$, $\hat{n} \in L^{2}\left(0,T; H^{1}(\Omega)\right) \cap L^{4}\left(0,T; L^{4}(\Omega)\right)$, and $u$, $c$, $n$ are the solutions of the problem (2.2), (2.3) and (2.4), respectively. Then $n \geq 0$, $n\in L^{\infty}(0, T; H^1(\Omega)) \cap L^2(0, T; H^2(\Omega))$ and $n_t\in L^2(0,T; L^2(\Omega))$.

Proof. Firstly, we verify the nonnegativity of $n$. We examine the set $A(t) = \{x : n(x, t)<0\}$. Along with (2.4), we get

Since $\frac{\partial n}{\partial \nu} \geq 0$ on $\partial\{n<0\}$, from this we deduce that the right hand side is nonnegative. Integrating this equality on $[0,t]$ gives

Then, we get $n \geq 0$.

Next, multiplying the first equation of (2.4) by $n$, and integrating it over $\Omega$, we get

So, we derive that

Multiplying the first equation of (2.4) by $-\Delta n$, and integrating it over $\Omega$, we get

Straightforward calculations yield

Multiplying the first equation of (2.4) by $n_t$, and combining with above inequality, we have

The proof is complete.

Introduce the spaces

Define a map

where the $(u, n)$ is the solution of the decoupled linear problem

Next, we use fixed point method to prove the local existence of solutions of the problem (1.1).

Lemma 2.6. The map $\mathcal{F}: X_{u} \times X_{n}\rightarrow X_{u} \times X_{n}$ is well defined and compact.

Proof. Let $(\hat{n},\hat{u})\in X_{u} \times X_{n}$, by Lemmas 2.3, 2.4, 2.5 we deduce that $(n,u) = \mathcal{F}(\hat{n},\hat{u})$ is bounded in $Y_u\times Y_n$. Note that the embeddings $H^{2}(\Omega) \hookrightarrow H^{1}(\Omega)$ is compact and interpolating between $L^{\infty}(0, T; H^1(\Omega))$ and $L^2(0, T; H^2(\Omega))$. It is easy to get that $u$ is bounded in $L^{4}\left(0,T; L^{4}(\Omega)\right)$ and $n$ is bounded in $L^{4}\left(0,T; L^{4}(\Omega)\right) \cap L^{2}\left(0,T; H^{1}(\Omega)\right)$. Therefore, the operator $\mathcal{F}:X_{u} \times X_{n}\rightarrow X_{u} \times X_{n}$ is a compact operator.

3.   Existence and uniqueness of strong solution of system

From Lemma 2.6, $(n,u)\in Y_n\times Y_u$ satisfies pointwisely a.e. in $Q$ the following problem

In order to prove the existence of solution, we first give some a priori estimates.

Lemma 3.1. Let $(n,c,u)$ be a local solution to (3.1). Then, it holds that

Proof. With Lemma 2.5 in hand, we get $n\geq 0$. Integrating the first equation of (3.1) over $\Omega$, we see that

Solving this differential inequality, we obtain that

Multiplying the third equation of (3.1) by $u$, and integrating it over $\Omega$, we get

Therefore, we see that

By the Gagliardo-Nirenberg interpolation inequality, we deduce that

Multiplying the third equation of (3.1) by $\Delta u$, and integrating it over $\Omega$, we get

Thus, we know

Multiplying the second equation of (3.1) by $c$, and integrating it over $\Omega$, we have

Then, we have

Multiplying the second equation of (3.1) by $-\Delta c$, and integrating it over $\Omega$, we get

Further, we have

The proof is complete.

Lemma 3.2. Let $(n,c,u)$ be a local solution to (3.1). Then, it holds that

Proof. We rewrite the first equation of (3.1) as

Multiplying the above equation by $\ln(n + 1)$ and integrating the equation, we have

For $I_1$, integrating by parts and using Young's inequality with small $\delta$, we get

For the term $I_2$, we have

For the rest term $I_3$, straightforward calculations yield

Combining $I_1$, $I_2$ with $I_3$, we conduct that

Multiplying the second equation of (3.1) by $\Delta c$, and integrating it over $\Omega$, we get

Straightforward calculations yield

Combing (3.6) and (3.7), it follows that

Taking $\delta$ small enough, and solving this differential inequality, we obtain that

The proof is complete.

Lemma 3.3. Assume $f\in L^2(0,T;H^1(\Omega))$, let $(n,c,u)$ be a local solution to (3.1). Then, it holds that

Proof. Taking the $L^2$-inner product with $n$ for the first equation of (3.1) implies

Here, we note that

From Lemma 2.2 and (3.2), it follows that

As an immediate consequence

Applying $\nabla$ to the first equation of (3.1), multiplying it by $\nabla \Delta c$, and integrating over $\Omega$ give

For $I_4$, by using the Gagliardo-Nirenberg interpolation inequality, we get

For the term $I_5$, we have

Along with $I_4$ and $I_5$, we conclude

Combining (3.9) and (3.10), it follows that

By choosing $\delta$ small enough and using (3.3) and (3.5), we have

The proof is complete.

Lemma 3.4. Assume $f\in L^2(0,T;H^1(\Omega))$, let $(n,c,u)$ be a local solution to (3.1). Then, it holds that

Proof. Taking the $L^2$-inner product with $-\Delta n$ for the first equation of (3.1) implies

For the term $I_6$, with the estimate (3.3), we have

For the term $I_7$, taking (3.8) into considering, we conduct that

For the term $I_8$, thanks to the nonnegativity of $n$, we see that

Combine the estimates about $I_6$, $I_7$ and $I_8$, it follows that

By taking $\delta$ small enough, we get

Therefore, this proof is complete.

Lemma 3.5. The operator $\mathcal{F}:X_{u} \times X_{n}\rightarrow X_{u} \times X_{n}$, is continuous.

Proof. Let $\{(\hat{n}_m,\hat{u}_{m})\}_{m\in \mathbb{N}}$ be a sequence of $X_{u} \times X_{n}$, Then, with Lemmas 2.3, 2.4 and 2.5 in hand, we conduct that $\{(n_m,u_m) = \mathcal{F}(\hat{n}_m,\hat{u}_{m})\}_{m\in \mathbb{N}}$ is bounded in $Y_u\times Y_n$. Taking the compactness of $Y_u\times Y_n$ in $X_{u} \times X_{n}$ into consider, we see that $\mathcal{F}$ is a compact operator, which means there exists a subsequence of $\{\mathcal{F}(\hat{n}_m,\hat{u}_{m})\}_{m\in \mathbb{N}}$, for convenience, still denoted as $\{\mathcal{F}(\hat{n}_m,\hat{u}_{m})\}_{m\in \mathbb{N}}$, and exists an element $(\hat{n},\hat{u})$ in $Y_u\times Y_n$ such that

Let $m\rightarrow \infty$ and take the limit, it is clear that $(n,u) = \mathcal{F}(\hat{n}_m,\hat{u}_{m})$ and $(\hat{n}_m,\hat{u}_{m}) = (\hat{n},\hat{u})$, this means that $\mathcal{F}(\hat{n}_m,\hat{u}_{m}) = (\hat{n}_m,\hat{u}_{m})$. Since uniqueness of limit, the map $\mathcal{F}$ is continuous.

Theorem 3.1. Let $u_0\in H^1(\Omega)$, $n_0\in H^1(\Omega)$, $c_0\in H^2(\Omega)$ with $n_0\geq 0$ in $\Omega$, and $f\in L^2(0,T;H^1(\Omega))$, then (1.1) exists unique strong solution $(n,c,u)$. Moreover, there exists a positive $C$ constant such that

Proof. From Lemmas 3.1, 3.3 and 3.4, it is easy to verify the existence of solution and (3.11). Therefore, we will prove the uniqueness of the solution in the following part. For convenience, we set $n = n_1-n_2$, $c = c_1-c_2$ and $u = u_1-u_2$, where $(n_i, c_i, u_i)$ is the strong solution of the system, where $i = 1, 2$. Thus, we obtain the following system

Taking the $L^2$-inner product with $n$ for the (3.13) implies

For the term $I_9$, due to the estimates (3.3) and (3.8), we have

For the term $I_{10}$, with the estimate (3.8) and (3.11), we get

For the term $I_{11}$,

With the use of estimates $I_i(i = 9,10, 11,12)$, we have

Taking the $L^2$-inner product with $c$ for the (3.14) implies

Then, we get

Taking the $L^2$-inner product with $c$ for the (3.15) implies

Straightforward calculations yield

Then, a combination of (3.19), (3.20) and (3.21) yields

By choosing $\delta$ small enough, we get

Applying Gronwall's lemma to the resulting differential inequality, we finally obtain the uniqueness of the solution.

4.   Existence of an optimal control

In this section, we will prove the existence of the optimal solution of control problem. The method we use for treating this problem was inspired by some ideas of Guillén-González et al [9]. Assume $\mathcal{U}\subset L^2(0,T;H^1(\Omega_c))$ is a nonempty, closed and convex set, where control domain $\Omega_c\subset \Omega$, and $\Omega_d\subset \Omega$ is the observability domain. We adjust the external source $f$, so that the bacterial density $n$, oxygen concentration $c$ and fluid velocity $u$ are as close as possible to a desired state $n_d$, $c_d$ and $u_d$, and at the final moment $T$ is as close as possible to a desired state $n_{\Omega}$, $c_{\Omega}$ and $u_{\Omega}$. We consider the optimal control problem as follows

Minimize the cost functional

subject to the system (1.1). Moreover, the nonnegative constants $\beta_i,i = 1,2,\cdots,7$ are given but not all zero, the functions $n_d$, $c_d$, $u_d$ represents the desired states satisfying

The set of admissible solutions of optimal control problem (4.1) is defined by

The function space $\mathcal{H}$ is given by

where $Y_c = L^{\infty}\left(0, T; H^{2}(\Omega)\right) \cap L^{2}\left(0, T; H^{3}(\Omega)\right)$.

Now, we prove the existence of a global optimal control for problem (1.1).

Theorem 4.1. Suppose $f\in \mathcal{U}$ is satisfied, and $n_0\geq 0$, then the optimal control problem (4.1) admits a solution $(\bar{n},\bar{c},\bar{u},\bar{f})\in \mathcal{S}_{a d}$.

Proof. Along with Theorem 3.1, we conduct that $\mathcal{S}_{a d} \neq \emptyset$, then there exists the minimizing sequence $\{(n_m,c_m,u_m,f_m)\}_{m\in \mathbb{N}}\in \mathcal{S}_{ad}$ such that

According to the definition of $\mathcal{S}_{ad}$, for each $m\in \mathbb{N}$ there exists $(n_m,c_m,u_m,f_m)$ satisfying

Observing that $\mathcal{U}$ is a closed convex subset of $L^2(0,T;H^1(\Omega_c))$. According to the definition of $\mathcal{S}_{ad}$, we deduce that there exists $(\bar{n},\bar{c},\bar{u},\bar{f})$ bounded in $\mathcal{H}$ such that, for subsequence of $(n_m,c_m,u_m, f_m)_{m\in \mathbb{N}}$, for convenience, still denoted by $(n_m,c_m,u_m,f_m)$, as $m\rightarrow +\infty$

According to the Aubin-Lions lemma [16] and the compact embedding theorems, we obtain

Since $\nabla \cdot\left(n_{m} \nabla c_{m}\right) = \nabla n_{m} \cdot \nabla c_{m}+n_{m} \Delta c_{m}$ is bounded in $L^2(0,T;L^2(\Omega))$, then

Recalling that

Therefore, we get that $\chi = \nabla(\bar{n}\nabla\bar{c})$. Owing to $(\bar{n},\bar{c},\bar{u},\bar{f})\in \mathcal{H}$, we see that $(\bar{n},\bar{c},\bar{u},\bar{f})$ is solution of the system (1.1), along with (4.2) implies that

On the other hand, we deduce from the weak lower semicontinuity of the cost functional

Therefore, this implies that $(\bar{n},\bar{c},\bar{u},\bar{f})$ is an optimal pair for problem (1.1).

5.   The first-order necessary optimality condition

In order to derive the first-order necessary optimality conditions for a local optimal solution of problem (4.1). To this end, we will use a result on existence of Lagrange multipliers in Banach spaces ([20]). First, we discuss the following problem

where $J:X\rightarrow \mathbb{R}$ is a functional, $G:X\rightarrow Y$ is an operator, $X$ and $Y$ are Banach spaces, and nonempty closed convex set $\mathcal{H}$ is subset of $X$ and nonempty closed convex cone $\mathcal{N}$ with vertex at the origin in $Y$.

$A^+$ denotes its polar cone

We consider the following Banach spaces

where

and the operator $G = (G_1,G_2,G_3,G_4,G_5,G_6):X\rightarrow Y$, where

which are defined at each point $s = (n,c,u,f)\in X$ by

The function spaces are given as follows

We see that $\mathcal{H}$ is a closed convex subset of $X$ and $\mathcal{N} = \{0\}$, and rewrite the optimal control problem

Taking the differentiability of $J$ and $G$ into consider, it follows that

Lemma 5.1. The functional $J:X\rightarrow R$ is Fréchet differentiable and the Fréchet derivative of $J$ in $\bar{s} = (\bar{n},\bar{c},\bar{u},\bar{f})\in X$ in the direction $r = (\tilde{n},\tilde{c},\tilde{u}, \tilde{f})$ is given by

Lemma 5.2. The operator $G:X\rightarrow Y$ is continuous-Fréchet differentiable and the Fréchet derivative of $J$ in $\bar{s} = (\bar{n},\bar{c},\bar{u},\bar{f})\in X$ in the direction $r = (\tilde{n},\tilde{c},\tilde{u}, \tilde{f})$, is the linear operator

defined by

Lemma 5.3. Let $\bar{s} = (\bar{n},\bar{c},\bar{u},\bar{f})\in \mathcal{S}_{ad}$, then $\bar{s}$ is a regular point.

Proof. For any fixed $(\bar{n},\bar{c},\bar{u},\bar{f})\in \mathcal{S}_{ad}$, we set $(g_n,g_c,g_u,\tilde{n}_0, \tilde{c}_0, \tilde{u}_0)\in Y$. Since $0\in \mathcal{C}(\bar{f})$, it suffices to show the existence of $(\tilde{n}, \tilde{c}, \tilde{u})\in Y_n\times Y_c\times Y_u$ such that

Next, we use Leray-Schauder's fixed point method to prove the existence of solutions of the problem (5.5), the operator $T:(\dot{n},\dot{u})\in X_n\times X_u\rightarrow (\tilde{n},\tilde{u})\in Y_n\times Y_u$ with $(\tilde{n}, \tilde{c},\tilde{u})$ solving the decoupled problem:

The system (5.6) is complemented by the corresponding Neumann boundary and initial conditions. Similar to the proof of Lemmas 2.3, 2.4, 2.5 and 2.6, we conduct that operator $T:X_n\times X_u\rightarrow X_n\times X_u$ is well-defined and compact.

Similar to the proof of Theorem 3.1, $(\tilde{n},\tilde{u})$ solves the coupled problem $(\bar{n},\bar{c},\bar{u},\bar{f})$$\in \mathcal{S}_{ad}$, and we set $(g_n,g_c,g_u,\tilde{n}_0, \tilde{c}_0, \tilde{u}_0)\in Y$. Since $0\in \mathcal{C}(\bar{f})$, it suffices to show the existence of $(\tilde{n}, \tilde{c}, \tilde{u})\in Y_n\times Y_c\times Y_u$ such that

complemented by the corresponding Neumann boundary and initial conditions.

Taking the $L^2$-inner product with $\tilde{u}$ for the third equation of (5.7) implies

By the Poincaré inequality and Young's inequality, we have

Taking the $L^2$-inner product with $\tilde{c}$ for the second equation of (5.7) implies

With the Poincaré inequality and Young's inequality in hand, we see that

Taking the $L^2$-inner product with $-\Delta \tilde{c}$ for the second equation of (5.7) implies

For the term $J_1$

For the term $J_2$, we see that

For the term $J_3$, we get

Therefore, combining $J_1$, $J_2$ and $J_3$, we have

Taking the $L^2$-inner product with $\tilde{n}$ for the first equation of (5.7) implies

For the term $J_4$, by Gagliardo-Nirenberg interpolation inequality, we have

For the term $J_5$,

For the term $J_6$,

For the term $J_7$,

Therefore, by choosing $\delta$ small enough, from $J_4$, $J_5$, $J_6$ and $J_7$, it follows that

By choosing $\delta$ small enough and combining (5.8)-(5.11), we get

Applying Gronwall's lemma to the resulting differential inequality, we obatin

Taking the $L^2$-inner product with $-\Delta\tilde{u}$ for the third equation of (5.7) implies

With the use of the Gagliardo-Nirenberg interpolation inequality, we derive

and

For the term $J_{10}$, we deduce

By choosing $\delta$ small enough, with the estimates $J_8$, $J_9$ and $J_{10}$, we have

Applying $\nabla$ to the first equation of (5.7), multiplying it by $\nabla \Delta \tilde{c}$, and integrating over $\Omega$ give

For the first term $J_{11}$, we have

Similarly, for the term $J_{12}$,

For the rest term $J_{13}$, we see

By choosing $\delta$ small enough, we get

From (5.13) and (5.14), along with $\delta$ small enough, it follows that

Applying Gronwall's lemma to the resulting differential inequality, we know

Taking the $L^2$-inner product with $-\Delta\tilde{n}$ for the first equation of (5.7) implies

With the Gagliardo-Nirenberg interpolation inequality in hand, we can estimate $J_{14}$ as follows

Similar to above estimates, we see

Similarly, we derive

and

For the rest terms, we know

Therefore, Taking $\delta$ small enough and together with $J_{14}-J_{18}$, we see that

Applying Gronwall's lemma to the resulting differential inequality, we know

Therefore, from Leray-Schauder theorem, we derive the existence of solution for (5.5). Along with the regularity of $(\tilde{n}, \tilde{c}, \tilde{u})$, the uniqueness of solution can easily get, so we omit the process.

Theorem 5.1. Assume that $\bar{s} = (\bar{n},\bar{c},\bar{u},\bar{f})\in \mathcal{S}_{a d}$ be an optimal solution for the control problem (5.3). Then, there exist Lagrange multipliers $(\lambda, \eta,\rho,\xi,\varphi,\omega)\in L^2(Q)\times (L^2(0,T;H^1(\Omega)))^{\prime}\times L^2(Q)\times (H^1(\Omega))^{\prime}\times(H^2(\Omega))^{\prime}\times(H^1(\Omega))^{\prime}$ such that for all $(\tilde{n}, \tilde{c}, \tilde{u},\tilde{f})\in V_n\times V_c\times V_u\times\mathcal{C}(\bar{f})$ has

where $\mathcal{C}(\bar{f}) = \{\theta(f-\bar{f}):\theta\geq 0, f\in \mathcal{U}\}$.

Proof. With the Lemma 5.3 in hand, we get that $\bar{s}\in\mathcal{S}_{ad}$ is a regular point. Then, togather with Theorem 3.1 in [20], it follows that there exist Lagrange multipliers $(\lambda, \eta,\rho,\xi,\varphi,\omega)\in L^2(Q)\times (L^2(0,T;H^1(\Omega)))^{\prime}\times L^2(Q)\times (H^1(\Omega))^{\prime}\times(H^2(\Omega))^{\prime}\times(H^1(\Omega))^{\prime}$ such that

for all $r = (\tilde{n}, \tilde{c}, \tilde{u},\tilde{f})\in V_n\times V_c\times V_u\times\mathcal{C}(\bar{f})$. Hence, the proof follows from Lemmas 5.1 and 5.2.

Corollary 5.1. Assume that $\bar{s} = (\bar{n},\bar{c},\bar{u},\bar{f})\in \mathcal{S}_{a d}$ be an optimal solution for the control problem (5.3). Then, there exist Lagrange multipliers $(\lambda, \eta,\rho)\in L^2(Q)\times (L^2(0,T;H^1(\Omega)))^{\prime}\times L^2(Q)$, satisfying

which corresponds to the linear system

subject to the following boundary and final conditions

and the following identities hold

Proof. By taking $(\tilde{c},\tilde{u},\tilde{f}) = (0,0,0)$ in (5.15), then it follows that the equation (5.16) holds. In light of an analogous argument, and in light of the (5.15), it guarantees that (5.17) and (5.18) hold. On the other hand, let $(\tilde{n},\tilde{c},\tilde{u}) = (0,0,0)$, as an immediate consequence we obtain

By choosing $\tilde{f} = f-\bar{f}\in \mathcal{C}(\bar{f})$ for all $\bar{f}\in \mathcal{U}$, thus we achieve (5.20).

Theorem 5.2. Under the assumptions of Theorem 5.1, system (5.19) has a unique weak solution such that

Proof. For convenience, we set $\tilde{\lambda} = \lambda(T-t)$, $\tilde{\eta} = \eta(T-t)$, $\tilde{\rho} = \rho(T-t)$, in order to simplify notations, we still write $\lambda$, $\eta$, $\rho$ instead of $\tilde{\lambda}$, $\tilde{\eta}$, $\tilde{\rho}$, then the adjoint system (5.19) can be written as follow

subject to the following boundary and final conditions

Following an analogous reasoning as in the proof of Lemma 5.3, we omit the process and just give a number of a priori estimates as follows.

Taking the $L^2$-inner product with $\lambda$ for the first equation of (5.21) implies

Then, we have

Taking the $L^2$-inner product with $-\Delta\eta$ for the first equation of (5.21) implies

Thus, we get

Taking the $L^2$-inner product with $\eta$ for the second equation of (5.21) implies

As an immediate consequence, we obtain

Taking the $L^2$-inner product with $\rho$ for the third equation of (5.21) implies

Therefore, we see that

Combining (5.22)-(5.25) and taking $\delta$ small enough, we have

Applying Gronwall's lemma to the resulting differential inequality, we know

The proof is complete.

Acknowledgment

The authors would like to express their deep thanks to the referee's valuable suggestions for the revision and improvement of the manuscript.