Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js

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

  • Received: 01 August 2021 Revised: 01 September 2021 Published: 26 October 2021
  • Primary: 92C17, 49J20; Secondary: 49K20, 35K51

  • This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.

    Citation: Yunfei Yuan, Changchun Liu. Optimal control for the coupled chemotaxis-fluid models in two space dimensions[J]. Electronic Research Archive, 2021, 29(6): 4269-4296. doi: 10.3934/era.2021085

    Related Papers:

    [1] Yunfei Yuan, Changchun Liu . Optimal control for the coupled chemotaxis-fluid models in two space dimensions. Electronic Research Archive, 2021, 29(6): 4269-4296. doi: 10.3934/era.2021085
    [2] Zhonghua Qiao, Xuguang Yang . A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28(3): 1207-1225. doi: 10.3934/era.2020066
    [3] Hyung-Chun Lee . Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, 2021, 29(3): 2533-2552. doi: 10.3934/era.2020128
    [4] Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao . Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29(3): 2489-2516. doi: 10.3934/era.2020126
    [5] Guanrong Li, Yanping Chen, Yunqing Huang . A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28(2): 821-836. doi: 10.3934/era.2020042
    [6] Ling Xue, Min Zhang, Kun Zhao, Xiaoming Zheng . Controlled dynamics of a chemotaxis model with logarithmic sensitivity by physical boundary conditions. Electronic Research Archive, 2022, 30(12): 4530-4552. doi: 10.3934/era.2022230
    [7] Jingshi Li, Jiachuan Zhang, Guoliang Ju, Juntao You . A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations. Electronic Research Archive, 2020, 28(2): 977-1000. doi: 10.3934/era.2020052
    [8] Nisachon Kumankat, Kanognudge Wuttanachamsri . Well-posedness of generalized Stokes-Brinkman equations modeling moving solid phases. Electronic Research Archive, 2023, 31(3): 1641-1661. doi: 10.3934/era.2023085
    [9] Linlin Tan, Bianru Cheng . Global well-posedness of 2D incompressible Navier–Stokes–Darcy flow in a type of generalized time-dependent porosity media. Electronic Research Archive, 2024, 32(10): 5649-5681. doi: 10.3934/era.2024262
    [10] José Luis Díaz Palencia, Saeed Ur Rahman, Saman Hanif . Regularity criteria for a two dimensional Erying-Powell fluid flowing in a MHD porous medium. Electronic Research Archive, 2022, 30(11): 3949-3976. doi: 10.3934/era.2022201
  • This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.



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

    {nt+un=Δn(nc)+γnμn2,in Q(0,T)×Ω,ct+uc=Δcc+n+f,in Q,ut+uu=Δuπ+nφ,in Q,u=0,in Q,nν=cν=0,u=0,on (0,T)×Ω,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),in Ω, (1.1)

    where ΩR2 is a bounded domain with smooth boundary Ω. ν is the outward normal vector to Ω, and γ, μ are positive constants. n, c denote the bacterial density, the oxygen concentration, respectively. u, π 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 U. We observe that in the subdomains where f0 we inject oxygen, and conversely where f0 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 ΩR3 without the term uu.

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

    nt+un=Δn(nc)+n(γμn), (1.2)
    ct+uc=Δcc+n, (1.3)
    ut=Δuπ+nφ, (1.4)
    u=0, (1.5)
    nν=cν=0,u=0. (1.6)

    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 μ>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 nd, cd and ud, and at the final moment T is as close as possible to a desired state nΩ, cΩ and uΩ. The main difficulties for treating the problem are caused by the nonlinearity of uu. 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.

    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 ΩR2. 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 0l<k. Suppose that 1q, r, p>0 and lka1 such that

    1plN=a(1qkN)+(1a)1r. (2.1)

    Then, for any uWk,q(Ω)Lr(Ω), there exist two positive constants C1 and C2 depending only on Ω, q, k, r and N such that the following inequality holds

    DluLpc1DkuaLqu1aLr+c2uLr

    with the following exception: If 1<q< and klNq is a non-negative integer, the (2.1) holds only for a satisfying lka<1.

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

    Lemma 2.2. Let ΩR2 be a bounded domain with smooth boundary. Then for all non-negative uH1(Ω), there holds

    u3L3(Ω)δu2H1(Ω)(u+1)log(u+1)L1(Ω)+p(δ1)uL1(Ω),

    where δ is any positive number, and p() is an increasing function.

    We first consider the existence of solutions to the linear problem of system (1.1). Assume functions u0H1(Ω), ˆuL4(0,T;L4(Ω)),ˆnL2(0,T;L2(Ω)), and consider

    {utΔu+ˆuu=π+ˆnφ,in Q,u=0,in Q,u=0,on Ω,u(x,0)=u0(x),in Ω. (2.2)

    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 u0H1(Ω), ˆuL4(0,T;L4(Ω)), ˆnL2(0,T;H1(Ω)),φL(Q), and u be the solution of the problem (2.2), then uL(0,T;H1(Ω))L2(0,T;H2(Ω)) and utL2(0,T;L2(Ω)).

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

    12ddtΩu2dx+Ω|u|2dx+Ωu2dx=Ωˆnφudx+Ωu2dxˆnL2uL2+u2L2C(ˆn2L2+u2L2).

    By Gronwall's inequality, we have

    u2L2+T0u2H1dτC(T0ˆn2L2dτ+u02L2).

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

    ut+Au+P(ˆuu)=P(ˆnφ),

    where A:=PΔ is called Stokes operator, which is an unbounded self-adjoint positive operator in L2 with compact inverse, for more properties of Stokes operator, we refer to [10]. Note that u=0, that is Pu=u, PΔu=Δu, Put=ut. So, in following calculations, we ignore the projection operator P. Multiplying this equation by Δu, and integrating it over Ω, we get

    12ddtΩ|u|2dx+Ω|Δu|2dx+Ω|u|2dx=ΩP(ˆuu)ΔudxΩP(ˆnφ)Δudx+Ω|u|2dx.

    For the terms on the right, we have

    ΩP(ˆuu)ΔudxΩP(ˆnφ)Δudx+Ω|u|2dxˆuL4uL4ΔuL2+ˆnL2ΔuL2+u2L2ˆuL4u1/2L2Δu3/2L2+ˆuL4uL2ΔuL2+ˆnL2ΔuL2+u2L212Δu2L2+C(ˆu4L4+ˆu2L4+1)u2L2+ˆn2L2.

    Therefore, we get

    ddtu2L2+u2H1C(ˆu4L4+ˆu2L4+1)u2L2+Cˆn2L2+C.

    By Gronwall's inequality, we derive

    u2L2+T0u2H1dτC.

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

    T0Ω|ut|2dxdtC.

    Summing up, we complete the proof.

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

    {ctΔc+uc+c=ˆn++f,in Q,cν=0,on (0,T)×Ω,c(x,0)=c0(x),in Ω. (2.3)

    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 c0H2(Ω), ˆnL2(0,T;H1(Ω)), fL2(0,T;H1(Ω)), u be the solution of the problem (2.2), and c be the solution of (2.3). Then cL((0,T),H2(Ω))L2((0,T),H3(Ω)) and ctL2(0,T;L2(Ω)).

    Proof. Multiplying the first equation of (2.3) by c, and integrating it over Ω, we infer from Ωc(uc)=12Ωc2udx=0 that

    12ddtΩc2dx+Ω|c|2dx+Ωc2dxˆnL2cL2+fL2cL2.

    Therefore, we have

    c2L2+c2H1C(c02L2+t0(ˆn2L2+f2L2)dτ).

    Multiplying the first equation of (2.3) by Δc, and integrating it over Ω, we get

    12ddtΩ|c|2dx+Ω|Δc|2dx+Ω|c|2dx=ΩucΔcdxΩΔcˆndxΩΔcfdx.

    Using the Young inequality and the Hölder inequality, we obtain

    ΩucΔcdxΩΔcˆndxΩΔcfdxuL4cL4ΔcL2+ˆnL2ΔcL2+fL2ΔcL2CuH1(c12L2Δc12L2+cL2)ΔcL2+ˆnL2ΔcL2+fL2ΔcL2=CuH1c12L2Δc32L2+CcL2ΔcL2+ˆnL2ΔcL2+fL2ΔcL212Δc2L2+Cu4H1c2L2+C(ˆn2L2+f2L2).

    Combining this and above inequalities, we conclude

    ddtc2L2+c2H1Cu4H1c2L2+C(ˆn2L2+f2L2).

    We therefore verify that

    c2L2+t0c2H1C(t0ˆn2L2dτ+t0f2L2dτ).

    Applying to the first equation of (2.3), multiplying it by Δc, and integrating over Ω give

    12ddtΩ|Δc|2dx+Ω|Δc|2dx+Ω|Δc|2dx=Ω(uc)ΔcdxΩˆn+ΔcdxΩfΔcdx.

    For the terms on the right, we obtain

    Ω(uc)ΔcdxΩˆn+ΔcdxΩfΔcdxΔcL2(uL4ΔcL4+uL4cL4)+ˆnL2ΔcL2+fL2ΔcL2ΔcL2(uL4Δc12L2Δc12L2+uL4ΔcL2+u12L2Δu12L2c12L2Δc12L2+uL2c12L2Δc12L2+u12L2Δu12L2cL2+uL2cL2)+ˆnL2ΔcL2+fL2ΔcL212Δc2L2+C(1+Δc2L2+Δu2L2+ˆn2L2+f2L2).

    Straightforward calculations yield

    Δc2L2+t0Δc2H1dτC(1+t0ˆn2H1dτ+t0f2H1dτ).

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

    T0Ω|ct|2dxdtC,

    and thereby precisely arrive at the conclusion.

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

    {ntΔn+un+n=(nc)+(1+γ)ˆn+μˆn+n,in Q,nν|Ω=0,n(x,0)=n0(x),in Ω. (2.4)

    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 0n0H1(Ω), ˆnL2(0,T;H1(Ω))L4(0,T;L4(Ω)), and u, c, n are the solutions of the problem (2.2), (2.3) and (2.4), respectively. Then n0, nL(0,T;H1(Ω))L2(0,T;H2(Ω)) and ntL2(0,T;L2(Ω)).

    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

    ddtA(t)ndxA(t)nνds+A(t)ndx=(1+γ)A(t)ˆn+dxμA(t)ˆn+ndx.

    Since nν0 on {n<0}, from this we deduce that the right hand side is nonnegative. Integrating this equality on [0,t] gives

    A(t)ndxdτ+t0A(t)ndxdτ=0.

    Then, we get n0.

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

    12ddtΩn2dx+Ω(n2+|n|2)dx+μΩˆn+n2dx=Ωncndx+(1+γ)Ωnˆn+dxnL4cL4nL2+(1+γ)ˆnL2nL2C(n12L2n12L2+nL2)cH2nL2+(1+γ)ˆnL2nL2C(n2L2c4H2+n2L2c2H2+ˆnL2)+12n2H1.

    So, we derive that

    n2L2+T0n2H1dtC(1+T0ˆn2L2dt).

    Multiplying the first equation of (2.4) by Δn, and integrating it over Ω, we get

    12ddtΩ|n|2dx+Ω|Δn|2dx+Ω|n|2dx=ΩunΔndx+Ω((nc)Δn(1+γ)ˆn+Δn+μˆn+nΔn)dxuL4nL4ΔnL2+nL4ΔcL4ΔnL2+nL4cL4ΔnL2+(1+γ)ˆnL2ΔnL2+μnL4ˆnL4ΔnL2CuH1(n12L2Δn12L2+nL2)ΔnL2+nL4(Δc12L2Δc12L2+ΔcL2)ΔnL2+μnL4ˆnL4ΔnL2+(n12L2Δn12L2+nL2)cH1ΔnL2+(1+γ)ˆnL2ΔnL212Δn2L2+C(n2L2+n4L4+Δc4L2+Δc2L2+ˆn2L2+ˆn4L4)12Δn2L2+C(1+n2L2+n4L2+n2L2n2L2+Δc2L2+ˆn2L2+ˆn4L4).

    Straightforward calculations yield

    n2L2+T0Ω(|Δn|2+|n|2+ˆn+|n|2)dxdtC.

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

    T0Ω|nt|2dxdtC.

    The proof is complete.

    Introduce the spaces

    Xu=L4(0,T;L4(Ω)),Xn=L4(0,T;L4(Ω))L2(0,T;H1(Ω)),Yu=L(0,T;H1(Ω))L2(0,T;H2(Ω)),Yn=L(0,T;H1(Ω))L2(0,T;H2(Ω)).

    Define a map

    F:Xu×XnXu×Xn,F(ˆu,ˆn)=(u,n),

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

    {ntΔn+un+n=(nc)+(1+γ)ˆn+μˆn+n,in (0,T)×ΩQ,ctΔc+uc+c=ˆn++f,in (0,T)×ΩQ,utΔu+ˆuu=π+ˆnφ,in (0,T)×ΩQ,u=0,in (0,T)×ΩQ,nν=cν=0,u=0,on (0,T)×Ω,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),in Ω.

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

    Lemma 2.6. The map F:Xu×XnXu×Xn is well defined and compact.

    Proof. Let (ˆn,ˆu)Xu×Xn, by Lemmas 2.3, 2.4, 2.5 we deduce that (n,u)=F(ˆn,ˆu) is bounded in Yu×Yn. Note that the embeddings H2(Ω)H1(Ω) is compact and interpolating between L(0,T;H1(Ω)) and L2(0,T;H2(Ω)). It is easy to get that u is bounded in L4(0,T;L4(Ω)) and n is bounded in L4(0,T;L4(Ω))L2(0,T;H1(Ω)). Therefore, the operator F:Xu×XnXu×Xn is a compact operator.

    From Lemma 2.6, (n,u)Yn×Yu satisfies pointwisely a.e. in Q the following problem

    {ntΔn+un+n=(nc)+α(1+γ)nμn2,in Q,ctΔc+uc+c=n+αf,in Q,utΔu+uu=π+αnφ,in Q,u=0,in Q,nν=cν=0,u=0,on (0,T)×Ω,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),in Ω. (3.1)

    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

    nL1+t0(nL1+nL2)dτC, (3.2)
    u2L2+t0u2H1dτC, (3.3)
    c2L2+t0c2H1dτC. (3.4)

    Proof. With Lemma 2.5 in hand, we get n0. Integrating the first equation of (3.1) over Ω, we see that

    ddtΩndx+Ωndx+μΩn2dx=α(1+γ)Ωndxμ2Ωn2dx+C.

    Solving this differential inequality, we obtain that

    nL1+t0(nL1+nL2)dτC.

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

    12ddtΩu2dx+Ω|u|2dx+Ωu2dx=αΩnφudx+Ωu2dxnL2uL2+u2L2C(n2L2+u2L2).

    Therefore, we see that

    u2L2+t0uH1dτC.

    By the Gagliardo-Nirenberg interpolation inequality, we deduce that

    t0u4L4dτCt0(u2L2u2L2d+u2L2)τu2L2t0u2L2dτ+t0u2L2dτC.

    Multiplying the third equation of (3.1) by Δu, and integrating it over Ω, we get

    ddtu2L2+u2H1C(u4L4+u2L4+1)u2L2+Cn2L2+C.

    Thus, we know

    u2L2+t0u2H1dτC.

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

    12ddtΩc2dx+Ω|c|2dx+Ωc2dxnL2cL2+αfL2cL2.

    Then, we have

    Multiplying the second equation of (3.1) by , and integrating it over , we get

    Further, we have

    The proof is complete.

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

    (3.5)

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

    Multiplying the above equation by and integrating the equation, we have

    For , integrating by parts and using Young's inequality with small , we get

    For the term , we have

    For the rest term , straightforward calculations yield

    Combining , with , we conduct that

    (3.6)

    Multiplying the second equation of (3.1) by , and integrating it over , we get

    Straightforward calculations yield

    (3.7)

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

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

    The proof is complete.

    Lemma 3.3. Assume , let be a local solution to (3.1). Then, it holds that

    (3.8)

    Proof. Taking the -inner product with 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

    (3.9)

    Applying to the first equation of (3.1), multiplying it by , and integrating over give

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

    For the term , we have

    Along with and , we conclude

    (3.10)

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

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

    The proof is complete.

    Lemma 3.4. Assume , let be a local solution to (3.1). Then, it holds that

    (3.11)

    Proof. Taking the -inner product with for the first equation of (3.1) implies

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

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

    For the term , thanks to the nonnegativity of , we see that

    Combine the estimates about , and , it follows that

    By taking small enough, we get

    Therefore, this proof is complete.

    Lemma 3.5. The operator , is continuous.

    Proof. Let be a sequence of , Then, with Lemmas 2.3, 2.4 and 2.5 in hand, we conduct that is bounded in . Taking the compactness of in into consider, we see that is a compact operator, which means there exists a subsequence of , for convenience, still denoted as , and exists an element in such that

    Let and take the limit, it is clear that and , this means that . Since uniqueness of limit, the map is continuous.

    Theorem 3.1. Let , , with in , and , then (1.1) exists unique strong solution . Moreover, there exists a positive constant such that

    (3.12)

    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 , and , where is the strong solution of the system, where . Thus, we obtain the following system

    (3.13)
    (3.14)
    (3.15)
    (3.16)
    (3.17)
    (3.18)

    Taking the -inner product with for the (3.13) implies

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

    For the term , with the estimate (3.8) and (3.11), we get

    For the term ,

    With the use of estimates , we have

    (3.19)

    Taking the -inner product with for the (3.14) implies

    Then, we get

    (3.20)

    Taking the -inner product with for the (3.15) implies

    Straightforward calculations yield

    (3.21)

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

    By choosing small enough, we get

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

    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 is a nonempty, closed and convex set, where control domain , and is the observability domain. We adjust the external source , so that the bacterial density , oxygen concentration and fluid velocity are as close as possible to a desired state , and , and at the final moment is as close as possible to a desired state , and . We consider the optimal control problem as follows

    Minimize the cost functional

    (4.1)

    subject to the system (1.1). Moreover, the nonnegative constants are given but not all zero, the functions , , represents the desired states satisfying

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

    The function space is given by

    where .

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

    Theorem 4.1. Suppose is satisfied, and , then the optimal control problem (4.1) admits a solution .

    Proof. Along with Theorem 3.1, we conduct that , then there exists the minimizing sequence such that

    (4.2)

    According to the definition of , for each there exists satisfying

    (4.3)

    Observing that is a closed convex subset of . According to the definition of , we deduce that there exists bounded in such that, for subsequence of , for convenience, still denoted by , as

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

    Since is bounded in , then

    Recalling that

    Therefore, we get that . Owing to , we see that 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 is an optimal pair for problem (1.1).

    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

    (5.1)

    where is a functional, is an operator, and are Banach spaces, and nonempty closed convex set is subset of and nonempty closed convex cone with vertex at the origin in .

    denotes its polar cone

    We consider the following Banach spaces

    where

    and the operator , where

    which are defined at each point by

    (5.2)

    The function spaces are given as follows

    We see that is a closed convex subset of and , and rewrite the optimal control problem

    (5.3)

    Taking the differentiability of and into consider, it follows that

    Lemma 5.1. The functional is Fréchet differentiable and the Fréchet derivative of in in the direction is given by

    (5.4)

    Lemma 5.2. The operator is continuous-Fréchet differentiable and the Fréchet derivative of in in the direction , is the linear operator

    defined by

    Lemma 5.3. Let , then is a regular point.

    Proof. For any fixed , we set . Since , it suffices to show the existence of such that

    (5.5)

    Next, we use Leray-Schauder's fixed point method to prove the existence of solutions of the problem (5.5), the operator with solving the decoupled problem:

    (5.6)

    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 is well-defined and compact.

    Similar to the proof of Theorem 3.1, solves the coupled problem , and we set . Since , it suffices to show the existence of such that

    (5.7)

    complemented by the corresponding Neumann boundary and initial conditions.

    Taking the -inner product with for the third equation of (5.7) implies

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

    (5.8)

    Taking the -inner product with for the second equation of (5.7) implies

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

    (5.9)

    Taking the -inner product with for the second equation of (5.7) implies

    For the term

    For the term , we see that

    For the term , we get

    Therefore, combining , and , we have

    (5.10)

    Taking the -inner product with for the first equation of (5.7) implies

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

    For the term ,

    For the term ,

    For the term ,

    Therefore, by choosing small enough, from , , and , it follows that

    (5.11)

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

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

    (5.12)

    Taking the -inner product with for the third equation of (5.7) implies

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

    and

    For the term , we deduce

    By choosing small enough, with the estimates , and , we have

    (5.13)

    Applying to the first equation of (5.7), multiplying it by , and integrating over give

    For the first term , we have

    Similarly, for the term ,

    For the rest term , we see

    By choosing small enough, we get

    (5.14)

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

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

    Taking the -inner product with for the first equation of (5.7) implies

    With the Gagliardo-Nirenberg interpolation inequality in hand, we can estimate as follows

    Similar to above estimates, we see

    Similarly, we derive

    and

    For the rest terms, we know

    Therefore, Taking small enough and together with , 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 , the uniqueness of solution can easily get, so we omit the process.

    Theorem 5.1. Assume that be an optimal solution for the control problem (5.3). Then, there exist Lagrange multipliers such that for all has

    (5.15)

    where .

    Proof. With the Lemma 5.3 in hand, we get that is a regular point. Then, togather with Theorem 3.1 in [20], it follows that there exist Lagrange multipliers such that

    for all . Hence, the proof follows from Lemmas 5.1 and 5.2.

    Corollary 5.1. Assume that be an optimal solution for the control problem (5.3). Then, there exist Lagrange multipliers , satisfying

    (5.16)
    (5.17)
    (5.18)

    which corresponds to the linear system

    (5.19)

    subject to the following boundary and final conditions

    and the following identities hold

    (5.20)

    Proof. By taking 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 , as an immediate consequence we obtain

    By choosing for all , 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 , , , in order to simplify notations, we still write , , instead of , , , then the adjoint system (5.19) can be written as follow

    (5.21)

    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 -inner product with for the first equation of (5.21) implies

    Then, we have

    (5.22)

    Taking the -inner product with for the first equation of (5.21) implies

    Thus, we get

    (5.23)

    Taking the -inner product with for the second equation of (5.21) implies

    As an immediate consequence, we obtain

    (5.24)

    Taking the -inner product with for the third equation of (5.21) implies

    Therefore, we see that

    (5.25)

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

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

    The proof is complete.

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



    [1] The Debye system: Existence and large time behavior of solutions. Nonlinear Anal. (1994) 23: 1189-1209.
    [2] B. Chen and C. Liu, Optimal distributed control of a Allen-Cahn/Cahn-Hilliard system with temperature, Applied Mathematics and Optimization, 2021. doi: 10.1007/s00245-021-09807-2
    [3] B. Chen, H. Li and C. Liu, Optimal distributed control for a coupled phase-field system, Discrete and Continuous Dynamical Systems Series B. doi: 10.3934/dcdsb.2021110
    [4] Optimal control for a conserved phase field system with a possibly singular potential. Evol. Equ. Control Theory (2018) 7: 95-116.
    [5] Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity (2017) 30: 2518-2546.
    [6] Reaction terms avoiding aggregation in slow fluids. Nonlinear Anal. Real World Appl. (2015) 21: 110-126.
    [7] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
    [8] Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential. Appl. Math. Optim. (2020) 81: 899-931.
    [9] F. Guillén-González, E. Mallea-Zepeda and M. Rodríguez-Bellido, Optimal bilinear control problem related to a chemo-repulsion system in 2D domains, ESAIM Control Optim. Calc. Var., 26 (2020), 21pp. doi: 10.1051/cocv/2019012
    [10] On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. (2009) 344: 381-429.
    [11] C. Jin, Large time periodic solutions to coupled chemotaxis-fluid models, Z. Angew. Math. Phys., 68 (2017), 24pp. doi: 10.1007/s00033-017-0882-9
    [12] Large time periodic solution to the coupled chemotaxis-Stokes model. Math. Nachr. (2017) 290: 1701-1715.
    [13] Optimal distributed control for a new mechanochemical model in biological patterns. J. Math. Anal. Appl. (2019) 478: 825-863.
    [14] Optimal control of a new mechanochemical model with state constraint. Math. Methods Appl. Sci. (2021) 44: 9237-9263.
    [15] Optimal control of Keller-Segel equations. J. Math. Anal. Appl. (2001) 256: 45-66.
    [16] Compact sets in the space . Ann. Mat. Pura Appl. (1987) 146: 65-96.
    [17] Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system. Z. Angew. Math. Phys. (2015) 66: 2555-2573.
    [18] Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system. Z. Angew. Math. Phys. (2016) 67: 1-23.
    [19] Optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state constraint. Appl. Math. Optim. (2020) 82: 721-754.
    [20] Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. (1979) 5: 49-62.
  • This article has been cited by:

    1. Hui Tang, Yunfei Yuan, Optimal control for a chemotaxis–haptotaxis model in two space dimensions, 2022, 2022, 1687-2770, 10.1186/s13661-022-01661-7
    2. Yunfei Yuan, Changchun Liu, Optimal control for a fully parabolic singular chemotaxis model with indirect signal consumption in two space dimensions, 2022, 0003-6811, 1, 10.1080/00036811.2022.2139244
    3. Sida Lin, Lixia Meng, Jinlong Yuan, Changzhi Wu, An Li, Chongyang Liu, Jun Xie, Sequential adaptive switching time optimization technique for maximum hands-off control problems, 2024, 32, 2688-1594, 2229, 10.3934/era.2024101
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2134) PDF downloads(230) Cited by(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog