ON A DISTRIBUTED CONTROL PROBLEM FOR A COUPLED CHEMOTAXIS-FLUID MODEL

. In this paper we analyze an optimal distributed control problem where the state equations are given by a stationary chemotaxis model coupled with the Navier-Stokes equations. We consider that the movement and the interaction of cells are occurring in a smooth bounded domain of R n ,n = 2 , 3 , subject to homogeneous boundary conditions. We control the system through a distributed force and a coeﬃcient of chemotactic sensitivity, leading the chemical concentration, the cell density, and the velocity ﬁeld towards a given target concentration, density and velocity, respectively. In addition to the existence of optimal solution, we derive some optimality conditions.


1.
Introduction. Chemotaxis is understood as the biological process in which the presence of living organisms activates the production of a certain chemical substance. In this process, the movement of organisms in response to a chemical stimulus can be given towards a higher or lower concentration of the chemical substance (positive or negative chemotaxis, respectively). The typical example for chemotaxis is the amoebae Dictyostelium, which is a specie of soil-living amoeba belonging to the phylum Mycetozoa. When they are running out of nutrients, they produce a chemical, cyclic Adenosine Monophosphate, attracting other amoebae in order to perform some kind of transition to a multicellular organism [11]. The most classical model in the framework of chemotactical movements is the Patlak-Keller-Segel system [18,19], which is given by in Ω, t > 0, ∂c ∂n = ∂ρ ∂n = 0 on ∂Ω, t > 0, c(x, 0) = c 0 (x) > 0, ρ(x, 0) = ρ 0 (x) > 0 in Ω, (1) where c is the chemical concentration and ρ denotes the cell density. The function αρ − βc models the production-consumption rate of the chemical; α is a positive constant denoting the rate of attractant production, and β > 0 is a parameter which measures the self-degradation of the chemical. The chemical and the cell fluxes are given respectively by J c = −D c ∇c and J ρ = χρ∇c − D ρ ∇ρ, where D c > 0, D ρ > 0 and χ are constants. Therefore, the cells perform a biased random walk in the direction of the chemical gradient, and the chemical diffuses (it is produced by the cells and it degrades) [11]. The term χρ∇c models the transport of cells towards the higher concentrations of chemical signal if χ > 0, and towards the lower concentrations of chemical signal if χ < 0; χρ is the so called sensitivity function. Model (1) has been modified in last decades with the aim of improving its consistency with biological reality and understand the implications of chemotaxis in different processes, as for instance, pathological and ecological processes, aggregation patterns, stability of nonconstant stationary states, blow up phenomena, and so on [3,14,16,20,28,31,32].
Stationary Patlak-Keller-Segel models related to (1) have been widely analyzed (see, for instance, [9,17,22,26,27,30] and references therein). In this case, since Ω ρ(x, t) dx = Ω ρ 0 (x) dx for all t > 0, by virtue of (1) 2 and (1) 3 , the stationary problem associated to (1) reads D c ∆c = βc − αρ in Ω, D ρ ∆ρ = ∇ · (χρ∇c) in Ω,      c > 0, ρ > 0 in Ω, ∂c ∂n = ∂ρ ∂n = 0 on ∂Ω, Ω ρ(x) dx = m 0 > 0, where m 0 is a given constant. On the other hand, in nature, cells frequently live in a viscous incompressible fluid and the chemical substances are transported by the fluid. In this context, it will be interesting to analyze the situation in which the chemical and cell fluxes are given by J c = −D c ∇c+cu and J ρ = χρ∇c−D ρ ∇ρ+ρu, respectively. Thus, we are led to the following nonlinear stationary model System (2) is completed with homogeneous Neumann conditions for the concentration and the density, and no-slip boundary condition for the velocity, that is, Here the unknowns are the chemical concentration c, the cell density ρ, the velocity u and the pressure π of the fluid; the equations for (u, π) are described by the Navier-Stokes equations. The coupling of chemotaxis and the fluid is realized through the terms u · ∇c, u · ∇ρ and −ρf, representing the transport of chemical substances, the transport of cells, and the forcing term exerted on the fluid by cells, respectively. The constant ν > 0 determines the viscosity of the fluid, D c and D ρ are positive constants representing the chemical diffusion and cell diffusion coefficients, respectively; χ measures the chemotactic sensitivity, and m 0 is a given positive constant. As it was said at the beginning, α is a constant denoting the rate of attractant production, and β > 0 is a parameter which measures the self-degradation of the chemical. We consider that the movement and interaction of cells are occurring in a bounded domain Ω of R n , n = 2, 3, with boundary ∂Ω smooth enough. Non-stationary chemotaxis models, including the coupling with the fluid velocity, have been recently addressed (see, for instance, [23,34,36] and references therein).
In this paper we deal with the mathematical formulation and analysis of a distributed optimal control problem of a chemotaxis process under the effect of a viscous and incompressible fluid. We consider the minimization of a general cost functional subject to constraints, where the state equations are given by the stationary model (2)-(3). We control the system through a distributed force and a coefficient of chemotaxis sensitivity leading the chemical concentration, the cell density and the velocity field towards a given target concentration, density and velocity, respectively. Additionally, the objective functional contains two penalty terms which are given by the norms of the controls in their respective spaces (see Theorems 2.2, 2.4 and 2.5 below). The exact mathematical formulation will be given in Section 2. Regularity in L p -spaces for elliptic problems with homogeneous Neumann boundary conditions are used in order to obtain a solution for the minimization problem (10). The control of velocity, the proliferation of organisms and the concentration of chemicals in diverse enviroments have significant applications in science and biological processes. In fact, in several applications, the respective biological setting requires to control the proliferation and death of cells, for example, bacterial pattern formation [33,35] or endothelial cell movement and growth in response to a chemical substance known as tumor angiogenesis factor (TAF), which have a significant role in the process of cancer cell invasion of neighboring tissue [5,6,25].
In past years, significative progress has been made in mathematical analysis and numerics of optimal control problems for viscous flows described by the Navier-Stokes equations and related models (see for instance, [1,13,15,21,24] and references therein). However, as far as we know, this kind of optimal control problems related to chemotaxis-fluid model have not been studied previously. In spite of, we remark that in [7] and [8] the authors study some results related to the controllability for the nonstationary Keller-Segel system (similar to model (1)) and the nonstationary chemotaxis-fluid model with consumption of chemoattractant substance associated to a system of type (2)-(3), based on Carleman estimates for the solutions of the adjoint systems. But clearly, the aims, the methods used and the analysis developed there differ markedly from those used here.
We will prove the solvability of the optimal control problem and state firstorder optimality conditions by using the Lagrange multipliers method; we derive some optimality conditions satisfied by the optimal controls. In order to obtain these conditions, we will use a penalty method. This technique has been used in [1,15,21] in order to derive optimality conditions for optimal control problems associated to the stationary state of the Navier-Stokes and Boussinesq equations, when the relation control-state is multivalued. Through this method, we introduce a family of approximate control problems which approximates the initial control problem; then, we analyze their optimality conditions and, finally, we pass to the limit in the parameter of approximation in order to derive the desired optimality conditions.
The remaining of this paper is arranged as follows. Having established the optimal distributive control problem, in Subsection 2.1 we analyze the existence of an optimal solution. In Subsection 2.2 we obtain an optimality system to the control problem (10). We prove the existence of Lagrange multipliers through a penalty method.

2.
Optimal distributive control problem. The aim of this work is the study of an optimal distributive control problem for (2)-(3). First of all, we recall some functional spaces which will be used throughout this paper. We will consider the usual Sobolev spaces H m (Ω) and Lebesgue spaces L p (Ω), 1 ≤ p ≤ ∞, with the usual notations for norms · H m and · p , respectively. If H is a Hilbert space, we denote its inner product by ·, · H ; in particular, the L 2 (Ω)-norm and the L 2 (Ω)inner product, will be represented by · and (·, ·), respectively. Corresponding Sobolev spaces of vector valued functions will be denoted by H 1 (Ω), L 2 (Ω), and so on. We also use the solenoidal Banach space V = {v ∈ H 1 0 (Ω) : ∇ · v = 0}, with the norm u V = ∇u . If Z is a general Banach space, its topological dual will be denoted by Z . In the paper, the letter K will denote different positive constants (independent of the coefficients of the model: D c , D ρ , ν, α, β and χ) which may change from line to line or even within the same line. Now we introduce the notion of weak solution and establish the distributive control problem related to the chemotaxis-fluid system.
The set of admissible solutions of problem (10) is defined by (8) includes several cases of objective functionals depending on the value of the coefficients γ i , i = 1, ..., 5. In particular, we can consider γ 1 = γ 3 = 0, and obtain an optimal density controlled by χ and f . In this case, given the optimal ρ, implicitly, by uniqueness in the equation for c, we also obtain a controlled concentration. On the other hand, following our analysis, it is possible to consider the case in which the only control is χ (in this case, J does not depend on f , i.e. γ 4 = 0), or the case in which the only control is f (in this case, J does not depend on χ, i.e. γ 5 = 0). The distributive control on χ can be interesting from a physical point of view because it allows to control the chemotactic sensitivity in order to force the movement of cells towards the increasing or decreasing chemical gradient. Notice that χ could provide minima with different meanings: a positive χ provides a model where the cells perform a biased random walk towards the increasing chemical gradient, while a negative χ provides a model where the cells perform a biased random walk towards the decreasing chemical gradient.
and γ4 2 f 2 in the functional J in (8) can be taken in another suitable L p -spaces. However, in our analysis, the norm γ2 6 ρ − ρ d 6 6 must be considered in the L 6 -space in order to obtain a uniform L 2 estimate for ∇ρ m , where ρ m is a minimizing density sequence whose limit reaches the minimum density (see estimate (12) below).

2.2.
Necessary optimality conditions and an optimality system. This section is devoted to obtain an optimality system to problem (10). We wish to use the method of Lagrange multipliers to turn the constrained optimization problem (10) into an unconstrained one. In order to prove the existence of Lagrange multipliers, we use a penalty method (see [1,15,21]). This method consists in the introduction of a family of penalized problems (P δ ) δ∈(0, 1) , which have at least one optimal solution; we can derive first-order necessary optimality conditions for (P δ ) and then, we obtain first-order necessary optimality conditions for the problem (10). First, we introduce the following operators Observe that A is the well-known Stokes operator. In order to simplify the notation, let us denote by M the set We consider the following family of penalized extremal problems (P δ ): where N > 0 is a given constant andŝ = (ĉ,ρ,û,f,χ) ∈ M is a solution of problem (10).