One very interesting feature of living organisms is their interaction with the environment in which they reside. Frequently, the form of interaction involves the movement of living organisms generated by an external stimulus, the response to such stimulus is called taxis. The process which leads to taxis it is divided into three steps [1]: First, the cell detects the extracellular signal by specific receptors on its surface then, the cell processes the signal and, finally, the cell alters its motile behavior. There exist different types of taxis, which depend on the nature of the stimulus (see, for instance, [2]), one of them is chemotaxis.

The chemotaxis phenomenon is understood as the movement of living organisms induced by the presence of certain chemical substances. In 1970, Keller and Segel [3] proposed a mathematical model that describes the chemotactic aggregation of cellular slime molds which move preferentially towards relatively high concentrations of a chemical secreted by the amoebae themselves. Such phenomenon is called chemo-attraction. In contrast, the phenomenon is called chemo-repulsion, if a region of high chemical concentration generate a repulsive effect on the organisms. The most classical model in the framework of chemotactic movements is the Keller-Segel system [3,4], which is given by the following system of partial differential equations: where $\Omega \subset {\mathbb{R}}^n$, $n=2,3$, is a bounded domain with smooth boundary $\partial \Omega$, $(0,T)$ is a time interval with $0<T<\infty$, $Q:=(0,T)\times \Omega$ is the time-space region, and $\mathbf{n}$ denotes the outward unit normal vector to $\partial \Omega$. The unknowns are the cell density $u:=u(t,x)\ge 0$ and a chemical concentration $v:=v(t,x)\ge 0$. The cell flux and chemical are given respectively by $\chi u\nabla v-{\alpha }_u\nabla$ and ${\alpha }_v\nabla v$, where ${\alpha }_u,{\alpha }_v>0$ and $\chi \ne 0$ are real 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) [5]. The term $\chi u\nabla v$ models the transport of cells; if $\chi >0$ towards the higher concentrations of chemical (chemo-attraction) and if $\chi <0$ towards the lower concentrations of chemical (chemo-repulsion). The term $-\beta v+h\left(u\right)$ models the consumption-production rate of the chemical, where $\beta$ is a real parameter which measures the self-degradation of the chemical, and the function $h\left(u\right)$ is the cells production term, which is nonnegative when $u\ge 0$.

In this paper we study a bilinear optimal control problem related to the parabolic-elliptic system associated to problem (1), considering nonlinear chemical signal production and a proliferation/degradation coefficient acting on a control subdomain $\omega$. Specifically, we consider a bounded domain $\Omega \subset {\mathbb{R}}^2$ with smooth boundary $\partial \Omega \in C^{2,1}$ and a time interval $(0,T)$, with $0<T<\infty$. Then, we study an optimal control problem related to the following coupled system of partial differential equations in the time-space region $Q:=(0,T)\times \Omega$: where $\omega \subset \Omega$ is the control subdomain, f denotes a bilinear control that acts on the subdomain $\omega$ and $1_{\omega }$ is the characteristic function of $\omega$. The control lies in a nonempty, closed and convex set $\mathcal{F}$. We observe that, in the subdomain $\omega$, where $f\ge 0$ the we inject chemical substance and, where $f\le 0$ we extract chemical substance; thus, we can see the control f as proliferation/degradation coefficient acting on the subregion $\omega$. The term, $u^p$, for $p>1$, is the nonlinear chemical signal production term.

The system (2) is completed with the initial condition and the Neumann boundary conditions

Studies on the existence of solutions related to system (2)-(4) can be consulted in [6,7,8,9,10,11,12,13,14,15]. In particular, the case when $f\equiv 0$ and $p=1$ has been analyzed by Mock in [6,7], in which the author proved the existence and uniqueness of global-in-time classical solutions and that the respective solutions are uniformly bounded and converge an exponential rate to steady-state. The parabolic-parabolic system related to problem (2)-(4) has been studied in [8,9,10,11,12,13,14]. In [8,9] considering linear production and $f\equiv 0$ has been studied, where Ciéslak et al. [8] proved the existence and uniqueness of smooth classical solutions in two-dimensional domains, as well as, the existence of weak solutions in spaces of dimension 3 and 4. In [9], the author delimits his analysis to a n-dimensional convex domain ($n\ge 3$) and changes the chemotactic term $\nabla ·(u\nabla v)$ by $\nabla ·\left(h\right(u)\nabla v)$, where $h\left(u\right)$ is an adequate smooth function. With this modification, the author proved the existence of a unique global-in-time classical solution and that the pair solution $(u,v)$ converges to $\frac{1}{|\Omega |}({\int }_{\omega }{u}_0,{\int }_{\omega }{u}_0)$, as t goes to ∞. Moreover, in [10,11], the authors proved, for a quadratic production term ($p=2$), the existence of weak solutions in 3D domains and global-in-time strong solutions assuming a regularity criteria in 1D and 2D domains; furthermore, they analyzed some numerical schemes to approximate the weak solutions. In [12], for $p=1$ and $f\not\equiv 0$, the authors proved the existence and uniqueness of strong solutions in 2D domains, and deduced that the solution $(u,v)$ does not blow-up at finite time. The same authors in [13] extend the results obtained in [12] to 3D domains, and proved the existence of weak solutions and established a regularity criterion to get global-in-time strong solutions. In [14], consider, in a two-dimensional domain, the nonlinear case for $p\in (1,2]$ and $f\not\equiv 0$ and proved the existence and uniqueness of strong solutions. The existence and uniqueness of strong solutions for problem (2)-(4), considering $p\in (1,2)$ and $f\in L^q\left(\omega \right)$, has been proved by Ancoma-Huarachi et al. [15]. For the stationary case and linear production term, we can refer to recent study developed by Lorca et al. [16].

In the context of the optimal control problems related to chemotaxis system, we can refer to [12,13,14,16,17,18,19,20,21,22] in all these works it proved the existence of at least one global optimal solution and derived an optimality system, in particular obtain first-order necessary optimality conditions. In [19] it studied an optimal control problem with state equations drive by a chemoattractive-Navier-Stokes evolution system in 3D domains where stated first-order necessary optimality conditions, using that the state is differentiable with respect to the control. Rodríguez-Bellido et al. [20] analyzed a distributed optimal control problem related to a stationary chemotaxis model coupled with the Navier-Stokes equations. Also, they derived an optimality system through a penalty method, because the relation control-to-state is multivalued. In [22] consider a 2D chemotaxis model with logistic source and proved the existence of weak solutions for the dynamical equation, the solvability of the optimal control problem and derive an optimality system using a generic result on the existence of Lagrange multipliers. The works [12,13,14] are dedicated to study control problems related to chemo-repulsion models and considered the parabolic-parabolic system associated with problem (2)-(4). In [12], the authors explored a bilinear optimal control problem in 2D domains, and proved the existence of global optimal solutions and derived an optimality system. The same authors in [13] studied the 3D version of [12]. Guillén-González et al. [14] extended the results of [13] for superlinear production term; that is, for $p\in (1,2]$. Recently, in [16] a bilinear optimal control problem related to a stationary version system of (2)-(4) has been studied for n-dimensional domains, with $n=1,2,3$. In this work the authors proved the existence of global optimal solutions and derived first-order necessary optimality conditions for local optimal solutions. As far as we known, optimal control problems related with system (2)-(4) has not been considered in the literature.

The paper is organized as follows: In Section 2 we fix some notations, introduce the function spaces which will be used through the work, give the concept of strong solutions of problem (2)-(4), present a result concerning to the existence and uniqueness of global-in-time strong solutions of (2)-(4), and establish two regularity (parabolic and elliptic) results for heat-Neumann problem that will used to achieve our results. Finally, in Section 3 we analyze the bilinear optimal control problem and obtain the several important results, which includes the existence of global optimal solutions, the derivation of the an optimality system for a local optimal solution, via a result on existence of Lagrange multipliers in Banach spaces, and obtain some extra regularity properties of the Lagrange multipliers.

In this section we establish some notations, definitions and preliminaries results that will be used throughout this work. We will use the classical Lebesgue spaces $L^s:=L^s(\Omega )$, for $1\le s\le \infty$, with norm denoted by ${\parallel ·\parallel }_{L^s}$. In particular, for $s=2$, the $L^2$-norm and the respective $L^2$-inner product will be denoted by $\parallel ·\parallel$ and $(·,·)$. Moreover, we use the Sobolev spaces $W^{m,s}:=W^{m,s}(\Omega )=\{u\in L^2:\parallel {\partial }^{\alpha }{u\parallel }_{L^s},\forall \left|\alpha \right|\le m\}$, with norm denoted by ${\parallel ·\parallel }_{W^{m,s}}$. When $s=2$, we denote by $H^m:=W^{m,2}$ and the respective norm by ${\parallel ·\parallel }_{H^m}$. Also, we will use the space $W_{\mathbf{n}}^{m,s}:=\{u\in W^{m,s}:\frac{\partial u}{\partial \mathbf{n}}=0on\partial \Omega \}$, for $m>1+\frac{1}{s}$, with nor denoted by ${\parallel ·\parallel }_{W_{\mathbf{n}}^{m,s}}$. Moreover, if X is a Banach space, we will denote by $L^s\left(X\right)$ the space of functions $u:[0,T]\to X$ that are integrable in the Bochner sense, and its norm will be denoted by ${\parallel ·\parallel }_{L^s\left(X\right)}$. For simplicity, we will denote $L^s\left(Q\right):=L^s\left(L^s\right)$ and its norm by ${\parallel ·\parallel }_{L^s\left(Q\right)}$. Also, $C\left(X\right)$ denotes the space of continuous functions $u:[0,T]\to X$, where X is a Banach space, and its respective norm by ${\parallel ·\parallel }_{C\left(X\right)}$. The topological dual space of a Banach space X will be denoted by $X^{\prime }$, and the respective duality for a pair X and $X^{\prime }$ by ${\langle ·,·\rangle }_{X^{\prime }}$ or simply by $\langle ·,·\rangle$ unless this leads ambiguity. Finally, as usual, the different letters $C,K,C_1,K_1,...,$ denote positive constants independent of $(u,v)$, but its value may change from line to line.

We are interested in studying a bilinear optimal control problem related with the strong solutions of problem (2)-(4). The following definition establishes the concept of strong solutions of system (2)-(4), more details can be consulted in [15].

Some properties that can be extracted directly from system (2)-(4) and that are key to obtaining the existence of strong solutions are the following:

Now, we present a result related to the existence and uniqueness of strong solutions to problem (2)-(4). This result is valid only when $p\in (1,2)$. For this reason, we restrict our analysis to $1<p<2$ (se [15], for more details).

Throughout this paper, we frequently use the following equivalent norms in the spaces $H^1$ and $H^2$ (see, for instance, [23]): and the classical 2D interpolation inequality

Moreover, we will apply the following results concerning to parabolic and elliptic regularity for the heat-Neumann problem:

This section is dedicated to the study of a bilinear optimal control problem related with the strong solutions of system (2)-(4). Firstly, we establish the statement of the bilinear control problem under analysis. Indeed, we assume that the controls set is $\mathcal{F}$ which is a nonempty, closed and convex subset of $\mathcal{B}\left(\widehat{K}\right)$, where $\mathcal{B}\left(\widehat{K}\right)\subset L^q\left(\omega \right)$, for $2<q<\infty$, is the open ball where $\beta$ and $\widehat{K}$ are the constants given in (9) (see Theorem 1 above) and $\omega \subset \Omega$ is the control domain. We consider the initial data $u_0\in H^1$ with $u_0\ge 0$ and the function $f\in \mathcal{F}$ that describes a bilinear control acting on the chemical equation (2)${}_2$.

Furthermore, we consider the Banach spaces the functional $J:\mathbb{X}\to \mathbb{R}$ defined by and the operator $R:=(R_1,R_2,R_3):\mathbb{X}\to \mathbb{Y}$, where $R_i:\mathbb{X}\to \mathbb{Y}$, for $i=1,2,3$, are defined at each point $s:=(u,v,f)\in \mathbb{X}$ by

In the functional J, defined in (15), the pair $(u_d,v_d)\in L^2\left(Q\right)\times L^2\left(Q\right)$ represents the desired states and the nonnegative real numbers ${\alpha }_u,{\alpha }_v$ and ${\alpha }_f$ measure the cost of the states $(u,v)$ and the control f, respectively. These real numbers are nonzero simultaneously. The functional J describes the deviation of the cell density u from a cell density $u_d$ and the deviation of the chemical concentration v from a desired chemical $v_d$, plus the cost of the control measured in the $L^q$-norm.

Then, taking $S:=S_u\times S_v\times \mathcal{F}$ we formulate the following bilinear optimal control problem: notice that $S\subset \mathbb{X}$ is a closed and convex set and that the functional J is weakly lower-semicontinuous on S. The set of the admissible solutions of control problem (17) is given by which, by virtue of Theorem 1, is a nonempty set.

We are interested in proving the existence of global optimal solutions to problem (17) and derive the so-called first-order necessary optimality conditions for any local optimal solution of control problem (17). In the following definitions we present the concepts of global optimal solutions and local optimal solutions of problem (17), respectively.