Bilinear Optimal Control of the Keller–Segel Logistic Model in 2D-Domains

An optimal control problem associated to the Keller–Segel with logistic reaction system is studied in 2D domains. The control acts in a bilinear form only in the chemical equation. The existence of an optimal control and a necessary optimality system are deduced. The main novelty is that the control can be rather singular and the state (cell density u and the chemical concentration v) remains only in a weak setting, which is not usual in the literature.


The Controlled Model
In this work we study an optimal control problem for the (attractive or repulsive) Keller-Segel model in a 2D domain ⊂ R 2 with logistic source term and bilinear control acting on the chemical equation: Here, f : Q c := (0, T ) × c → R is the control with c ⊂ ⊂ R 2 the control domain (denoting 1 c the characteristic function in c ), and the states u, v : Q := (0, T ) × → R 2 + are the cellular density and chemical concentration, respectively. Moreover, r , μ > 0 are coefficients of the logistic reaction, and κ ∈ R is the chemotaxis coefficient (κ > 0 models attraction and κ < 0 repulsion). We are interested in studying of optimal control problem associated to the weak solution setting for system (1), see Definition 1 below.

Previous Results
In the last decades, there has been a surge of activity on the study of the chemotaxis model describing the motion of cells directed by the concentration gradient of a chemical substance. Moreover, it is important to consider the biological situation where the bacterial population may proliferate according to a logistic law and the chemical signal is produced by cells. On the other hand, the chemotaxis-fluid systems, which is basically the chemotaxis model coupled with the Navier-Stokes equations, appear when the interactions between cells and the chemical signal is also extended with liquid environments. For more details, see the excellent review [3] and the references therein.
Plenty of analytical results have been obtained for the "uncontrolled" problem (1), that is, for f ≡ 0. Many of these results are based on classical in time solutions of such systems following Amann's works (see, for instance [2]). Amongst the many articles related to this uncontrolled system, let us mention those on existence of weak and strong solutions in R 2 . In this case, without considering logistic reaction (i.e. r = μ = 0), the existence of global weak solutions was provided by Liu and Lorz [16]. In two-dimensional bounded convex domains, the existence of (global) classical solutions was obtained by Winkler [25]. In the presence of a logistic source, the existence of global weak solutions (and their long time behavior) has been analyzed in [14] by Lankeit. In this case, the existence of global mild solutions was examined in [8]. For 3D domains, we also refer [26] and the references therein.
It is important to mention that remarkable progress has been made in the mathematical and numerical analysis of optimal control problems for viscous flows described by the Navier-Stokes equations and other related models, see e.g., [1,4,19]. However, the literature related to optimal control for chemotaxis problems is still scarce. The reader can consult distributed linear control in [7] for a mathematical model of cancer invasion and [21] for the Keller-Segel system. In [17], the authors established the existence of an optimal control for a parabolic attraction-repulsion chemotaxis model with logistic source in 2D by introducing a linear distributed (positive) control in the chemical equation. The case of a Neumann boundary linear control for a chemotaxis system is treated in [22] for a one-dimensional problem. In these cases, positivity of the control needs to be imposed to guarantee the positivity of the states. As far as we know, the case of distributed bilinear control is only treated in [10] for a one-dimensional system of Keller-Segel type, acting either in the equation for the cell density or the chemical concentration. In [20], an optimal (distributed) control problem is studied constrained to a stationary chemotaxis model coupled with the Navier-Stokes equations. We note that in [5,6] some results are provided related to the controllability for the Keller-Segel system and the chemotaxis-fluid model with consumption of chemoattractant substance, respectively. These results are based on Carleman-type estimates for the solutions of the adjoint system. Recently, a bilinear optimal control problem associated to the chemotaxis-Navier-Stokes model (without logistic source) in bounded 3D domains was examined in [18]. For the chemo-repulsion case, this problem was studied in [11,13] for 2D and 3D domains respectively, and in [12] for 2D domains with a potential nonlinear production term, by changing the production term u in the v equation of (1) by u p , with 1 < p ≤ 2.

Main Contributions of the Paper
We state the definition of weak solutions and then we will obtain the existence and uniqueness of such solutions (u, v) of (1) which are bounded with respect to the control f .
Hereafter, L 2+ means L 2+ε for small enough ε. Notice that, since we are in 2D bounded domains, v ∈ C([0, T ]; W 1+,2+ ( )) implies v ∈ L ∞ (0, T ; L ∞ ( )), hence using that f ∈ L 2+ (Q c ) one has f v ∈ L 2+ (Q). That means that the maximal regularity expected is v ∈ X 2+ . The previous weak regularity for u ∈ W 2 will be enough to solve the optimal control problem formulated in (3), which represents an improvement over previous optimal control results that needed the strong solution setting to obtain the first order necessary optimality system (5).
There exists a unique weak solution (u, v) of system (1) in the sense of Definition 1. Moreover, there exists a positive constant where we denote Finally, for any r , μ, κ, , The second main result of this paper will be the existence of a global optimal solution for the following problem: Here, the pair (u d , v d ) ∈ L 2 (Q) 2 represents the target states and the nonnegative numbers γ u , γ v and γ f measure the cost of the states and control, respectively. With respect to the control constraint, we assume F ⊂ L 2+ (Q c ) to be a nonempty, closed and convex set.
The functional J defined in (3) describes the deviation of the cell density u and the chemical concentration v from a target cell density u d and chemical concentration v d , respectively, plus the cost of the control f measured in the L 2+ -norm.
, then the bilinear optimal control problem (3) has at least one global optimal solution (ũ,ṽ,f ).
Finally, we obtain the existence and uniqueness of Lagrange multipliers associated to any local optimal control of (3): Theorem 3 Lets = (ũ,ṽ,f ) ∈ S ad be a local optimal solution of (3). Then, there exists a unique Lagrange multiplier (λ, η) ∈ X 2 × W 2 satisfying the optimality system Remark 1 If γ f > 0 and F ≡ L 2+ (Q c ) (that is, no convexity constraints on the control are imposed), then optimality condition (6) becomes the equality The rest of the paper is organized as follows. The proofs of Theorems 1, 2 and 3 are given in Sects. 2, 3 and 4, respectively. Conclusions will be made at Sect. 5.
Along this manuscript, the following result on L p regularity will be considered.
admits a unique solution u such that Moreover, there exists a positive constant C := C( p, , T ) such that

Proof of Theorem 1
We prove the existence via the Leray-Schauder fixed point theorem (the precise statement of this result can be consulted, for instance, in [13], Theorem 2) and the uniqueness by a comparison argument.

Existence
Let us introduce the auxiliary spaces whereū + := max{ū, 0} ≥ 0,v + := max{v, 0} ≥ 0. In fact, first we compute v and after u. In the following lemmas, we will prove that the hypotheses of the Leray-Schauder fixed point theorem are satisfied. (7). Considering the linear parabolic u-problem in (7), one has u ∈ W 2 owing to v ∈ X 2+ , hence ∇v ∈ L 4+ (Q) and then

Lemma 6 The set
with M independent of α, such that endowed with the corresponding initial and boundary conditions. Therefore, it suffices to look for a bound of (u, v) in W 2 × X 2+ independent of α. This bound is carried out into six steps: Step 1: Non-negativity: u, v ≥ 0.
Integrating directly over (0, T ) for a fixed T > 0 in (11), and using (13), we obtain which implies that Step 4: Taking v as test function in (10) 2 and using that α ∈ (0, 1], we obtain where we have used the following standard inequality in 2D domains Therefore, taking δ small enough in (14), we get From Gronwall's lemma, and due to the boundedness of u and f in Adding (14) to (15) and taking δ small enough, we obtain By testing (10) 1 by u, after a few computations, we get Adding u 2 L 2 to both sides of this inequality, we arrive at Therefore, applying the Gronwall lemma and using Step 4, we obtain that u is bounded in L ∞ (0, T ; L 2 ( )) ∩ L 2 (0, T ; H 1 ( )).

By interpolation, from
Step 4 and Step 5 we also have bounds for v in L ∞− (Q) and u in L 4 (Q), respectively. Therefore, u + f v ∈ L 2+ (Q). Then, the heat regularity result in Theorem 4 allows us to deduce that v ∈ X 2 + and to obtain the corresponding bound on X 2 + depending on v 0 W 1 + ,2 + ( ) and the bound of This finishes the proof of Lemma 6.
The proof is similar to Lemma 3.4 in [11]. Consequently, from Lemmas 5, 6 and 7, the Leray-Schauder fixed point theorem implies that the map R(ū,v) has at least one fixed point R(u, v) = (u, v) which is a weak solution to system (1) in (0, T ).
Finally, we observe that estimate (2) is shown following the same steps given in the proof of Lemma 6 above (now for the case α = 1).

Uniqueness of Solution
This proof follows the same argument as in [11], but it is included here for the reader convenience. Let (u 1 , v 1 ), (u 2 , v 2 ) ∈ W 2 × X 2 be two weak solutions of system (1). Substracting equations (1) , we obtain the following system Testing (16) 1 by u ∈ L 2 (H 1 ) and (16) Note that the term μ u 2 (u 1 + u 2 ) dx has the good sign.

Proof of Theorem 2
The admissible set for the optimal control problem (3) is defined by From the definition of J and the assumption that either γ f > 0 or F is bounded in From (8)- (9), there exists C > 0, independent of m, such that Therefore, from (22), (23), and taking into account that F is a closed convex subset of L 2+ (Q c ) (hence it is weakly closed in L 2+ (Q c )), there existss = (ũ,ṽ,f ) ∈ W 2 × X 2+ × F such that, for some subsequence of {s m } m∈N , still denoted by {s m } m∈N , the following convergences hold, as m → +∞: u m →ũ weakly in L 2 (H 1 ) and weakly* in L ∞ (L 2 ), (24) v m →ṽ weakly in L 2+ (W 2,2+ ) and weakly* in L ∞ (W 1+,2+ ), From convergences (24)- (27), using Sobolev embeddings and Aubin-Lions compactness results (see, for instance, [15,23]), one has In particular, using (28), (30) and (31), the limit of the nonlinear terms of (11) can be controlled as follows: Moreover, from convergence (29) (1), that is,s ∈ S ad . Therefore, On the other hand, since J is lower semicontinuous on S ad , one has J (s) ≤ lim inf m→+∞ J (s m ), which jointly with (34), implies thats is a global optimal control.

A Generic Lagrange Multipliers Theorem
We consider the Lagrange multipliers theorem given in [27] (see also [24,Chapter 6], for more details) that we will apply to get first-order necessary optimality conditions for any local optimal solution (ũ,ṽ,f ) of problem (3). First, we consider the following (generic) optimization problem: where J : X → R is a functional, G : X → Y is an operator, X and Y are Banach spaces, and M is a nonempty closed and convex subset of X. Definition 3 (Lagrange multiplier) Lets ∈ S be a local optimal solution for problem (35). Suppose that J and G are Fréchet differentiable ins. Then, any ξ ∈ Y is called a Lagrange multiplier for (35) at the points if where C(s) = {θ(s −s) : s ∈ M, θ ≥ 0} is the conical hull ofs in M.

Definition 4
Lets ∈ S. It will be said thats is a regular point if

Application of the Lagrange Multiplier Theory
Now, in order to reformulate the optimal control problem (3) in the abstract setting (35), we introduce the Banach spaces Thus, the optimal control problem (3) is reformulated as follows where with (û,v) the global weak solution of (1) without control, i.e.,f = 0, F is defined in (4) and Remark 2 From Definition 2, the Lagragian associated to the optimal control problem (37) is the functional L : The set M defined in (38) is a closed convex subset of X and the admissible set of control problem (37) is Concerning to the differentiability of the functional J and the constraint operator G, one has the following results.
Proof From Definition 4, one has thats = (ũ,ṽ,f ) ∈ S ad is a regular point if for any To this end, we will use the Leray-Schauder fixed point Theorem for the operator where (U , V ) is the solution of the decoupled problem (first V and after U ) Let us show that S satisfies the hypothesis of the Leray-Schauder Theorem.
Step 1 (S is well-defined, continuous and bounded). We prove that S maps bounded sets in In particular, using that problem (42) is linear, it is not difficult to prove the continuity of S from Applying L 2+ -regularity to the heat equation (43) 2 (Theorem 4), one has V ∈ X 2+ and Taking ϕ = U in (43) 1 , we arrive at Finally, using 2D interpolation estimates, we have Then, using (44), Gronwall's Lemma applied to (45) guarantees the bound for U in W 2 .
Step 2 (compactness): Using that W 2 × X 2+ is compactly embedded in L 4− (Q) × L ∞ (Q), it follows that the operator S is compact.
Step 4 (conclusion): Applying the Leray-Schauder fixed point theorem, one has the existence of (U , V ) ∈ W 2 × X 2+ , a solution of problem (42). Its uniqueness is directly deduced from the linearity of problem (42).

Existence of Lagrange Multipliers
Now, the existence of Lagrange multiplier for problem (3) associated to any local optimal solutions = (ũ,ṽ,f ) ∈ S ad will be shown.
From Theorem 12, an optimality system for problem (3) can be derived.

Corollary 13 Lets = (ũ,ṽ,f ) ∈ S ad be a local optimal solution for the control problem (3). Then any Lagrange multiplier
and the optimality condition Thus, choosing F = θ( f −f ) ∈ C(f ) for all f ∈ F and θ ≥ 0, (53) is deduced.

Regularity of Lagrange Multipliers
Theorem 14 Lets = (ũ,ṽ,f ) ∈ S ad be a local optimal solution for problem (3). Then, problem (5) has a unique solution (λ, η) such that In order to prove the existence of a solution for (54), the Leray-Schauder fixed point Theorem can be applied as before, now for the operator where (λ, η) = T (λ,η) solves the decoupled problem (first computing λ and after μ) The proof follows the same lines as before and it will be omitted. Indeed, the key point is to show that the set of possible fixed points is bounded in X 2 ×W 2 (with respect to α). In fact, if (λ, η) ∈ T α , then (λ, η) ∈ X 2 ×W 2 and it solves the coupled linear problem Now, taking λ− λ ∈ L 2 (Q) as test function in (55) 1 and η ∈ L 2 (H 1 ) as test function in (55) 2 , the following bound is obtained via Gronwall's Lemma: Therefore, applying Leray-Schauder fixed point theorem, the existence of a solution of problem (5), (λ, η) ∈ X 2 × W 2 , is obtained. Its uniqueness is directly deduced from the linearity of problem (5).
In the following result, more regularity and uniqueness of the Lagrange multiplier (λ, η) given by Theorem 12 will be obtained via the uniqueness of problem (5).

Conclusions
The existence and uniqueness of a weak solution for problem (1) in 2D-domains allows one to deduce the existence of (at least) a global optimal solution of (3), leading the system near to the desired stated of cellular density and chemical concentration. The existence of a unique and regular Lagrange multiplier characterized by its optimality system (5)- (6) is also proven. The fact of using only weak solutions for problem (1) is a novelty with respect to the previous results in related models appearing in [11][12][13].

Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
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