Optimal bilinear control problem related to a chemo-repulsion system in 2D domains

ESAIM: COCV 26 (2020) 29 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2019012 www.esaim-cocv.org OPTIMAL BILINEAR CONTROL PROBLEM RELATED TO A CHEMO-REPULSION SYSTEM IN 2D DOMAINS Francisco Guill´ en-Gonz´ alez1,*, Exequiel Mallea-Zepeda2 and Mar´ ıa ´ Angeles Rodr´ ıguez-Bellido1 Abstract. In this paper, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a bidimensional domain. The existence, uniqueness and regularity of strong solutions of this model are deduced, proving the existence of a global optimal solution. Afterwards, we derive first-order optimality conditions by using a Lagrange multipliers theorem. Mathematics Subject Classification. 35K51, 35Q92, 49J20, 49K20. Received June 14, 2018. Accepted March 12, 2019. 1. Introduction In biology, the chemotaxis phenomenon is understood as the movement of living organisms induced by the presence of certain chemical substances. In 1970 Keller and Segel [12] proposed a mathematical model that describes chemotactic aggregation of cellular slime molds which move preferentially towards relatively high concentrations of a chemical substance secreted by the amoebae themselves. Such phenomenon is called chemoattraction with production. In contrast, if regions of high chemical concentration generate a repulsive effect on the organisms, the phenomenon is called chemorepulsion. We are interested in studying a chemorepulsion model given by the following system of partial differential equations          ∂tu−∆u=∇·(u∇v) in (0, T)×Ω≡Q, ∂tv−∆v+v=h(u) in (0, T)×Ω≡Q, u(0, x) = u0(x), v(0, x) = v0(x) in Ω, ∂u ∂n= 0,∂v ∂n= 0 on (0, T)×∂Ω, (1.1) where Ω ⊂R2, is a bounded domain with smooth boundary ∂Ω, ndenotes the outward unit normal vector to ∂Ω and (0, T) is a time interval. The unknowns are cell density u(t, x)≥0 and chemical concentration v(t, x)≥0. The function h(u) represents the production term, which must be nonnegative when u≥0. Keywords and phrases: Chemorepulsion-production model, strong solutions, bilinear control, optimality conditions. 1Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico and IMUS, Universidad de Sevilla, Sevilla, Spain. 2Departamento de Matem´atica, Universidad de Tarapac´a, Arica, Chile. *Corresponding author: [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2020 2F. GUILL´ EN-GONZ´ ALEZ ET AL. System (1.1), when the production term is linear, that is h(u) = u, was studied by Ci´eslak et al in [7]. The authors, based on the abstract theory for quasilinear parabolic problems (see [2]), proved the global existence and uniqueness of smooth classical solution in 2D domains, and global existence of weak solutions in spaces of dimension 3 and 4. Tao [25], in a bounded convex domain Ω ⊂Rn(n≥3), studies system (1.1) with h(u) = u and a modification in the density-dependent chemotactic sensitivity function, that is, the term ∇·(u∇v) is changed by ∇·(g(u)∇v), where g∈C2([0,+∞]), g(0) = 0,0< g(u)≤C(u+ 1)αfor all u > 0 with some C > 0 and α > 0.The author proves that, under assumptions of initial data 0 6≡ u0∈C0(Ω) and v0∈C1(Ω) are nonnegative and that α < 4 n+2 , there exists a unique global in time classical solution of (1.1) and the corresponding solution (u, v) converges to (u0, u0) as time goes to +∞, where u0:= 1 |Ω|RΩu0. In this work we study a control problem subject to this chemorepulsion with linear production model in which a bilinear control acts injecting or extracting chemical substance on a subdomain of control Ωc⊂Ω. Specifically, we consider Ω ⊂R2be a bounded domain of class C2, then we study a control problem associated to the following system in Q:= (0, T)×Ω, ∂tu−∆u=∇·(u∇v), ∂tv−∆v+v=u+fv, (1.2) with initial conditions u(0, x) = u0(x)≥0, v(0, x) = v0(x)≥0 in Ω,(1.3) and boundary conditions ∂u ∂n= 0,∂v ∂n= 0 on (0, T)×∂Ω.(1.4) Here, the function fdenotes a bilinear control that acts on chemical concentration, which lies in a closed convex set F. We observe that in the subdomains where f≥0 we inject chemical substance, and conversely where f≤0 we extract chemical substance. There is a wide collection of publications dealing with optimal control of PDEs. See, for example, [1,4,11,13,15,17,20,24,28,29] and the references therein. In all previous publications, the control variable enters the state equation either on the right-hand side (distributed controls) or is part of the boundary conditions (boundary controls). As far as we know, the literature related to optimal control problems with bilinear control is scarce, see [3,10,14,27]. The main difficulty is that the solution of the state equation depends nonlinearly on the control and state variables (see the second equation in (1.2)). In the context of optimal control problems associated to chemotaxis models, the literature is also scarce, see [8,10,19,21,22]. In [8] the authors study a distributed optimal control for a two-dimensional model of cancer invasion. Using the Leray-Schauder fixed point theorem, they prove the existence of weak solutions of state system. Also, they prove the existence of optimal control and derive an optimality system. The works [10,21] delimit their study to a one-dimensional domain. In [10] two extreme problems on a chemoattractant model are analyzed; one involves harvesting the actual cells and the other depicts removing a proportion of the chemical substance. The control is bilinear (total) and acts on a portion of the cells or chemical substance. They prove the existence of optimal solutions and derive an optimality system. Also, they design a numerical scheme for the optimality system and present some examples. In the problem studied in [21], the control acts on the boundary conditions for the chemical substance. The existence of optimal solutions is proved. In the recent work [19], the authors analyze a distributive optimal control problem where the state equations are given by a stationary chemotaxis model coupled with the Navier-Stokes equations (chemotaxis-fluid system). They prove the existence of an optimal solution. In addition, they derive an optimality system through a penalty OPTIMAL BILINEAR CONTROL PROBLEM 3 method, because the relation control-state is multivalued. Finally, in [22], on a 2D domain, the authors study a problem in which the control variable is distributed, and acts on the equation for the chemical substance. They prove the existence of optimal solutions. Furthermore, using the fact that the state is differentiable with respect to the control, they derive an optimality system. Other studies related to controllability for the nonstationary Keller-Segel system and nonstationary chemotaxis-fluid model can be consulted in [5,6], respectively. The outline of this paper is as follows: In Section 2, we fix the notation, introduce the functional spaces to be used, give the definition of strong solution for system (1.2)–(1.4) and we state a parabolic regularity result that will be used throughout this work. In Section 3, we prove the existence (and uniqueness) of strong solution of (1.2)–(1.4) using the Leray-Schauder fixed point theorem. In Section 4, we establish the optimal control problem, proving the existence of an optimal solution and we obtain the first-order optimality conditions based on a Lagrange multipliers theorem in Banach spaces. Finally, we obtain a regularity result for Lagrange multipliers. 2. Preliminaries In order to establish the control problem, we will introduce some notations. We will use the Lebesgue space Lp(Ω), 1 ≤p≤+∞, with norm denoted by k · kLp. In particular, the L2(Ω) norm and its inner product will denoted by k·k and (·,·), respectively. We consider the usual Sobolev spaces Wm,p(Ω) = {u∈Lp(Ω) : k∂αukLp<+∞,∀|α| ≤ m}, with norm denoted by k·kWm,p . When p= 2, we write Hm(Ω) := Wm,2(Ω) and we denote the respective norm by k·kHm. Also, we use the space Wm,p n(Ω) = {u∈Wm,p(Ω) : ∂u ∂n= 0 on ∂Ω} (m > 1 + 1/p), with norm denoted by k·kWm,p n. If Xis a Banach space, we denote by Lp(0, T;X) the space of valued functions in Xdefined on the interval [0, T] that are integrable in the Bochner sense, and its norm will be denoted by k·kLp(X). For simplicity we denote Lp(Q) := Lp(0, T;Lp(Ω)) if p6= +∞and its norm by k·kLp(Q). In the case p= +∞,L∞(Q) means L∞((0, T)×Ω), and its norm is denoted by k·kL∞(Q). Also, we denote by C([0, T]; X) the space of continuous functions from [0, T] into a Banach space X, and its norm by k·kC(X). The topological dual space of a Banach space Xwill be denoted by X0, and the duality for a pair X and X0by h·,·iX0or simply by h·,·i unless this leads to ambiguity. Moreover, the letters C,K,C1,K1,..., are positive constants, independent of state (u, v) and control f, but its value may change from line to line. We are interested in the study of a control problem associated to strong solutions of system (1.2)–(1.4). In the following definition we give the concept of strong solution of (1.2)–(1.4). Definition 2.1. Let f∈L4(Q), u0∈H1(Ω), v0∈W3/2,4 n(Ω) with u0≥0 and v0≥0 a.e. in Ω, a pair (u, v) is called strong solution of problem (1.2)–(1.4) in (0, T), if u≥0 and v≥0 in Q, u∈ Yu:= {u∈L∞(0, T;H1(Ω)) ∩L2(0, T;H2 n(Ω)), ∂tu∈L2(Q)},(2.1) v∈ Yv:= {v∈L∞(0, T;W3/2,4 n(Ω)) ∩L4(0, T;W2,4 n(Ω)), ∂tv∈L4(Q)},(2.2) the system (1.2) hold pointwisely a.e. (t, x)∈Q, ∂tu−∆u=∇·(u∇v),(2.3) ∂tv−∆v+v=u+fv, (2.4) and the boundary and initial conditions (1.3) and (1.4) are satisfied, respectively. Remark 2.2. The problem (1.2)–(1.4) is conservative in u. In fact, integrating (1.2)1in Ω we have d dt ZΩ u= 0,i.e. ZΩ u(t) = ZΩ u0:= m0,∀t > 0.(2.5) 4F. GUILL´ EN-GONZ´ ALEZ ET AL. Also, integrating (1.2)2in Ω we deduce d dt ZΩ v+ZΩ v=m0+ZΩ fv. (2.6) We define the space c W2−2/p,p(Ω) as follows c W2−2/p,p(Ω) = W2−2/p,p(Ω) if p < 3, W2−2/p,p n(Ω) if p > 3.(2.7) In order to study the existence of solution of system (1.2)–(1.4), we will use the following regularity result for the heat equation (see [9], p. 344). Lemma 2.3. For Ω∈ C2, let 1<p<+∞(p6= 3) and g∈Lp(Q),u0∈c W2−2/p,p(Ω). Then the problem      ∂tu−∆u=gin Q, u(0, x) = u0(x)in Ω, ∂u ∂n= 0 on (0, T)×∂Ω, admits a unique solution uin the class u∈C([0, T]; c W2−2/p,p(Ω)) ∩Lp(0, T;W2,p(Ω)), ∂tu∈Lp(Q). Moreover, there exists a positive constant C=C(p, Ω, T )such that kukC( c W2−2/p,p)+k∂tukLp(Q)+kukLp(W2,p)≤C(kgkLp(Q)+ku0kc W2−2/p,p ).(2.8) In particular, the equation ∂tu−∆u=gis pointwisely satisfied a.e. in Q. Remark 2.4. In the case of p= 3, one concludes that u∈C([0, T]; X3,3)∩L3(0, T;W2,3(Ω)), ∂tu∈L3(Q), for a certain space X3,3(see [9], Thm. 10.22) whose description is not evident in terms of c W2−2/p,p(Ω) or another Sobolev space. Thorough this paper, we will use the following equivalent norms in H1(Ω) and H2(Ω), respectively (see [18] for details): kuk2 H1≃ k∇uk2+ZΩ u2 ,∀u∈H1(Ω),(2.9) kuk2 H2≃ k∆uk2+ZΩ u2 ,∀u∈H2 n(Ω),(2.10) and the classical interpolation inequality in 2Ddomains kukL4≤Ckuk1/2kuk1/2 H1,∀u∈H1(Ω).(2.11) 3. Existence and uniqueness of strong solution of system (1.2)–(1.4) In this section we will prove the existence (and uniqueness) of solution of (1.2)–(1.4) using the Leray-Schauder fixed point theorem. Specifically we will prove the following result: OPTIMAL BILINEAR CONTROL PROBLEM 5 Theorem 3.1. Let u0∈H1(Ω),v0∈W3/2,4 n(Ω) with u0≥0and v0≥0in Ω, and f∈L4(Q). There exists a unique strong solution (u, v)of system (1.2)–(1.4) in sense of Definition 2.1. Moreover, there exists a positive constant K1:= K1(m0, T, ku0kH1,kv0kW3/2,4 n ,kfkL4(Q)), such that k(∂tu, ∂tv)kL2(Q)×L4(Q)+k(u, v)kC(H1×W3/2,4 n)+kukL2(H2)+kvkL4(W2,4)≤ K1.(3.1) 3.1. Existence Let us introduce the “weak” spaces Xu:= C0([0, T]; L2(Ω)) ∩L8/3(0, T;W1,8/3(Ω)) and Xv:= C0([0, T]; C(Ω)).(3.2) We define the operator R:Xu×Xv→ Yu×Yv,→ Xu×Xvby R(¯u, ¯v) = (u, v) the solution of the decoupled linear problem          ∂tu−∆u=∇·(¯u+∇v), ∂tv−∆v+v= ¯u++f¯v+, u(0) = u0, v(0) = v0, ∂u ∂n= 0,∂v ∂n= 0, (3.3) where ¯u+:= max{¯u, 0} ≥ 0, ¯v+:= max{¯v, 0} ≥ 0. In fact, first we find vand after u. In the following lemmas we will prove the hypotheses of Leray-Schauder fixed point theorem. Lemma 3.2. The operator R:Xu×Xv→ Xu×Xvis well defined and compact. Proof. Let (¯u, ¯v)∈ Xu× Xv. Since Xu,→L4(Q), Xv⊂L∞(Q) and f∈L4(Q), then ¯u++f¯v+∈L4(Q). By applying Lemma 2.3 (for p= 4), there exists a unique solution v∈ Yvof (3.3)2such that kvkL4(W2,4)+k∂tvkL4(Q)+kvkC(W3/2,4 n)≤C(k¯ukL4(Q)+k¯vkL∞(Q)kfkL4(Q)+kv0kW3/2,4 n ) ≤C(kv0kW3/2,4 n ,kfkL4(Q)).(3.4) Now, using the fact that v∈ Yv, in particular we have ∇v∈L∞(0, T;L4(Ω)) ∩L4(0, T;W1,4(Ω)) ,→L8(Q), and taking into account that ∇¯u+∈L8/3(Q), ∆v∈L4(Q), ¯u+∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)) ,→L4(Q) we have ∇·(¯u+∇v) = ¯u+∆v+∇¯u+·∇v∈L2(Q).Thus, again by Lemma 2.3 (for p= 2), we conclude that there exist a unique u∈ Yusolution of (3.3)1such that kukL2(H2)+k∂tukL2(Q)+kukC(H1)≤C(k¯ukL4(Q)k∆vkL4(Q)+k∇¯ukL8/3(Q)k∇vkL8(Q)+ku0kH1) ≤C(ku0kH1,kv0kW3/2,4 n ,kfkL4(Q)).(3.5) In the last inequality of (3.5), estimate (3.4) has been used. Therefore, Ris well defined from Xu× Xvto Yu×Yv. Moreover, the compactness of Ris consequence of estimates (3.4) and (3.5), and the compact embedding Yu×Yv,→ Xu×Xv. Indeed, let (¯u, ¯v) be in a bounded set of Xu×Xvand consider (u, v) = R(¯u, ¯v). From (3.5), u is bounded in L∞(0, T;H1(Ω)) and ∂tuis bounded in L2(Q). Because of H1(Ω) ,→L2(Ω) in a compact way, from Th´eor`eme 5.1 of [23] one can deduce that Yuis compactly embedded in C0([0, T]; L2(Ω)). Moreover, interpolating 6F. GUILL´ EN-GONZ´ ALEZ ET AL. between L∞(0, T;H1(Ω)) and L2(0, T;H2(Ω)), uis bounded in L8/3(0, T;H7/4(Ω)) (see [9] or [26]). Using that H7/4(Ω) ,→W1,p(Ω), p < 8, in a compact way, and that ∂tuis bounded in L2(Q), again from Th´eor`eme 5.1 of [23] one has in particular that Yuis compactly embedded in L8/3(0, T;W1,8/3(Ω)). Similarly, starting from (3.4), vis bounded in L∞(0, T;W3/2,4 n(Ω)) and ∂tvis bounded in L4(Q). Taking into account that W3/2,4 n(Ω) ,→C0(Ω) in a compact way, one can deduce that Yvis compactly embedded in C0([0, T]; C0(Ω)) = C0([0, T]×Ω). Lemma 3.3. The set Tα={(u, v)∈ Yu×Yv: (u, v) = αR(u, v)for some α∈[0,1]}(3.6) is bounded in Xu×Xv(independently of α∈[0,1]). In fact, Tαis also bounded in Yu×Yv, i.e. there exists M=M(m0, T, ku0kH1,kv0kW3/2,4 n ,kfkL4(Q))>0,(3.7) with Mindependent of α, such that all pairs of functions (u, v)∈Tαfor α∈[0,1] satisfy k(u, v)kL2(H2)×L4(W2,4)+k(∂tu, ∂tv)kL2(Q)×L4(Q)+k(u, v)kC(H1)×C(W3/2,4 n)≤M. Proof. Let (u, v)∈Tαfor α∈(0,1] (the case α= 0 is trivial). Then, owing to Lemma 3.2, (u, v)∈ Yu×Yvand satisfies pointwisely a.e. in Qthe following problem: ∂tu−∆u=∇·(u+∇v), ∂tv−∆v+v=αu++αfv+(3.8) endowed with the corresponding initial and boundary conditions. Therefore, it suffices to look a bound of (u, v) in Yu×Yvindependent of α. This bound is carried out into five steps: Step 1: u, v ≥0 and ZΩ u(t) = m0. By testing (3.8)1by u−:= min{u, 0} ≤ 0, and considering that u−= 0 if u≥0, ∇u−=∇uif u≤0, and ∇u−= 0 if u > 0, we have 1 2 d dtku−k2+k∇u−k2=−(u+∇v, ∇u−)=0, thus u−≡0 and, consequently, u≥0. Similarly, testing (3.8)2by v−:= min{v, 0} ≤ 0 we obtain 1 2 d dtkv−k2+k∇v−k2+kv−k2=α(u+, v−) + α(fv+, v−)≤0, which implies v−≡0, then v≥0. Therefore (u+, v+) = (u, v). Finally, integrating (3.8)1in Ω and using (2.5) we obtain ZΩ u(t) = m0. Step 2: vis bounded in L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)). We observe that, thanks to the positivity of u, we have 0 ≤ln(u+ 1) ≤u. Then ZΩ|ln(u+ 1)|2≤ZΩ|u|2.(3.9) OPTIMAL BILINEAR CONTROL PROBLEM 7 We also note that ZΩ|∇ln(u+ 1)|2=ZΩ∇u u+ 1 2 ≤ZΩ|∇u|2.(3.10) Taking into account that u∈L2(0, T;H1(Ω)), from (3.9) and (3.10) we deduce that ln(u+ 1) ∈L2(0, T;H1(Ω)). Then, since u∈ Yu, testing (3.8)1by αln(u+ 1) ∈L2(0, T;H1(Ω)) and (3.8)2by −∆v∈L4(Q), and integrating by parts, we have d dtαZΩ (u+ 1) ln(u+ 1) + 1 2k∇vk2+ 4αk∇√u+ 1k2+k∆vk2+k∇vk2 =−αZΩ u u+ 1∇v·∇u+αZΩ∇u·∇v−αZΩ fv∆v =αZΩ 1 u+ 1∇u·∇v−αZΩ fv∆v. (3.11) Applying the H¨older and Young inequalities we obtain αZΩ 1 u+ 1∇u·∇v≤α 2ZΩ |∇u|2 u+ 1 +α 2ZΩ |∇v|2 u+ 1 ≤2αk∇√u+ 1k2+α 2k∇vk2,(3.12) −αZΩ fv∆v≤αkfkL4kvkL4k∆vk ≤ δkvk2 H2+α2Cδkfk2 L4kvk2 H1.(3.13) Moreover, integrating (3.8)2in Ω and using (2.5) and (2.6), we have d dtZΩ v+ZΩ v=α m0+αZΩ fv. Multiplying this equation by ZΩ vand using the H¨older and Young inequalities we obtain 1 2 d dtZΩ v2 +ZΩ v2 =α m0ZΩ v+αZΩ fvZΩ v ≤1 2ZΩ v2 +Cα2m2 0+Cα2kfk2kvk2.(3.14) Replacing (3.12)–(3.14) in (3.11), and taking into account that α≤1, we obtain d dtαZΩ (u+ 1) ln(u+ 1) + 1 2kvk2 H1+ 2αk∇√u+ 1k2+C1kvk2 H2≤C(m2 0+kfk2 L4kvk2 H1),(3.15) where the constants C, C1are independent of α. From (3.15) and Gronwall lemma we have kvk2 L∞(H1)≤exp(kfk2 L2(L4))ku0k2+kv0k2 H1+Cm2 0T:= K0(m0, T, ku0k,kv0kH1,kfkL2(L4)).(3.16) 8F. GUILL´ EN-GONZ´ ALEZ ET AL. Now, integrating (3.15) in (0, T) and using (3.16) we obtain kvk2 L2(H2)≤Cku0k2+kv0k2 H1+m2 0T+kvk2 L∞(H1)kfk2 L2(L4) := K1(m0, T, ku0k,kv0kH1,kfkL2(L4)).(3.17) Therefore, from (3.16) and (3.17) we conclude that vis bounded in L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)). Step 3: uis bounded in L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)). Testing (3.8)1by u, applying the H¨older and Young inequalities, and using (2.11), we obtain 1 2 d dtkuk2+k∇uk2=−(u∇v, ∇u)≤ kukL4k∇vkL4k∇uk ≤ Ckuk1/2k∇vkL4kuk3/2 H1 ≤Ckuk2k∇vk4 L4+1 2kuk2 H1. Thus, taking into account that m2 0=ZΩ u(t)2 and the equivalent norm of H1(Ω) given in (2.9), we have d dtkuk2+kuk2 H1≤Ck∇vk4 L4kuk2+ 2m2 0.(3.18) On the other hand, using (2.11), jointly (3.16) and (3.17), k∇vk4 L4(Q)≤CK0K1. Therefore, we can apply the Gronwall lemma in (3.18), obtaining kuk2 L∞(L2)≤exp(CK0K1)(ku0k2+ 2m2 0) := K2(m0, T, ku0k,kv0kH1,kfkL2(L4)).(3.19) Also, integrating (3.18) in (0, T) we have kuk2 L2(H1)≤ ku0k2+ 2m2 0T+CK0K1kuk2 L∞(L2)≤ ku0k2+ 2m2 0T+CK0K1K2 := K3(m0, T, ku0k,kv0kH1,kfkL2(L4)).(3.20) Therefore, from (3.19) and (3.20) we deduce that uis bounded in L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)). Step 4: vis bounded in Yv. Taking into account that f∈L4(Q) and v∈L∞(0, T;H1(Ω)), in particular αfv ∈L7/2(Q). Then using Lemma 2.3 (for p= 7/2) in (3.8)2we conclude that vsatisfies the following inequality kvkL7/2(W2,7/2)+k∂tvkL7/2(Q)+kvkC(W10/7,7/2 n)≤C(αku+fvkL7/2(Q)+kv0kW10/7,7/2 n ) ≤C(kukL4(Q)+kfkL4(Q)kvkL28(Q)+kv0kW10/7,7/2 n ).(3.21) From (3.16), kvkL28 (Q)≤K0. Using (2.11) and taking into account (3.19) and (3.20) we have kuk4 L4(Q)≤Ckuk2 L∞(L2)kuk2 L2(H1)≤CK2K3.(3.22) Therefore, from (3.21) one has kvkC(W10/7,7/2 n)is bounded (independently of α). In particular, by Sobolev embeddings, we obtain kvkL∞(Q)is also bounded. OPTIMAL BILINEAR CONTROL PROBLEM 9 Then, from (3.22) and using again Lemma 2.3 (for p= 4), we obtain that vsatisfies the estimate kvkL4(W2,4)+k∂tvkL4(Q)+kvkC(W3/2,4 n)≤C(αku+fvkL4(Q)+kv0kW3/2,4 n ) ≤C(kukL4(Q)+kvkL∞(Q)kfkL4(Q)+kv0kW3/2,4 n ) ≤K4(m0, T, ku0k,kv0kW3/2,4 n ,kfkL4(Q)).(3.23) Therefore vis bounded in Yv. Step 5: uis bounded in Yu. Testing (3.8)1by −∆u∈L2(Q) we have 1 2 d dtk∇uk2+k∆uk2=−(∇·(u∇v),∆u) = −(u∆v+∇u·∇v, ∆u).(3.24) By the H¨older and Young inequalities, and using interpolation inequality (2.11), we obtain −(u∆v+∇u·∇v, ∆u)≤(kukL4k∆vkL4+k∇ukL4k∇vkL4)k∆uk ≤δk∆uk2+Cδkuk2 L4k∆vk2 L4+Ck∇uk1/2k∇vkL4kuk3/2 H2 ≤δk∆uk2+Cδkuk2 L4k∆vk2 L4+Cδk∇uk2k∇vk4 L4+δkuk2 H2.(3.25) Replacing (3.25) in (3.24), choosing δsmall enough to absorb the k∆uk2and kuk2 H2terms, and taking into account that ZΩ u(t)2 =m2 0and (2.9) and (2.10), we have d dtkuk2 H1+Ckuk2 H2≤Ckuk2 L4k∆vk2 L4+Ck∇uk2k∇vk4 L4+Cm2 0.(3.26) Then, from (3.22) and (3.23), and applying Gronwall lemma to (3.26), we deduce kuk2 L∞(H1)≤K5(m0, T, ku0kH1,kv0kW3/2,4 n ,kfkL4(Q)).(3.27) Finally, integrating (3.26) in (0, T ) we obtain kuk2 L2(H2)≤K6(m0, T, ku0kH1,kv0kW3/2,4 n ,kfkL4(Q)).(3.28) Then, from (3.8)1, (3.23), (3.27) and (3.28) we have k∂tukL2(Q)=k∆u+u∆v+∇u·∇vkL2(Q) ≤ k∆ukL2(Q)+kukL4(Q)k∆vkL4(Q)+k∇ukL4(Q)k∇vkL4(Q) ≤K7(m0, T, ku0kH1,kv0kW3/2,4 n ,kfkL4(Q)),(3.29) which implies that uis bounded in Yu. Finally, from (3.23) and (3.27)-(3.29) we conclude that the elements of Tαare bounded in Yu× Yvfor α∈(0,1]. The radius Min (3.7) follows from (3.23) and (3.27)-(3.29). Lemma 3.4. The operator R:Xu×Xv→ Xu×Xv, defined in (3.3), is continuous. 16 F. GUILL´ EN-GONZ´ ALEZ ET AL. Proof. Fixed (˜u, ˜v, ˜ f)∈ Sad, let (gu, gv, U0, V0)∈Y. Since 0 ∈ C(˜ f), it suffices to show the existence of (U, V )∈ Yu×Yvsuch that              ∂tU−∆U−∇·(U∇˜v)−∇·(˜u∇V) = guin Q, ∂tV−∆V+V−U−˜ fV =gvin Q, U(0) = U0, V (0) = V0in Ω, ∂U ∂n= 0,∂V ∂n= 0 on (0, T)×∂Ω. (4.32) In order to prove the existence of solution of (4.32), we can use the Leray-Schauder’s fixed point argument over the operator S: ( ¯ U, ¯ V)∈ Xu×Xv7→ (U, V )∈ Yu×Yvwith (U, V ) the solution of the decoupled problem: (∂tU−∆U=∇·(˜u∇V) + ∇·(¯ U∇˜v) + guin Q, ∂tV−∆V+V=¯ U+˜ f¯ V+gvin Q, endowed with the corresponding initial and boundary conditions. In fact, first we find Vand after U. Adapting Section 3.1, we can prove that operator Sis well-defined from Xu×Xvto Yu×Yvand compact from Xu×Xv to itself, due to the regularity (gu, gv)∈L2(Q)×L4(Q), (U0, V0)∈H1(Ω) ×W3/2,4 n(Ω), (˜u, ˜v)∈ Yu×Yvand ˜ f∈L4(Q). It suffices to prove Lemma 3.3 but now defining Tα={(U, V )∈ Yu×Yv: (U, V ) = α S(U, V ) for some α∈[0,1]}, which is very similar to the proof of Lemma 3.3. In fact, now Steps 2 and 3 are easier and can be proved jointly. With this objective, if (U, V )∈Tα, then (U, V ) solves the coupled problem (∂tU−∆U−∇·(˜u∇V) = α∇·(U∇˜v) + αguin Q, ∂tV−∆V+V=αU +α˜ fV +αgvin Q, (4.33) endowed with the corresponding initial and boundary conditions. By testing by (U, −∆V), one has 1 2 d dtkUk2+k∇Vk2+k∇Uk2+k∇Vk2+k∆Vk2≤ |(˜u∇V, ∇U)|+α|(U∇˜v, ∇U)|+α|(U, ∆V)| +α|(˜ fV, ∆V)|+α|(gu, U)|+α|(gv,∆V)|.(4.34) Applying the H¨older and Young inequalities to the terms on the right side of (4.34) and taking into account (2.11), we have |(˜u∇V, ∇U)|≤k˜ukL4k∇VkL4k∇Uk ≤ Cδk˜uk2 L4k∇Vkk∇VkH1+δk∇Uk2 ≤δ(k∇Vk2 H1+k∇Uk2) + Cδk˜uk4 L4k∇Vk2,(4.35) α|(U∇˜v, ∇U)| ≤ αkUkL4k∇˜vkL4k∇Uk ≤ C α kUk1/2k∇˜vkL4kUk3/2 H1 ≤δkUk2 H1+Cδα2k∇˜vk4 L4kUk2,(4.36) α|(gu, U)| ≤ αkguk2+kUk2,(4.37) α|(U, ∆V)| ≤ δk∆Vk2+Cδα2kUk2,(4.38) α|(˜ fV, ∆V)| ≤ αk˜ fkL4kVkL4k∆Vk ≤ δk∆Vk2+Cδα2k˜ fk2 L4kVk2 H1,(4.39) α|(gv,∆V)| ≤ δk∆Vk2+Cδα2kgvk2.(4.40) OPTIMAL BILINEAR CONTROL PROBLEM 17 On the other hand, testing (4.33)2by Vone has 1 2 d dtkVk2+kVk2 H1≤α|(U, V )|+α|(˜ fV, V )|+α|(gv, V )| ≤δkVk2+kVk2 H1+Cδα2kUk2 +Cδα2k˜ fk2 L4kVk2+Cδα2kgvk2.(4.41) Summing the inequalities (4.34) and (4.41), and then adding kUk2to both sides of the obtained inequality and considering (4.36)–(4.40), taking δsmall enough and any α∈[0,1], we have d dt(kUk2+kVk2 H1) + C(kUk2 H1+kVk2 H2)≤C(1 + k∇˜vk4 L4)kUk2+C(kguk2+kgvk2) +C(k˜uk4 L4+k˜ fk2 L4)kVk2 H1.(4.42) By applying the Gronwall Lemma in (4.42) we conclude that there exists a positive constant C0that depends on T, kU0k,kV0kH1,k˜ukL4(Q),k∇˜vkL4(Q),k˜ fkL2(L4),kgukL2(Q)and kgvkL2(Q),such that k(U, V )kL∞(L2×H1)∩L2(H1×H2)≤C0.(4.43) Now, following Step 4 in the proof of Lemma 3.3, we obtain that Vis bounded in Yv, because the following estimate holds kVkL4(W2,4(Ω)) +k∂tVkL4(Q)+kVkC(W3/2,4 n)≤C1(C0,kV0kW3/2,4 n ,kgvkL4(Q),k˜ fkL4(Q)).(4.44) Now, we follow Step 5 in the proof of Lemma 3.3 with small modifications. By testing (4.33)1by −∆U, using the H¨older and Young inequalities, and considering the interpolation inequality (2.11), we obtain 1 2 d dtk∇Uk2+k∆Uk2≤C α2(kguk2+kUk2 L4k∆˜vk2 L4+k∇Uk2k∇˜vk4 L4) +C(k˜uk2 L4k∆Vk2 L4+k∇˜uk2 L4k∇Vk2 L4) + δ(k∆Uk2+kUk2 H2).(4.45) On the other hand, from (4.32)1we deduce d dtZΩ U=αZΩ gu, which implies 1 2 d dtZΩ U2 =αZΩ guZΩ U≤Cδα2ZΩ gu2 +δZΩ U2 ,(4.46) and ZΩ U(t) 2 =ZΩ U0+αZt 0ZΩ gu 2 ≤C. (4.47) Summing inequalities (4.45)–(4.47), taking δsmall enough and α∈[0,1], accounting (2.9), (2.10) and (4.44), we can obtain the estimate kUkL∞(H1)∩L2(H2)≤C. Finally, the estimate k∂tUkL2(L2)≤Cis deduced as in the proof of Lemma 3.3. Therefore, we can deduce the existence of solution for (4.32) from Leray-Schauder fixed-point theorem. The uniqueness of (U, V ) follows directly from the regularity of (U, V ) and the linearity of system (4.32). Thus, we conclude the proof. 18 F. GUILL´ EN-GONZ´ ALEZ ET AL. Now we show the existence of Lagrange multipliers. Theorem 4.11. Let ˜s= (˜u, ˜v, ˜ f)∈ Sad be a local optimal solution for the control problem (4.28). Then, there exist Lagrange multipliers (λ, η, ξ, ϕ)∈L2(Q)×L4/3(Q)×(H1(Ω))0×(W3/2,4 n(Ω))0such that for all (U, V, F)∈ Wu×Wv×C(˜ f)one has αuZT 0ZΩd (˜u−ud)Udxdt+αvZT 0ZΩd (˜v−vd)Vdxdt+NZT 0ZΩc (˜ f)3Fdxdt −ZT 0ZΩ∂tU−∆U−∇·(U∇˜v)−∇·(˜u∇V)λdxdt −ZT 0ZΩ∂tV−∆V+V−U−˜ fV ηdxdt −ZΩ U(0)ξ dx −ZΩ V(0)ϕ dx +ZT 0ZΩc F˜vη dxdt≥0.(4.48) Proof. From Lemma 4.10, ˜s∈ Sad is a regular point, then from Theorem 4.6 there exist Lagrange multipliers (λ, η, ξ, ϕ)∈L2(Q)×L4/3(Q)×(H1(Ω))0×(W3/2,4 n(Ω))0such that J0(˜s)[r]−hG0 1(˜s)[r], λi−hG0 2(˜s)[r], ηi−hG0 3(˜s)[r], ξi−hG0 4(˜s)[r], ϕi ≥ 0,(4.49) for all r= (U, V, F)∈ Wu×Wv×C(˜ f). Thus, the proof follows from (4.29)–(4.30). From Theorem, 4.11 we derive an optimality system for which we consider the following spaces Wu0:= {u∈ Wu:u(0) = 0},Wv0:= {v∈ Wv:v(0) = 0}.(4.50) Corollary 4.12. Let ˜s= (˜u, ˜v, ˜ f)be a local optimal solution for the optimal control problem (4.28). Then the Lagrange multiplier (λ, η)∈L2(Q)×L4/3(Q), provided by Theorem 4.11, satisfies the system ZT 0ZΩ∂tU−∆U−∇·(U∇˜v)λdxdt−ZT 0ZΩ Uη dxdt =αuZT 0ZΩd (˜u−ud)Udxdt∀U∈ Wu0,(4.51) ZT 0ZΩ∂tV−∆V+Vηdxdt−ZT 0ZΩc ˜ fV η dxdt−ZT 0ZΩ∇·(˜u∇V)λdxdt =αvZT 0ZΩd (˜v−vd)Vdxdt∀V∈ Wv0,(4.52) which corresponds to the concept of very weak solution of the linear system          ∂tλ+ ∆λ−∇λ·∇˜v+η=−αu(˜u−ud)χΩdin Q, ∂tη+ ∆η+∇·(˜u∇λ)−η+˜ fηχΩc=−αv(˜v−vd)χΩdin Q, λ(T)=0, η(T)=0 in Ω, ∂λ ∂n= 0,∂η ∂n= 0 on (0, T)×∂Ω, (4.53) OPTIMAL BILINEAR CONTROL PROBLEM 19 and the optimality condition ZT 0ZΩc (N(˜ f)3+ ˜vη)(f−˜ f) dxdt≥0,∀f∈ F.(4.54) Proof. From (4.48), taking (V, F) = (0,0), and taking into account that Wu0is a vectorial space, we have (4.51). Similarly, taking (U, F) = (0,0) in (4.48), and considering that Wv0is a vectorial space we obtain (4.52). Finally, taking (U, V ) = (0,0) in (4.48) we have NZT 0ZΩc (˜ f)3Fdxdt+ZT 0ZΩc ˜vηF dxdt≥0,∀F∈ C(˜ f). Therefore, choosing F=f−˜ f∈ C(˜ f) for all f∈ F in the last inequality, we obtain (4.54). In the following result we show that the Lagrange multiplier (λ, η), provided by Theorem 4.11, has some extra regularity. Theorem 4.13. Under of conditions of Theorem 4.11, system (4.53) has a unique strong solution (λ, η)such that λ∈L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)), ∂tλ∈L2(Q),(4.55) η∈L∞(0, T;W2−2/p,p(Ω)) ∩Lp(0, T;W2,p(Ω)), ∂tη∈Lp(Q),for any p < 2.(4.56) Proof. Let s=T−t, with t∈(0, T) and ˜ λ(s) = λ(t), ˜η(s) = η(t). Then system (4.53) is equivalent to            ∂s˜ λ−∆˜ λ+∇˜ λ·∇˜v−˜η=αu(˜u−ud)χΩdin Q, ∂s˜η−∆˜η−∇·(˜u∇˜ λ) + ˜η−˜ f˜ηχΩc=αv(˜v−vd)χΩdin Q, ˜ λ(0) = 0,˜η(0) = 0 in Ω, ∂˜ λ ∂n= 0,∂˜η ∂n= 0 on (0, T)×Ω. (4.57) Following an analogous reasoning as in the proof of Lemma 4.10, we can obtain the energy inequality d ds(k˜ λk2 H1+k˜ηk2) + C(k˜ λk2 H2+k˜ηk2 H1)≤C(k˜ηk2+k˜ λk2) + C(k˜u−udk2+k˜v−vdk2) +Ck˜ λk2 H1k∇˜vk4 L4+Ck˜uk4 L4k∇˜ λk2+Ck˜ fk4 L4k˜ηk2 ≤C(1 + k˜ fk4 L4)k˜ηk2+C(k˜u−udk2+k˜v−vdk2) +C(1 + k˜uk4 L4+k∇˜vk4 L4)k˜ λk2 H1.(4.58) Thus, we deduce that ˜ λ∈L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)), ˜η∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)), hence in particular (4.55) holds. Now, since ˜ f∈L4(Qc) and ˜η∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)) ,→L4(Q) we have ˜ f˜η∈L2(Q).(4.59) 20 F. GUILL´ EN-GONZ´ ALEZ ET AL. Also, taking into account that ˜u∈ Wu, where Wuis defined in (4.25), and ˜ λ∈L∞(0, T;H1(Ω)) ∩ L2(0, T;H2(Ω)), we obtain ∇·(˜u∇˜ λ) = ˜u∆˜ λ+∇˜u·∇˜ λ∈Lp(Q)∀p < 2.(4.60) Therefore, from (4.57)2, (4.59), (4.60) and Lemma 2.3 we conclude (4.56). Corollary 4.14. (Optimality System) Let ˜s= (˜u, ˜v, ˜ f)∈ Sad be a local optimal solution for the control problem (4.28). Then, the Lagrange multiplier (λ, η)satisfies the regularity (4.55)and (4.56)and the following optimality system                    ∂tλ+ ∆λ−∇λ·∇˜v+η=−αu(˜u−ud)χΩda.e. (t, x)∈Q, ∂tη+ ∆η+∇·(˜u∇λ)−η+˜ fηχΩc=−αv(˜v−vd)χΩda.e. (t, x)∈Q, λ(T)=0, η(T) = 0 in Ω, ∂λ ∂n= 0,∂η ∂n= 0 on (0, T)×∂Ω, ZT 0ZΩc (N(˜ f)3+ ˜vη)(f−˜ f) dxdt≥0,∀f∈ F. (4.61) Remark 4.15. If F ≡ L4(Qc), that is, there is no convexity constraint on the control, then, (4.61)5becomes N(˜ f)3χΩc+ ˜vηχΩc= 0. Thus, the control ˜ fis given by ˜ f=−1 N˜vη1/3 χΩc.(4.62) Remark 4.16. All the results obtained in this work hold when the control fbelong to Lq(Q), for q > 2. Indeed, we obtain the existence of pointwise strong solutions (u, v) of (1.2)-(1.4), where the regularity for uremains fixed, that is, u∈L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)) with ∂tu∈L2(Q), and v∈L∞(0, T;W2−2/q,q(Ω)) ∩ Lq(0, T;W2,q(Ω)) with ∂tv∈Lq(Q). We fix q= 4 only for simplicity in the notation. Acknowledgements. F. 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