LOCAL EXACT CONTROLLABILITY TO POSITIVE TRAJECTORY FOR PARABOLIC SYSTEM OF CHEMOTAXIS

. In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani’s ﬁxed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is shown to be in L ∞ ( Q ), which is estimated by using the methods of maximal regularity and L p - L q estimate for parabolic equations.


BAO-ZHU GUO AND LIANG ZHANG
be equipped with their graph norms, where H 1 (Ω) * stands for the dual space of H 1 (Ω). The duality between H 1 (Ω) * and H 1 (Ω) is represented by ·, · . In this paper, we are concerned with the following controlled parabolic system with the state functions u ≡ u(x, t) and v ≡ v(x, t) : where ∂ t = ∂/∂t, ∂ ν = ∂/∂ν denotes the derivative with respect to the outer normal ν of ∂Ω, 1 ω represents the characteristic function of ω, f ≡ f (x, t) is the control function, u 0 (·) and v 0 (·) are the initial values, and χ, γ, and δ are given positive constants.
is said to be a weak solution to Equation (1), if for all ϕ ∈ L 2 (0, T ; H 1 (Ω)), the following identities hold: We write the free system (1) (i.e. in the absence of f ) as follows: in Q, ∂ t v = ∆v − γv + δu in Q, ∂ ν u = 0, ∂ ν v = 0 on Σ, u(x, 0) = u 0 (x) v(x, 0) = v 0 (x) x ∈ Ω. (2) The system (2) is a prototype chemotaxis system, called the Keller-Segel model which describes the aggregation process of slime mold resulting from chemotactic attraction. In Equation (2), u(·, ·) represents the density of the cellular slime mold and v(·, ·) is the density of the chemical substance ( [15]). In the last decade, there are a large number of works attributed to the mathematical analysis of the Keller-Segel system. Several topics on the Keller-Segel model for chemotaxis such as the aggregation, the blow-up solutions, and the chemotactic collapse have been addressed and some significant results have been achieved from different discipline perspectives. We refer to [11] and [12] and the references therein for a detailed introduction of mathematical problems on the Keller-Segel model of chemotaxis. Generally speaking, the blow-up of solutions of the Keller-Segel system in finite or infinite time depends strongly on the space dimension. In 1-d case, a finite time blow-up never occurs, and the global solution exists and converges to the stationary solution as times goes to infinity ( [20]). However, the blow-up may occur in finite or infinite time in n-dimensional case for n ≥ 3 ( [13]). For the 2-d case, several thresholds have been found for the existence of the global solution. When the mass of the initial data is below some threshold value, the solution exists globally and its L ∞ -norm is uniformly bounded for all time, while the mass of the initial data is larger than some threshold value, the solution would blow up either in finite or EXACT CONTROLLABILITY FOR PARABOLIC SYSTEM OF CHEMOTAXIS 145 in infinite time ([4, 8, 23]). For more results on chemotaxis equations, we refer to [19,26].
Since the densities of the cell and the chemical substance are usually positive, it is reasonable to consider the positivity of the solutions. In fact, it is easy to see that when (u 0 , v 0 ) ≥ 0, the corresponding solution (u, v) ≥ 0 holds for Equation (2) ( [4,13]). In addition, the total mass of u is conserved: The strict positive solution to Equation (2) is claimed by the following Theorem 1.2.
On account of the blow-up feature of solutions of the Keller-Segel model, it is significant to consider controllability for system (1). Definition 1.3. Let (u, v) be a positive trajectory of system (2) corresponding to an initial value (u 0 , v 0 ). We say that the system (1) is locally exactly controllable to the trajectory (u, v) at time T , if there is a neighborhood O of (u 0 , v 0 ) such that for any initial value (u 0 , v 0 ) ∈ O, the corresponding solution (u, v) of Equation (1) driven by some control function f satisfies Remark 1. The Theorem 1.2 implies the existence of local solution to Equation (1). We take T = T (u 0 , v 0 ) < T max in Theorem 1.2, where T max is the maximal time for the existence of solution to Equation (2). When t → T max , the solution of (2) may blow up. If system (1) is locally exactly controllable, then we can drive the state of system (1) by some control force to a given trajectory at time T before T max to avoid blow-up.
Controllability for parabolic equations attracts intensive attention in the last few years. We refer [5], [7] and other literature. However, very few results, to the best of our knowledge, are available on the control problems of system (1). In [22], an optimal control problem for the system (1) with the control to be distributed on the second equation of Equation (1) is considered. We believe that the present paper is a first work on controllability for system (1). Compared with the coupled parabolic systems aforementioned, the mathematical difficulty for the control problems of system (1) is brought by the chemotactic term −χ∇ · (u∇v). The techniques presented in this paper would be useful for other forms of chemotaxis system such as the parabolic-elliptic chemotaxis system and even for other coupled systems like drift-diffusion equations from the semiconductor device.
The idea of obtaining the controllability of system (1) is somehow classical: We first establish null controllability for the linearized system and then apply a fixed point theorem. Now, we consider null controllability for the linearized system of system (1), which reads as follows: where a(·, ·) ∈ L ∞ (Q), B(·, ·) ∈ L ∞ (Q) N with B · ν = 0 on Σ, f (·, ·) ∈ L 2 (Q) is the control force, and y 0 (·), z 0 (·) ∈ L 2 (Ω) are given initial data. We state our first result on null controllability for linear parabolic system (4).
there exists a control f ∈ L ∞ (Q) such that the solution (y, z) of system (4) corresponding to f satisfies (y, z) ∈ V 1 (Q)×V 1 (Q) and y(x, T ) = z(x, T ) = 0 for almost all x ∈ Ω. In addition, the control f satisfies where C is a positive constant depending only on Ω and ω, and To the best of our knowledge, there are two methods to build L ∞ controls. The paper [24] (see also [6]) proposes a direct way to realize L ∞ control for a single heat equation by applying the L 1 observability inequality from an L 2 observability inequality and the well-posedness of adjoint system. Here we adopt a different way (see, e.g., [2]) that the L ∞ control is obtained by minimizing a cost function and applying the boot-strap method, which involves L p -L q estimate and maximal regularity of semigroups in order to improve the estimation by giving an explicit representation of the upper bound of controls with respect to T . This method can also be adopted to build control in space W 1,2 ([0, T ]; L 2 (Ω)) (see, e.g., [25]).
The main result of this paper is the following Theorem 1.5.
Theorem 1.5. Let p > N + 2 and let (u, v) be a positive trajectory of system (2) corresponding to (ū 0 ,v 0 ), which satisfies (3). Then, there exists a positive constant c 1 independent of T such that for each (u 0 , v 0 ) that satisfies u 0 ≥ 0, v 0 ≥ 0, and there is a control f ∈ L ∞ (Q) such that system (1) admits a unique solution (u, v) satisfying For notational simplicity, we use C or C(Ω, ω) to denote, throughout the paper, a positive constant that is independent of time T yet depends on Ω and ω without specification.
We proceed as follows. In Section 2, we give some preliminary results. Section 3 is devoted to the proofs of the main results.

2.
Preliminaries. In this section, we collect some results that are needed in next section. These results are particularly useful in establishment of the regularity of linear parabolic systems and the L ∞ -estimate of controls.
For p ∈ (1, ∞), let A := A p denote the sectorial operator given by Suppose that γ is a positive constant.
Proposition 1. Let A be given by (8). Then the following assertions hold, where (iv) is a direct consequence of succeeding Equations (11) and (13).
(ii). Let e −tA t≥0 and e −t(A+γ) t≥0 be the analytic C 0 -semigroups generated by −A and −(A+γ) on L p (Ω) (1 < p < ∞), respectively. By standard C 0 -semigroup theory ( [21]) where m(t) = min{1, t}, and (iii). Let α ≥ 0 and 1 < p < ∞. Then, for any ε > 0, there exists a constant C ε depending on Ω, ε, and p such that ([13, Lemma 2.1]) (iv). For any ε > 0, there exists a constant C ε depending on Ω, ε, and p such that where C is a positive constant independent of T and F . (15) is first established as Theorem 9.1 of [16] in Chapter IV, but the independency of C with respect to T is given later as Theorem 1.1 of [17].

BAO-ZHU GUO AND LIANG ZHANG
Now, we consider the well-posedness of the following linear parabolic system which covers system (4) as its special case: Then the following assertions hold.
Proof. The existence of solution with respect to (y 0 , z 0 , F ) in corresponding function spaces can be deduced similarly as [16] for which we omit here. We only show the required estimations with respect to time T . Since the proof for (17) is similar to (18), we need only show (18). Multiply the first equation of Equation (16) by |y| p−2 y and integrate over Ω to obtain d dt |y| In the same way, we can obtain, from the second equation of Equation (16), that d dt |z| Differentiate |∇z| p p with respect to t and take the second equation of Equation (16) into account again to obtain d dt |∇z| The inequalities (20)- (22) together with Gronwall's inequality lead to On the other hand, by the maximal regularity (15) for the second equation of Equation (16), it follows that which, together with Equation (23), yields (18). Now, we turn to the L ∞ -estimate (19). We first assume that y 0 ∈ C Ω and F ∈ C Q . Let A be defined by (8), and let e −tA t≥0 and e −t(A+γ) t≥0 be the analytic C 0 -semigroups generated by −A and −(A + γ) in L p (Ω), 1 < p < ∞, respectively. Then the solution (y, z) of Equation (16) can be represented as First, take C Ω -norm on both sides of (24) to obtain To estimate (26), we observe that the operator −A generates a bounded analytic semigroup on C Ω ( [1]). By the maximum principle, for any 0 ≤ s < t ≤ T , Since p > N + 2, we can take ε and α so that Then, with the help of (10) and (13), and the Hölder inequality, we obtain t 0 e −(t−s)A ∇ · (By + a∇z)) (·, s) By (26)-(28), and (23) Next, take W 1,p (Ω)-norm on both sides of (25) to derive that for 0 ≤ t ≤ T , To estimate (30), we first notice that which can be obtained by the same energy method used in proving (23). Let 1/2 < α < 1 − 1/p. By (9) and (12) with application of the Hölder inequality, we have This, together with (23), (30)-(31), and the Sobolev embedding Finally, let us consider the general case that y 0 ∈ L ∞ (Ω) and F ∈ L ∞ (Q). Let For each n, let (y n , z n ) be a solution of Equation (16) corresponding to (y 0n , z 0 , F n ), which satisfies the inequalities (17) and (19) with (y, z) replaced by (y n , z n ). We can extract a subsequence of (y n , z n ) such that it converges to (y, z) which is a weak solution of (16) corresponding to (y 0 , z 0 , F ). Moreover, (y, z) satisfies the inequality (19). This ends the proof of the proposition.
3. Proof of main results. To prove Theorem 1.2, we need the following Lemma 3.1.
Then there is a positive constant δ 0 depending on a ∞ and ∇ · B 2 ∞ such that the weak solution of the following equation x ∈ Ω. (33) Proof. We split the proof into two steps.
To this end, let u − = max{−u, 0}. Multiply both sides of Equation (33) by −u − and perform integration over Ω to give Apply Gronwall's lemma to above inequality to obtain Since (u 0 ) − = 0, it follows that u − = 0 and hence u(x, t) ≥ 0 as claimed.
Step 2. Suppose that a ∈ L ∞ (Q) and u 0 ≥ k 0 . Set u = (v + k 0 e −λt )e µt , where λ and µ are specified later. Then Equation (33) is transformed into Choose µ ≥ a ∞ and λ ≥ µ + a ∞ + ∇ · B 2 ∞ ≥ 2 a ∞ + ∇ · B 2 ∞ . Then the right-hand side of the first equation of Equation (34) is nonnegative and thus the conditions of Step 1 verify. Therefore, v(x, t) ≥ 0 and u(x, t) ≥ e −δ0T k 0 by taking Finally, if B 2 ≡ 0, then taking u in Step 2 as u = (v + k 0 )e µt , we can obtain u(x, t) ≥ k 0 for (x, t) ∈ Q with the same arguments.

BAO-ZHU GUO AND LIANG ZHANG
To study null controllability for system (4), we consider observability for the adjoint system of system (4): where φ T , θ T ∈ L 2 (Ω). In order to obtain null controllability for Equation (4), we need to establish an observability inequality, which can be derived as a consequence of a global Carleman inequality for the adjoint system (37).
Since the essential part of the proof for inequality (40) is very similar to [14], where instead of w = e sα u, we can introduce w = (sϕ) d e sα u and estimate in a similar way to obtain (40), we put the detailed proof in the Appendix.
For notation simplicity in the sequel, we introduce and To establish observability for the adjoint system (37), we need the following type of Carleman estimate (44). The key step to establishing this inequality is to bound the energy of θ by the partial energy of φ, for which we use the multiplier method and the fact that θ can be represented by φ. More precisely, θ = δ −1 (−∂ t φ−∆φ−B∇φ). If we substitute θ in the second equation of (37), then the adjoint system (37) is actually a non-local equation with respect to φ. satisfying γ(λ 1 ) ≥ λ 1 > 1 such that for any λ ≥ λ 1 , s ≥ γ(λ)(T + T 2 ) and φ T , θ T ∈ L 2 (Ω), the associated solution (φ, θ) to Equation (37) satisfies where C 1 = C 1 (Ω, ω , ω). (37) with d = 2 and the second one with d = 0, respectively, we obtain that there exist positive constants c 0 (Ω, ω ) and λ 0 1 satisfying
It is well known that null controllability for system (4) with L 2 (Q) control is equivalent to the "observability inequality" for system (37): for every solution (φ, θ) of Equation (37). However, in order to make the control in the space L ∞ (Q), we need to establish instead an "improved observability inequality" as the following Proposition 3.
Setting λ and s in above inequality as , we obtain (52). This completes the proof of the porposition. Now we are ready to prove Theorem 1.4 by showing that the linear system (4) is null controllability with L ∞ controls.
Proof of Theorem 1.4. For ε > 0, let us consider the following optimal control problem: subject to all f ∈ L 2 (Q), where (y, z) is the solution of Equation (4) corresponding to f . The existence of an optimal pair (f ε , y ε , z ε ) to the optimal control problem (54) follows from the standard argument. By the Pontryagin maximum principle where (φ ε , θ ε ) is the solution of the adjoint system following Here, (y ε , z ε ) is the solution of (4) with f = f ε . By Equations (4), (55), and (56) and Proposition 3, it follows that where and in the rest of the proof C denotes different positive constant depending only on Ω and ω. We can simply get, from (55) and (57), that the control function f ε satisfies f ε 2 ≤ e Cκ (|y 0 | 2 + |z 0 | 2 ) . Next we show that f ε can be taken in L ∞ (Q). To this end, let τ be a sufficiently small positive constant and let {τ j } M +1 j=0 be a finite increasing sequence such that 156 BAO-ZHU GUO AND LIANG ZHANG 0 < τ j < τ, j = 0, 1, . . . , M, τ M +1 = τ. Let {p i } M i=0 be another finite increasing sequence such that p 0 = 2, p M > (N + 2)/2 and, Moreover, let α 0 = min Ω α Then where γ(λ) is given by (39). We can assume at the first beginning that the parameters λ and s are large enough so that the "observability inequality" (52) and hold.
Proof of Theorem 1.5. Let (u, v) be a trajectory of system (2) with the initial value (u 0 , v 0 ) which satisfies (3).
Then, (y, z) solves the following parabolic system The local exact controllability of system (1) is equivalent to the local null controllability of system (70).
where δ 0 and k 0 are the positive constants defined in Theorem 1.2. For each η ∈ K, we consider the following linearized system: where a η = χ(u + η) and B = χ∇v.
Here and in what follows, we denote by (y, z) the solution to Equation (71) corresponding to f and η if there is no ambiguity. By (5), we see that the control functions are bounded: By (19) and (72), we have the following estimate: For η ∈ K, define a multi-valued mapping Λ : ∃f satisfying (72) such that (y, z) is the solution to Equation (71)  We apply Kakutani's fixed-point theorem ( [3, p.7]) to the map Λ to prove Theorem 1.5. First, it is clear that K is a convex subset of L 2 (Q). By the argument aforementioned, we see that Λ(η) is nonempty and convex for each η ∈ K. Moreover, by (73), Λ(η) is bounded in V 1 (Q) for each η ∈ K and hence Λ(η) is a compact subset of L 2 (Q) by the Aubin-Lions lemma ( [3, p.17]). Next, we show that Λ is upper semi-continuous. To this purpose, let {η n } ∞ n=1 be a sequence of functions in K such that η n → η strongly in L 2 (Q), and let y n ∈ Λ(η n ) for each n. Then, by the definition of Λ(η n ), there exists f n for each n such that (y n , z n ) solves the following equation: ∂ t y n = ∆y n − ∇ · (By n ) − ∇ · (a ηn ∇z n ) + 1 ω f n in Q, ∂ t z n = ∆z n − γz n + δy n in Q, ∂ ν y n = 0, ∂ ν z n = 0 on Σ, y n (x, 0) = y 0 (x) z n (x, 0) = z 0 (x) x ∈ Ω, and y n (x, T ) = z n (x, T ) = 0 for almost all x ∈ Ω. Moreover, the control f n satisfies By (75) and Proposition 2, we obtain By (75) and (76) and applying the Aubin-Lions lemma again, we can obtain that f ∈ L ∞ (Q), y ∈ V 1 (Q), and z ∈ V 2 (Q), and the subsequences of f n , y n , z n , still denoted by themselves, such that f n → f weak * in L ∞ (Q) and weakly in L 2 (Q); y n → y weakly in V 1 (Q) and strongly in L 2 (Q); z n → z weakly in V 2 (Q) and strongly in L 2 (0, T ; H 1 (Ω)).
Now, by the weak solution of (41) in the sense of transposition, we see that Therefore, by (101), (102), and Hölder's inequality, we obtain Q λ 4 (sϕ) 3+2d e 2sα z 2 dxdt By the weak solution of (100) again, we have This inequality, together with (103), leads to (40) with 2d being replaced by d. This completes the proof of Lemma 3.2.