A NULL CONTROLLABILITY PROBLEM WITH A FINITE NUMBER OF CONSTRAINTS ON THE NORMAL DERIVATIVE FOR THE SEMILINEAR HEAT EQUATION

We consider the semilinear heat equation in a bounded 
domain of R^m. We prove the null controllability of the system 
with a finite number of constraints on the normal derivative, when 
the control acts on a bounded subset of the domain. First, we 
show that the problem can be transformed into a null controllability 
problem with constraint on the control, for a linear system. 
Then, we use an appropriate observability inequality to solve the 
linearized problem. Finally, we prove the main result by means of 
a fixed-point method.


Introduction
Let m ∈ N\{0} and let Ω ⊂ R m be a bounded domain with boundary Γ of class C 2 .Let also ω be a non empty subdomain of Ω and Γ 0 a non empty part of Γ.For a time T > 0, set Q = Ω × (0, T ), Σ = Γ × (0, T ), Σ 0 = Γ 0 × (0, T ) and G = ω × (0, T ).Consider the following system of semilinear heat equation: (1) where f is a function of class C 1 on R, y 0 ∈ L 2 (Ω), v ∈ L 2 (G) represents the control function and χ ω is the characteristic function of ω, the set where controls are supported.The function f is assumed to be globally Lipschitz all along the paper, i.e. there exists K > 0 such that and assume for simplicity that (3) f (0) = 0.
We denote y by y(x, t, v) to mean that the solution y of (1) depends on the control v.
Null controllability problem with constraint on the control has been studied by O. Nakoulima in [1,2], for the parabolic evolution equation.Indeed, he solved in [2] the following null controllability problem with constraint on the control: Given a finite-dimensional subspace Y of L 2 (G) and y 0 ∈ H 1 0 (Ω), find a control v ∈ Y ⊥ , the orthogonal complement of Y in L 2 (G), such that the solution of Nakoulima to prove the existence of sentinels with given sensitivity in [3], and to solve a new type of controllability problem (see [4]): Given e i in L 2 (Q), 1 i M, and y 0 ∈ L 2 (Ω), find a control v ∈ L 2 (Q) such that the solution of (4) satisfies y(T ) = 0 in Ω and (5) We also refer to [5] where a boundary null controllability with constraints on the state for a linear heat equation is solved.G. M. Mophou in [6] showed the null controllability with a finite number of constraints on the state, for a nonlinear heat equation involving gradient terms.
The main result of this paper is as follows: Theorem 1.1.Let f be a globally Lipschitz function of class C 1 on R satisfying (3).Then for any y 0 ∈ L 2 (Ω) and e j ∈ H 1 0 (Σ) j = 1, . . ., m satisfying (6), there exists a unique control ṽ of minimal norm in L 2 (G), such that (ṽ, ỹ) satisfies the null controllability problem with constraint on the normal derivative (1), (7) and (8).Moreover there exists a positive constant The proof of this theorem will be the subject of the last section.The rest of the paper is organized as follows.In Section 2, we show that problem (1), ( 7), ( 8) is equivalent to a null controllability problem with constraint on the control for a linearized system derived from (1).In Section 3, we prove an observability estimate for the linearized system.In Section 4, we use this estimate to prove the null controllability of the linearized system.Section 5 is devoted to proving Theorem 1.1.

Equivalence with null controllability problem with constraint on the control for linearized system
We introduce the notation In view of the globally Lipschitz assumption (2) on f , K being the Lipschitz constant of f .Thus, system (1) may be rewritten in the form (11) Note that since y ∈ L 2 (0, T ; H 1 0 (Ω)) and ∆y ∈ H −1 (0, T ; L 2 (Ω)), we can define ∂y ∂ν on Γ and ∂y ∂ν ∈ H −1 (0, T ; H − 3 2 (Γ)), which is a subset of H −1 (Σ), the dual of H 1 0 (Σ).Consequently our aim is: such that the solution y of ( 12) satisfies (7) and (8).As we said in the introduction, we show in the rest of this section that problem (12), ( 7), ( 8) is equivalent to a null controllability problem with constraint on the control.For each e j , 1 j m, consider the adjoint of system (12): The following lemma holds: EJQTDE, 2012 No. 95, p. 4 Lemma 2.1.Under the hypothesis (6), the functions q j χ ω , 1 j m, are linearly independent for any z ∈ L 2 (Q).
Proof.Let γ j ∈ R, 1 j m, be such that (14) Since q j is solution of (13) for each j ∈ {1, . . ., m}, then m j=1 γ j q j := q satisfies: (15) γ j e j on Σ 0 .Combining the first equation of ( 15) with ( 14), we deduce that, according to a unique continuation property for the evolution equation, q = 0 in Q.Therefore, we have in particular q = 0 on Σ 0 .Since the second equation of (15) holds, the hypothesis (6) implies that γ j = 0 for all j ∈ {1, . . ., m} and the proof of Lemma (2.1) is complete.
In the sequel, we will denote by P the orthogonal projection operator from L 2 (G) into U.

Observability estimate
We prove in this section an observability estimate which is adapted to the constraint, deriving from a global Carleman inequality due to A. V. Fursikov and O. Yu.Imanuvilov [7].

But we have:
EJQTDE, 2012 No. 95, p. 10 Using (35) and (36), we deduce that: Consequently, there exist a subsequence of (σ n ) n (still denoted by Now in view of (33) and the definition of Extracting subsequences, we can deduce that: Therefore, (39) Since for any z ∈ L 2 (Q), q j (z), 1 j m is solution of ( 13) and e j ∈ H 1 0 (Σ), one can prove that q j (z) ∈ H 2,1 (Q).Moreover there exists a positive constant C such that (40) . By extracting subsequences we may deduce that there exist ψ j ∈ H 2,1 (Q) such that for j ∈ {1, . . ., m} As a consequence of the Aubin-Lions compactness Lemma, the injection from On the other hand, using (10), there exists a positive constant C = C(T, Ω) such that Consequently, there exist a subsequence of a 0 (z n ) (still denoted by Therefore in view of ( 40)-(42), ψ j , 1 j m is solution of (43) Since P n σ n ∈ U(z n ) and satisfies (36), we can apply Lemma 3.4 with On the other hand, it follows from (35) that (44) We can deduce that Hence from (38), we have: We conclude that σχ ω ∈ Span({ψ 1 χ ω , . . ., ψ m χ ω }).
Let us now give a proposition that we will need to prove estimation (9).The proof requires the following two lemmas: Lemma 3.6.Assume (6).Let θ be the function given by Proposition 2.2.Let q j , 1 j m and u 0 respectively defined by system (13) and (20).For any z ∈ L 2 (Q), set Then there exists δ > 0 such that for any z ∈ L 2 (Q), where Proof.To prove (47), we argue by contradiction.If (47) does not hold, then for any n ∈ N * , there exist a sequence (z n ) n of L 2 (Q) and a vector Consequently, there exist subsequences of Xj (z n ), 1 j m (still denoted by Xj (z n )) and Xj ∈ R such that for 1 j m, Then from (40), ( 41), (50) and Lemma 3.4, it follows that φn → m j=1 Xj ψ j := φ strongly in L 2 (Q).
EJQTDE, 2012 No. 95, p. 14 Then there exists a positive constant Proof.In view of (20), we have for any So (53) can be rewritten in the form Now applying Lemma 3.6 to the left-hand-side of (55), we get Using the Cauchy-Schwarz inequality for the right-hand-member of the latter identity, it follows that Since q j is solution of (13) for 1 j m, we have in addition to (40), the following energy inequality, Consequently, we obtain according to (56), .
Moreover, if follows from (54) that for any z ∈ L 2 (Q), , . Hence , , which ends the proof of the Proposition.

Null controllability of the linearized system
We begin by proving the existence of a solution for problem (16), (17), (18).We define on V × V the following symmetric bilinear form: In view of Proposition 3.5, this bilinear form is an inner product on V.
Let V = V be the completion of the pre-Hilbert space V with respect to the norm .
The completion V of V is a Hilbert space.
For every pair (u, y) of A, we define the functional (77) and we consider the optimal control problem: (78) inf{J ε (u, y)|(u, y) ∈ A}.
The problem is then reduced to finding a fixed point of S. Indeed, if z ∈ L 2 (Q) is such that S(z) = ỹ(ṽ) = z, the solution ỹ of ( 12) is actually solution of (11).Then, the control ṽ is the one we were looking for, since by construction, ỹ(ṽ) satisfies ( 7) and (8).
In order to conclude the existence of a fixed point of S, we can use EJQTDE, 2012 No. 95, p. 27 Schauder's fixed point Theorem.So it is sufficient to check the following three properties: 5.1.Continuity of S. We divide the proof into five steps.
Step 1: Let (z n ) n be a sequence of L 2 (Q) and assume that z n → z strongly in L 2 (Q).Then there exists a subsequence (z n k ) k such that z n k (x) → z(x) almost everywhere in Q. f being a function of class C 1 , the function a 0 is continuous and is such that a 0 (z n k (x)) → a 0 (z(x)) almost everywhere in Q.
Then, using (135) and the fact that α j q j strongly in L 2 (G).
Since ũn k ∈ U(z n k ) ⊥ , we have G ũn k q j (z n k )dxdt = 0, 1 j m.