Carleman Estimates and null controllability of coupled degenerate systems

In this paper, we study the null controllability of weakly degenerate coupled parabolic systems with two different diffusion coefficients and one control force. To obtain this aim, we develop first new global Carleman estimates for degenerate parabolic equations with weight functions different from the ones in the previous works.

This paper is organized as follows. Section 2 is devoted to the well-posedness of the coupled degenerate systems. In section 3, we establish our new Carleman estimates for degenerate parabolic equations and deduce similar estimates for the coupled degenerate systems. In section 4, we deduce observability inequality and null controllability results. In appendix, we give summarized proofs of Caccioppoli and Hardy-Poincaré inequalities.

Well-posedness
In order to study the well-posedness of the system (1.1)-(1.4), we introduce the weighted spaces . We define the operator (A i , D(A i )) by We recall the following properties of (A i , D(A i )). Proposition 2.1. ( [7], [13]). For i = 1, 2, the operator A i : D(A i ) −→ L 2 (0, 1) is closed, self-adjoint, negative and with dense domain.
In the Hilbert space H := L 2 (0, 1) × L 2 (0, 1), the system (1.1)-(1.4) can be transformed in the following Cauchy problem , and As the operator A is diagonal and since B(t) is a bounded perturbation, the following wellposedness and regularity results hold.
for a constant C T > 0.
• These weight functions are independent of the diffusion coefficient. This play a crucial role to study coupled system of non cascade form.
• The existence of the function σ was proved for example in [23] using Morse functions. But in 1-dimension one can show this easily using cut-off functions.
• For this choice of the parameters d, ρ and λ the weight functions ϕ and Φ satisfy the following inequalities which are needed in the sequel (3.14) • For nondegenerate problems one needs the following estimates see e.g. [23] lim t→O + and this is satisfied for all k ≥ 1 with . • For the degenerate case one needs in addition the estimate We begin by proving first a new Carleman estimate for the problem (1.5)-(1.7) with one equation.
Theorem 3.2. Let T > 0 and suppose that y 0 ∈ H 1 α . Then, for all β ∈ [α, 1) there exist two positive constants C and s 0 such that every solution y of (1.5)-(1.7) satisfies for all s ≥ s 0 Proof. For s > 0, let us introduce the function z := e sϕ y. We have We have ||f s || 2 It is easy to check that if y ∈ H 2 α (0, 1) then we have also z ∈ H 2 α (0, 1). So x α z ∈ H 1 (0, 1) ⊂ L ∞ (0, 1) by the Sobolev imbedding theorem. Then, using the facts that z(t, 0) = z(t, 1) = z t (t, 0) = z t (t, 1) = 0 and x α z x , x α , ψ, ψ x are bounded, we deduce that the first integral with boundary terms vanishes and We have then Now we will show that J 3 , J 4 and J 5 can be absorbed by J 1 and J 2 . For this, let ε > 0 fixed to be specified later. First, Since β ≥ α and |ΘΘ| ≤ CΘ 3 then for s large enough. In the other hand for J 4 we have Now we will use the Hardy-Poincaré inequality (6.56). We have 2α − β < 1 and we will show that 1 0 x 2α−β z 2 x dx < +∞. Using (3.18) and the fact that β < 1 we obtain, Then, we get from (3.19) The quantity εC 2α−β + 1 4ε is minimal for ε = and for all β ∈ [α, 1) we have The term J 4 can then be absorbed by J 2 . For the last term J 5 , since |Θ| ≤ c 4 Θ 2 and β ≥ α, we have by applying the Hardy-Poincaré inequality Therefore by choosing ε small enough, we obtain for s large enough. So replacing z by e sϕ y we deduce immediately the conclusion of the theorem.
Theorem 3.3. Let T > 0 and suppose that y 0 ∈ H 1 α . Then, for all β ∈ [α, 1) there exist two positive constants C and s 0 such that every solution y of (1.5)-(1.7) satisfies for all s ≥ s 0 Proof. Let us consider an arbitrary open subset Let z = ξy where y is the solution of (1.5)-(1.7). Then z satisfies the following system Therefore, applying the Carleman estimate (3.17) to the equation (3.21) we obtain So using the definition of ξ and the Cacciopoli's inequality, see Lemma 5.1, we obtain Then, there exists a constant ρ 0 > 0 such that for all ρ ≥ ρ 0 there exists s 0 (ρ) > 0 such that for each s ≥ s 0 (ρ) the solution v of the last problem satisfy the following estimate: where the functions Φ and φ are defined in Theorem 3.3.
Remark 3.5. The last estimate was showed in [23] for Θ(t) = 1 t(T −t) but by careful examination of the proof one can see easily that it remains valid for all Θ ∈ C 2 (0, T ) satisfying (3.15), see Remark 3.1.
To achieve the proof of the Theorem 3.7, let Z := ζy, where the function ζ is defined as ζ = 1 − ξ. Then Z is a solution of the following problem 1), Applying the classical Carleman estimate (3.27), it follows that for s large enough Therefore, using the Caccioppoli inequality and the definitions of Z and ζ we deduce Thanks to (3.14) there exists a constant c > 0 such that for all (t, Then, using (3.26), (3.28), (3.14), (3.15) and the fact that 1/2 ≤ ξ 2 + ζ 2 ≤ 1 we obtain the global estimate This ends the proof of Theorem 3.3.
The estimate in Theorem 3.3 was obtained for regular initial data. By density we deduce the following result for the general case: y 0 ∈ L 2 (0, 1). Corollary 3.6. Let T > 0 be given. Let β ∈ [α, 1) and µ ≥ max(0, 2 + 2α − 3β). Then there exist two positive constants C and s 0 such that every solution y of (1.5)-(1.7) satisfies for all s ≥ s 0 Proof. Let y 0 ∈ L 2 (0, 1). By the density of H 1 α (0, 1) in L 2 (0, 1), there exist a set (y n 0 ) n in H 1 α (0, 1) which converges to y 0 . Let y n the unique solution in the space Z T := C [0, T ], L 2 (0, 1) ∩ L 2 0, T ; H 1 α of the problem (1.5)-(1.7) associated to the initial data y n 0 . As in (2.13) one has for a constant C T > 0 (y m − y n )(t) Z T := sup Therefore the set (y n ) n has a limit y in the Banach space Z T . Using classical argument in semigroup theory it is easy to show that y is the solution of the problem (1.5)-(1.7) associated to the initial data y 0 . On the other hand since x α ≤ x 2α−β and x µ ≤ x 2+2α−3β on (0,1) then we deduce from Theorem 3.3 the estimate And since sΘe 2sϕ , s 3 Θ 3 e 2sϕ x µ and s 3 φ 3 e 2sΦ are bounded then one can pass to the limit and get the desired estimate.
Proof. Since U is solution of the problem x ∈ (0, 1), then applying the estimate (3.26) to this system we obtain Using the Hardy-Poincaré inequality (6.56) one has for s large enough So since β ≥ α, ξ x is supported in ω ′ and Θ is bounded below then for s large enough we havē Similarly, for s large enough we havē This gives an estimate on (0, a ′ ). As above, to obtain an estimate on (a ′ , 1), we apply (3.28) to each equation of the system (1.9)-(1.12), we use Hardy-Poincaré inequality and we obtain the estimate Consequently, using (3.38), (3.39) and (3.29) we deduce the global estimate This ends the proof.

Appendix 1
As in [2], [8], [1], we give the proof of the Caccioppoli's inequality for degenerate coupled systems with two different diffusion coefficients.
Lemma 5.1. Let ω ′ ⋐ ω. Then there exists a positive constant C such that x) e 2sϕ i dxdt.
6. Appendix 2 Proof. This result was proved by Cannarsa et al. in [2] for γ ∈ (0, 1), but by a careful examination of the proof one can see that it remains valid for all γ < 1. In fact let γ < 1 and δ = γ+1 2 . Using Holder inequality and Fubini's theorem one has This ends the proof.

Conclusion
In this paper, we studied the null controllability of linear degenerate systems with two different coefficients diffusion not necessarily of the cascade form. We developed new Carleman estimates. By a standard linearization argument and fixed point, see [1], [2], [9], [35], one can show easily the null controllability of semilinear degenerate coupled systems with two different diffusion coefficients. In this paper we studied coupled system of two weakly degenerate equations. The cases when one of the equation is strongly degenerate systems are open.