On Algebraic condition for null controllability of some coupled degenerate systems

In this paper we will generalize the Kalman rank condition for the null controllability to $n$-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and $m$-controls. For that we prove a global Carleman estimate of the solution of a scalar $2n$-order equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.


Introduction and Main result
In this work, we focus the following problem ½ ω denotes the characteristic function of ω, T > 0, D is a n × n matrix, B is a n × m matrix, v = (v 1 , · · · , v m ) * is the control and Y = (y 1 , · · · , y n ) * is the state. In the sequel we denote also Q ω := (0, T ) × ω. The operator M is defined by My = (ay x ) x for y ∈ D(M) ⊂ L 2 (0, 1). For Y = (y 1 , · · · , y n ) * , MY denotes (My 1 , · · · , My n ) * . The function a is a diffusion coefficient which degenerates at 0 (i.e., a(0) = 0) and which can be either weak degenerate (WD), i.e., x θ is nondecreasing near 0, if K > 1, ∃θ ∈ (0, 1)x → a(x) x θ is nondecreasing near 0, if K = 1. (1. 3) The boundary condition CY = 0 is either Y (0) = Y (1) = 0 in the weak degenerate case (W D) or Y (1) = (aY x )(0) = 0 in the strongly degenerate case (SD). It is well known that null controllability of non degenerate (a > 0) parabolic systems have been widely studied over the last 40 years and there have been a great number of results. In the case of one equation (n = 1), the result was obtained by A. V. Fursikov and O. Y. Imanuvilov [12] and G. Lebeau and L. Robbiano [17]. In the case of coupled systems n ≥ 2, M. Gonzalez-Burgos, L. de Teresa [14] provided a null controllability result for a cascade parabolic system. Recently, F. Ammar-Khodja et al. [4,5] obtained several results characterizing the null controllability of fully coupled systems with m-control forces by a generalized Kalman rank condition. For degenerate systems (e.g., a(0) = 0), null controllability of one equation was studied in [7,9] and the references therein. The case of two coupled equations (n = 2), cascade systems are considered in [10], and in [1,2] the authors have studied the null controllability of degenerate non-cascade parabolic systems. In the case n > 2, in a recent work [11], we have extended the null controllability results obtained by Ammar-Khodja et al. [5] to a class of parabolic degenerate systems (1.1) in the two following cases : (1) the coupling matrix A is a cascade one and the diffusion matrix D = diag(d 1 , · · · , d n ) where d i > 0, i = 1, · · · , n, (2) the coupling matrix A is a full matrix (non cascade) and the diffusion matrix D = dI n , d > 0.
In the present paper, we study the case where the coupling matrix A is a full matrix and the diffusion matrix D is a diagonalizable n × n matrix with positive real eigenvalues, i.e., D = P −1 JP, P ∈ L(R n ), det(P ) = 0, (1.4) where J = diag(d 1 , · · · , d n ), d i > 0, 1 i n. The strategy used in this case is quite different from the one used in [11], and follows the one used in [6]. To establish an observability inequality to the adjoint system of (1.1), we prove a global Carleman estimate for a degenerate scalar equation (3.28) of 2n order in space. This will lead to several Carleman estimates, and thus to an observability inequality, for our adjoint system. Another difference of [11] is that in the Carleman estimates used for one degenerate equation, here we need to establish ones involving the terms y t and (a(x)y x ) x in addition to the state y and its space derivative y x .
Let us introduce the following weighted spaces. In the (WD) case :
The adjoint system associated to the system (1.1) is the following (1.6) To study the null controllability of the system (1.1), we need to establish an observability inequality of the corresponding adjoint problem (1.6). Indeed, we must prove the existence of a positive constant C such that, for every ϕ 0 ∈ L 2 (0, 1) n , the solution ϕ ∈ C 0 ([0, T ], L 2 (0, 1) n ) of system (1.6) satisfies (1.7) The inequality (1.7) will be deduced from a global Carleman estimate satisfied by the solution of the The rest of the work is organized as follows: In section 2, we state some properties of the unbounded operator K and give a useful characterization of the Kalman condition Ker(K * ) = {0} by using the spectrum of operator M. Section 3 is devoted to show several intermediate Carleman estimates for scalar parabolic degenerate equations of order 2 and 2n in space. In Section 4, the proof of Theorem 1.1 is given in the end of Section 4.
All along the article, we use generic constants for the estimates, whose values may change from line to line.

Spectrum of operator M and some algebraic tools
This section will be devoted to prove two crucial properties of the Kalman operator K and to give an equivalent algebraic condition to the condition (1. a(x) It is known that the operator −M is a definite positive operator. We will use the fact that H 2 a (0, 1) is compactly embedded in L 2 (0, 1), see [8,18]. Thus, −M is a self-adjoint positive definite operator with compact resolvent. Therefore, there exists a Hilbertian basis (Φ n ) n∈N * of L 2 (0, 1) and a sequence (λ p ) p∈N * of real numbers with λ n > 0 and λ n −→ +∞, such that Remark 2.2. In the case a(x) = x α with 0 < α 1 as in [15] the eigenfunctions and eigenvalues of M can be explicitly given using Bessel's functions. Now, we give some algebraic tools. It is known that D := ∩ p 0 D(M p ) is dense in D(M p ) for every p 0 and D nm ⊂ D(K). Thus, D(K) = L 2 (0, 1) nm and K * is well defined from D(K * ) ⊂ L 2 (0, 1) n into L 2 (0, 1) nm . The formal adjoint of K, again denoted by K * is given by and it coincides with the adjoint operator of K on D n . Moreover, we note that when a ∈ C ∞ ([0, 1]), from [8,Proposition 3.8], D = C ∞ ([0, 1]). Thereafter, we recall some properties of the Kalman operator K as it is given in [6]. For any j, p ∈ N * , we consider the projection operator where (·, ·) stands for the scalar product in L 2 (0, 1). All along this paper, we denote by | · | the euclidian norm in R j . Thus, if j ∈ N * , we have the follwing characterization of D j For p ∈ N * , L p := −λ p D + A ∈ L(R n ) and We have the following equalities Since L and K are closed unbounded operators, one has and then D(K) = u ∈ L 2 (0, 1) nm : In a similar way, we obtain      We define also the operator KK * : The operator KK * is closed, and a simple computation provides As in [6], we obtain the following result.
Theorem 2.4. We have the following properties (1) there exists a constant C > 0 such that for all u ∈ D(M n−1 ) nm , Ku ∈ L 2 (0, 1) n and By adapting the proof of [6, Theorem 2.1] to our case and using the fact that the polynomial F (λ) := is either identically 0 or far from 0 for any λ sufficiently large, one can deduce the following corollary: Corollary 2.5. Either there exists p 0 ∈ N * such that rank K p = n for every p > p 0 or rank K p < n for every p ∈ N *

Carleman estimates
In this section we give a new global Carleman estimate for the adjoint problem (1.6). By the same way as in [6, Proposition 3.3], we can show the following result.
Proposition 3.1. If ϕ = (ϕ 1 , · · · , ϕ n ) * is the solution of problem (1.6) corresponding to initial data ϕ 0 ∈ X := D n , then ϕ ∈ C k ([0, T ]; D(M p ) n ) for every k, p 0, and In order to state our fundamental result, we need to show first some Carleman estimates in the case of a single parabolic degenerate equation.

Carleman estimate for one equation.
In this subsection we shall establish a new Carleman estimate for the solution of the following parabolic equation Let us consider the following time and space weight functions where the parameters c, ρ and λ are chosen as in [2] The following Carleman estimate will be crucial for the aim of this subsection. Note that the Carleman estimate needed in this work is different from the one showed in [7] and used in [11], since it involves in addition to u and u x the terms u t and Mu.
Theorem 3.2. let T > 0. Then there exist two positive constants C and s 0 such that, for all for all s ≥ s 0 .
Proof. Let u be the solution of equation (3.2). For s > 0, the function w = e sϕ u satisfies Moreover, from the Lemma 3.4, 3.5 and 3.6 in [7], we can deduce the following estimate to w in (0, T ) × (0, 1) Using the same technique as in [2] and [7] the term Qc 2 w 2 dx dt can be absorbed by the last two terms in the left side of inequality (3.5). Thus Using the previous estimate, we will bound the integral Therefore, Since the function x −→ x 2 a is nondecreasing, then one has Thanks to the Hardy-Poincaré inequality (2.1), we can estimate In a similar way, to bound the integral x 2 a w.
Since the function ψ is bounded on (0, 1) then Therefore, for s large enough From inequalities (3.6), (3.11) and (3.12), one obtains Consequently, we obtain the estimate (3.4) which completes the proof.

(3.21)
Hence, by inequality [12, Lemma 1.2], we have Again, from the definition of ζ and the Cacciopoli inequality [2, Lemma 6.1], we obtain for s large enough. Similarly since ζuxx = Zxx − ζxxu − 2ζxux and thanks to Cacciopoli inequality, we get The estimates (3.22)-(3.24) lead to Since a is continuous on (x ′ 1 , 1] and by using Mu = a ′ ux + auxx we obtain Again, Proposition 3.5 can be generalized as follows. Proposition 3.6. Let T > 0 and τ ∈ R. Then, there exist two positive constants C and s 0 such that for every u 0 ∈ L 2 (0, 1), the solution u of equation for all s s 0 , with ζ = 1 − ξ.
Now we examine the case of a scalar degenerate parabolic equation ??.

Carleman estimate for a scalar degenerate parabolic equation.
In this section we will consider z, with the monomial derivative M i ∂ j t z ∈ L 2 (0, T ; H 2 a (0, 1)) for every i, j ∈ N, a solution of the following scalar degenerate parabolic equation of order 2n in space.
where P i ≡ ∂ t + d i M, 1 i n, d i > 1 and α i1,··· ,ip , α i , α ∈ R depend only on the matrices D and A. The main result in this subsection is the following.
Theorem 3.7. Let us fix k 1 , k 2 ∈ N and τ 0 ∈ R. Then, there exist two positive constants C 0 and s 0 (only depending in ω, n, a, D, A, τ 0 , k 1 and k 2 ) and r = r(n) ∈ N such the following inequality holds for all s s 0 and for every solution φ of equation (3.28) that satisfies M i ∂ j t φ ∈ L 2 (0, T, H 2 a (0, 1)) for every i, j ∈ N. The terms J (τ, φ) and I(τ, z) are given by (3.30) Proof. Adapting the technique used by Ammar-Khodja et al. in [6] to our degenerate case, the proof will be divided in three steps. All along this proof C will be a generic constants that may depend on ω, n, a, D, A, τ 0 , k 1 and k 2 .
Step 1 : Let us denote and consider the following change of variables Having in mind the regularity assumptions on z, (3.28) and (3.31), one gets ψ i , F (z) ∈ L 2 (Q) for every i, 1 i n and Ψ = (ψ 1 , · · · , ψ n ) * satisfies the following cascade system (3.33) For i = 1, · · · , n − 1, applying respectively Proposition 3.4 and Proposition 3.6 and combining the two estimates obtained leads to And for i = n, we obtain for every s s 0 . Thus, a suitable combination of the above inequalities leads to Step 2 : For i = 1, · · · , n, let us introduce the following sequence (O i ) 1 i n of open sets and an associated family of truncation functions (χ i ) 1 i n such that Let l 3 and k ∈ {2, · · · , n}, we multiply the equation ∂ t ψ k−1 + d k−1 Mψ k−1 = ψ k , satisfied by ψ k−1 , by δ χ k ψ k with δ = (sθ) τ0+l e 2sΦ and integrate on Q, we obtain For every ν µ and (t, x) ∈ Q, we have (3.40) We have Since the function x → x 2 a(x) is bounded on O k−1 , we have Likewise, we get Therefore On the other hand, for I 2 we have where k(i) = max(5, 3(n − i + 1)). By iterating this operation (n − 1) times, there exist a positive constant C > 0 and an integer which in view of (3.32) implies Now, at this level the left-hand-side of (3.48) does not contains enough terms to absorb the term corresponding to F (z). So, in order to absorb the term F (z), let Π denote then any permutation of the set {1, 2, · · · , n} and consider, instead of (3.32), the new change of variable (3.49) Then system (3.33) becomes The same procedure as above leads to a similar estimate as (3.48) which reads then From the definition of F (z) (3.31), we deduce (3.52) Choosing s large enough such that C(sθ) τ0 Now, we will show the Carleman estimate for the adjoint problem (1.6). Recall D = ∩ ∞ p=0 D(M p ) which is dense in L 2 (0, 1). We have the following result see [6, Proposition 3.3.] Proposition 4.1. Let ϕ 0 ∈ D n and let ϕ = (ϕ 1 , · · · , ϕ n ) * be the corresponding solution of problem (1.6). Then, ϕ ∈ C k ([0, T ]; D(M p ) n ) for every k, p 0, and for every i (with 1 i n) ϕ i solves equation (3.28).
At present, using the condition Ker(K * ) = {0} we state the following global Carleman estimate for the solution of Problem (1.6) Corollary 4.3. In addition to the assumptions in Theorem 4.2, we assume the condition Ker(K * ) = {0}. Then, given τ ∈ R and k (n − 1)(2n − 1), there exist two positive constants C and σ such that for every ϕ 0 ∈ L 2 (0, 1) n the corresponding solution ϕ to the adjoint problem (1.6) satifies At present, we are ready to give the proof of the main result.