Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix

In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called \textit{Kalman's rank} condition.


Introduction and Main result
In this paper we focus on the controllability properties of some non-diagonalizable parabolic degenerate systems. D is a non-diagonalizable n × n matrix that satisfies the following assumptions : • there exists α 0 > 0 such that Dξ.ξ α 0 |ξ| 2 ∀ξ ∈ R n (1.2) • there exists a non-singular matrix P ∈ L(C n ) such that D = P JP −1 (1. 3) for some J ∈ L(C n ) of the form J = diag(J 1 , · · · , J p ), where the J i are the Jordan blocks associated to the eigenvalues d i of D.
Controlling coupling systems of partial differential equations attracted growing interest during the last decade, the main question is whether it is possible to control such systems with fewer controls (i.e the number of controls is less than the number of equations). For finite dimensional linear systems, the controllability can be characterized by algebraic rank condition on the matrices generating the dynamics and taking account of the control action. The theory has been adapted and extended to more general systems including infinite dimensional systems. At our knowledge, in the nondegenerate case, M. Gonzalez-Burgos, L. de Teresa [18] provided a null controllability result for a cascade parabolic system by one control force under a condition on the sub-diagonal of the coupling matrix. 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. In [16], the authors gave controllability results for a system in the case where the diffusion matrix is non diagonalizable.
For degenerate systems, the case of two coupled equations (n = 2), cascade systems are considered in [13,14] and in [1,2] the authors have studied the null controllability of degenerate noncascade parabolic systems.
In [15], we have extended the null controllability results obtained by Ammar-Khodja et al. [5] to a class of parabolic degenerate systems of PDEs in the two following cases (1) the coupling matrix A is a cascade one and the diffusion matrix the coupling matrix A is a full matrix (noncascade) and the diffusion matrix D = dI n , d > 0.
On the other hand, in [3] we study the null controllability of (1.1) in the case where de diffusion matrix D is diagonalizable n × n matrix with positive real eigenvalues, i.e., where J = diag(d 1 , · · · , d n ), d i > 0, 1 i n In the current paper, we assume that diffusion matrix D is non-diagonalizable. We use the same approach as [16] without imposing that Jordan's block sizes are bounded by 4. Thus our proof is also an improvement of the one given in [16] for the nondegenerate case.
In order to study the null controllability of system (1.1), we will consider the following corresponding adjoint problem in (0, 1). (1.8) Since the null controllability of system (1.1) is equivalent to the existence of a positive constant C such that, for every z 0 ∈ L 2 (0, 1) n , the solution z ∈ C 0 ([0, T ]; L 2 (0, 1) n ) to the adjoint system (1.8) satisfies the observability inequality : The strategy used in this case is slightly different from the one used in [3], although in both cases it is necessary to show Carleman estimates for a scalar PDE of order 2n in space.
Note that this technique failed in the case where M is a compact operator, since its spectrum admits zero as a point of accumulation, so the Kalman condition is no longer verified, which does not ensure a perfect coupling of the equations.
All along the article, we use generic constants for the estimates, whose values may change from line to line.
Let us remark that when A ∈ L(R n ) and B ∈ L(R m , R n ) are constant matrices, [A|B] ∈ L(R nm , R n ) is the matrix given by With this notation, we have the following main result. (1.10) The rest of this paper is organized as follows. In Section 2 prove the wellposedness of the problem (1.1). Section 3 is devoted to some controllability results for one parabolic equation. In Section 4, we prove some useful estimates under the assumption (1.10) and we establish Carleman estimates for a scalar PDE of order 2n in space. In Section 5, we give the proof of the main result. And finally, in Section 6 we study the null controllability for semilinear systems.

Wellposedness of the problem
The semigroup generated by the operator (M, D(M)) is analytic with angle π 2 ( see [9, Theorem 2.8]). In order to prove the wellposedness of the problem (1.1), it suffices to show that the operator DM generates a c 0 -semigroup. in fact, similarly like in [22] we prove, under the assumption that all eigenvalues of the diffusion matrix D have positive real part the operator DM is the generator of an analytic semigroup.
Thus, (1.1) is well posed in the sens of semigroup theory and the following global existence result holds.

Carleman estimates for one equation
In order to establish a Carleman estimate for the adjoint system (1.8), we are led to see Carleman estimates already established in the case of one single parabolic degenerate equation of order 2 in space [3,15].
For this purpose, let us consider the following time and space weight functions is an open subset of ω, and the parameters c, ρ and λ are chosen as in [15] such that Let ω ′ a subset of ω and set ω ′′ : The two following results have been proved in [3].
Proposition 3.1. Let T > 0 and τ ∈ R. Then there exists two positive constants C and s 0 such that, for all u 0 ∈ L 2 (0, 1), the solution u of equation for all s s0.

Some useful results
Since the number of control forces is less than the number of equations, we need to highlight the equation coupling tools. In deed the equations are coupled by means this algebraic condition rank[λ i D − A|B] = n ∀i 1. Let us introduce the following operators 4.1) K and K * are densely defined unbounded operators. we have the following estimate for any t ∈ [0, T ) and any k (n − 1) 2 , where R only depends on n, D and A.
Proof. We adapt the same argument as in [16] to our degenerate case. Let denote by K i the where a i ∈ R n and a i = 0 for all i ≥ p + 1 for some p ≥ 1, then Let us denote by η i j , for 1 ≤ j ≤ n, the real and nonnegative eigenvalues of K i K * i . Then we have There exists c 1 such that Indeed, let us set p(λ) = detK(λ)K(λ) * for all λ, with Thus, p(λ) is a polynomial function of degree 2n(n − 1), p(λ) 0 for all λ and p(λ i ) = 0 for all i. Since the roots of p(λ) = 0 are in a disk of radius R for some R > 0, then, there exists C 2 > 0 such that p(λ) C 2 for |λ| R. Moreover, for some ℓ, one has λ ℓ > R. Hence, • Or i ℓ and then λ i λ ℓ > R and detK where · 2 in the usual Euclidean norm in L(R n ). Then we infer Coming back to (4.5) we get As this is true for all f spanned by a finite amount of the w i , then we infer that this must also hold for all f ∈ L 2 ((0, 1); R n ) such that (−M) k K * f ∈ L 2 ((0, 1); R n ). In particular, we find (4.2).
From now on, we consider φ with the monomial derivative M i ∂ j t φ ∈ 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 (∂ t , M) is the operator defined by P (∂ t , M) = det(∂ t I d + D * M + A * ).
Now we prove all components of every solution of the adjoint system (1.8) are solutions of the scalar PDE (4.7). So, it will be necessary to establish Carleman estimate for the scalar PDE (4.7). First we recall The following result [3,6,16].  for any k ≥ 0 and j ≥ 0, we can find an integer m(k, j) ≥ 0, a constant C(k, j) > 0 and an open set ω(k, j) satisfying ω ⋐ ω(k, j) ⋐ ω 1 , such that where φ satisfies (4.7) and Proof. We will prove (4.8) by induction on k and j in two steps.
Step 1 : Proof of (4.8) for k = j = 0 Let us see that, if s large enough, one has for some m(0, 0), C(0, 0) and ω(0, 0). We assume that D satisfies the assumptions (1.2)-(1.4), then we have for some p ≥ 1 ii , the J * i are Jordan blocks, i.e. each of them is of the form (1.4) for some d i ∈ C and the A ij provide the corresponding block decomposition of A. Thus we can write (4.7) as follow (4.11) in the term F (φ) we find the composition of at most p − 2 operators of kind detH i (∂ t , M) applied to φ. Let us define the functions ψ i by Thus the equation (4.11) can be written as by hypothesis, we have Cφ = Cψ 2 = · · · = Cψ p = 0 on Σ Let us consider the first PDE of (4.12), assume that J p is a Jordan block of dimension r associated to the complex eigenvalue α with Re(α) > 0 and let denote by η 1 , · · · , η r the diagonal components of A pp . Then this PDE can be rewritten as where G(ψ p ) is a linear combination of partial derivatives of ψ p . Again, let us introduce the new variables Therefore, we can rewrite (4.12) as a first-order system for the ζ i : (4.14) with Cζ 1 = Cζ 2 = · · · = Cζ r = 0 on Σ.
Notice that |G(ψ p )| 2 is bounded by a sum of squares of derivatives of ψ p . More precisely, we have |G(ψ p )| 2 ≤ CI G (ψ p ), with The following I α,A (τ, z) term already used in [3] is defined by where A ∈ {ϕ, Φ} and α ∈ {ξ, ς} used in Proposition 3.1 and Proposition 3.2. Applying Carleman estimates (3.5) established in Proposition 3.1 to the first PDE in (4.14), we get And for the j th PDE in (4.14) where j = 2, · · · , r, we have Consequently, an appropriate linear combination of the terms in the left hand sides absorbes the global weighted integrals of |ζj | 2 for j = 2, · · · , r.
The next task will be to add some extra terms on the left hand side of the previous inequality. To this end, we reason as follow. We apply −M to the j th PDE in (4.14) where j = 2, · · · , r, we have −(∂t + αM + ηj )Mζj = −Mζj−1.

Using Proposition 3.1, we get
Then, we can add all these new terms to the left hand side of (4.16) and, for a new positive constant C, obtain r j=1 I ξ,ϕ (τ + 3(j − 1), ζj) + r j=2 I ξ,ϕ (τ + 3(j − 2) − 1, Mζj) We can continue the previous process and add better global terms on the left hand side of (4.17). Thus, if we apply (−M) 2 to the j th PDE in (4.14) where j = k + 1, · · · , r and k = 2, · · · , r − 1, we have (∂t + αM + ηj )(−M) k ζj = (−M) k ζj−1 we use again Proposition 3.1 for the previous equations Let us denote by J ξ,ϕ (τ, ζ) the following sum where ζ = (ζ1, · · · , ζr). Then, we can add all these new terms (4.18) to the left hand side of (4.17) and, for a new positive constant C, obtain with s sufficiently large. From now on, we fix s sufficiently large and we try to replace the local terms in (4.20) corresponding to ζ1, · · · , ζr−1 by a term of the form (ψp = ζr) using the same computation [15,Lemma 3.7], we can show the existence of a constant C > 0 and an integer ℓ1 such that : Since the operators (∂t + αM + ηj ), j = 1, · · · , r commute, we see that (4.13) can be rewritten equivalently in the form where σ is any permutation in Pr. Hence, we can introduce the new variables ζ σ r = ψp, ζ σ r−1 = (∂t + αM + η σ(r) )ζ σ r , · · · , ζ σ 1 = (∂t + αM + η σ(2) )ζ σ 2 . and we can also rewrite (4.13) as a first-order system for the ζ σ i : · · · · · · · · · (∂t + αM + η σ(r) )ζ σ r = ζ σ r−1 . Again, with Cζ σ 1 = Cζ σ 2 = · · · = Cζ σ r = 0 on Σ. As before, we obtain an estimate like (4.21) In this inequality we have used the notation ζ σ = (ζ σ 1 , · · · , ζ σ r ). Now, let us define I ξ,ϕ (τ, ζ) by we have Observe that all the terms in IG(ψp) are also in the left multiplied by weights of the form s κ θ κ e 2sϕ with κ > 0. Consequently, for sufficiently large s, these terms are absorbed and we find Let us now consider the second PDE in (4.12). Arguing in the same way, we deduce the following estimate for ψp−1 : (4.24) The corresponding similar estimate also holds for ψp−1, etc. Thus, after addition and taking into account that ψ1 = φ and the global integrals of ψp, · · · , ψ2 in the right hand side are smaller than the terms in the left, we get an estimate for all the ψi : Q ω ′ (sθ) ℓ 2 |ψi| 2 e 2sϕ dxdt . (4.25) Again, using the cascade structure of system (4.12), all the local integrals in the right can be absorbed by the left hand side, with the exception of the local weighted integral of |φ| 2 . All we have to do is to enlarge the open set 1 and argue like in the passage from (4.20) to (4.21). Therefore, the following is obtained: we see that, taking into account that the operators detHi(∂t, M), commute for i = 1, · · · , p. Then we can rewrite (4.11) in the form where σ is any permutation in Pn. Then we have the following equivalent formulation of (4.7) thus, we can also get an estimate of the same form where, now, we have in the left global weighted integrals of φ, ψ σ 2 , · · · , ψ σ p . Taking into account that F (φ) is a sum of terms where, at most, p − 2 operators of the kind detHj(∂t, M) are applied to φ. Since σ is arbitrary in Pn, using all these possible estimates together and arguing as above, it becomes also clear that the terms containing |F (φ)| 2 can be controlled by the terms in the left. This gives Likewise, by applying Proposition 3.2 we infer From (4.26) and (4.28) we deduce (4.28) wherel = max(ℓ2, ℓ3). This proves (4.9).
Step 2 : Induction on k and j.
Let us now assume that (4.8) is true for any k ′ = 0, 1, · · · , k any j ′ = 0, 1, · · · , j and any solution to (4.7) and let us prove (4.8) (for instance) with k replaced by k + 1; the proof with the same k and j replaced by j + 1 is essentially the same. Sinceφ := (−M)φ also satisfies (4.7), we have by hypothesis Thus, that there exist m(k + 1, j), C(k + 1, j) and ω(k + 1, j) such that This ends the proof.

Proof of the main result
Proof of Theorem 1.1. Let us first assume that (1.1) is null-controllable. If we have rank[λ i D + A|B] ≤ n − 1 for some i, then the associated ordinary differential system is not null-controllable. This means there exists z T ∈ R n \ {0} such that the solution to the Cauchy problem satisfies B * y(t) = 0. If we now set φ T = y T w i , where w i is an eigenfunction associated to λ i , we see that the corresponding solution to the adjoint system (1.8) cannot satisfy the observability inequality (1.9). Consequently, (1.10) must hold. Conversely, let us assume that (1.10) is satisfied, and let us prove that the system (1.1) is nullcontrollable.
Let z be the solution to the adjoint system (1.8) corresponding to a final data z T , by Proposition 4.1, for k (n − 1) 2 there exists a positive constant C depends on n, D and A such that for any t ∈ [0, T ). From (4.1) the components of K * z are appropriate linear combinations of the components of z and their second-order in space derivatives. Notice again that, for all t ∈ [0, T ), z(·, t) is regular enough to give a sense to (−M) k (K * z), which belongs to L 2 (0, 1).
By Proposition 4.2 z ∈ C k ([0, T ]; D(M p ) n ) for every k, p 0 and for every i the component z i of z solves equation (4.7). Thus, we can write (4.8) for any component of B * z. This gives the following inequality for all j, k ≥ 0 and all ℓ = 1, · · · , m From (5.2) and (5.3), we will easily deduce (1.9) and, therefore, the null controllability of (1.1).

Null controllability for semilinear systems
Now we consider the following semi-linear non-diagonalizable parabolic degenerate systems.
in (0, 1), where F is a globally Lipschitz function depending only on Y and F (0) = 0. Our goal is to prove the null controllability of the system (6.1). We will use a standard strategy, as in [24,7,10,1], which consists in using the linearization technique, the approximate null controllability, the variational approach and the Schauder fixed point theorem.
The system (6.1) can be written as follow We assume the following Let us recall the set X T (see Theorem 2.2) induced with the norm For a fixed Y in X T , consider the associated linear system in (0, 1), (6.3) and its adjoint system   in (0, 1), (6.4) Thus, from (6.2), it follows that the matrix A Y satisfies the algebraic condition (1.10).
In order to construct a suitable fixed point operator, we start at first by proving the uniqueness of the control with minimal norm. For a given ε > 0 and Y 0 ∈ L 2 (0, A) n we consider the following functional where Y is the solution of (6.3) with initial data Y 0 and Z is the solution of (6.4) with initial data Z 0 . By a classical arguments, minimization problems where Y ε, Y is the solution of (6.3) associated to the control v and Z ε, Y is the solution of the adjoint problem (6.4) with the initial data Z ε, Y 0 . Since J * ε, Y (Z ε, Y 0 ) 0 ,then from (6.6) and (6.8) we infer On the other hand, by the observability inequality (1.9) From the estimates (6.7) (6.9) and (6.10) we infer The uniqueness of the control v ε, Y allows to define the operator Any fixed point Y ε of K ε is a solution of the semilinear system (6.1) associated to v ε,Y ε and it satisfies Y ε 2 (L 2 (0,1)) n εC. (6.13) Indeed, suppose first that Y 0 ∈ (H 1 a ) n . From (2.3) and since the matrix A is constant, we have the following estimate Thus, by (6.11) we infer where Y T := H 1 0, T ; L 2 (0, 1) n ∩ L 2 0, T ; H 1 a n with the norm Thus, the range of K ε is include in the ball B(0, R) of X T with the radius R = C Y 0 (H 1 where C is the constant used in (6.15). Then K ε (B(0, R)) ⊂ B(0, R). Now let us prove that the operator K ε is continuous and compact. The compactness of K ε results from the compactness of the embedding Y T ֒→ X T (6.17) see [10,Theorem 4.4]. For the continuity, let us consider the sequence Y n that converge to Y in X T . To simplify, let denote Y ε, Yn and v ε, Yn respectively by Y n and v n (for a fixed ε). From (6. 16) we deduce that the sequence Y n is bounded in the space Y T . Thus we can extract a subsequence that converges weakly in Y T to Y and strongly in X T by dint of (6.17). Likewise, thanks to (6.11) we can assume that v n converges weakly to v. So Y is then the solution of (6.3) associated to Y and v. Therefore, in order to show that K ε ( Y ) = Y it suffices to prove that v = v ε,Y . From the definition of v n , we have for all v in L 2 (Q) m Y Yn,v (T ) 2 L 2 (0,1) n , (6.18) where Y Yn,v is the solution of (6.3) associated to Y n and v. Passing to the limit in the inequality (6.18), one has for all v in L 2 (Q) m This means that v minimizes J ε, Y . Consequently K ε ( Y ) = Y , Hence the continuity of K ε , Thus, the following result is then proved. Theorem 6.1. Assume (6.2) is fulfilled. For all Y 0 in (H 1 a ) n the semi-linear parabolic degenerate system (6.1) is approximatively null controllable. i.e. for all ε > 0 there exists a control v ε ∈ L 2 (Q) m for which the associated solution Y vε satisfies Y vε (T ) L 2 (0,1) n ε.
Proof. From Theorem 6.1 the set {v ε , ε > 0} is bounded in L 2 (Q) m , thus it contains a sequence (v n ε ) that converges (weakly) in L 2 (Q) m to a limit v 0 that satisfies v 0 The sequence (Y v n ε ) converges strongly to (Y v0 ) in X T . Moreover (Y v0 ) is the solution to (6.1) with v = v 0 . So according to (6.20), for all x ∈ (0, 1) we have Y v0 (T, x) = 0 thus, the semi-linear parabolic degenerate system (6.1) with regular initial data is null controllable. Now we are able to give the proof of the null controllability of the semi-linear parabolic degenerate system (6.1) with general initial data. As in [10,8,1], we can show also the following well posedness of degenerate parabolic semi-linear systems which is of great utility. admits a solution U in X T .