Comptes Rendus Mathématique

Uniqueness theorem for partially observed elliptic systems and application to asymptotic synchronization

Finally, denoting by I N the identity of R N , we define the following operators L and G of diagonal form: The first objective of the present paper is to find a simple and efficient characterization for the uniqueness of solution to the over-determined system with variable Φ = (φ (1) , . . ., φ (N ) ) T : associated with the condition of observation: where β ∈ R, D is a matrix of order N × M and A is a symmetric matrix of order N , both with constant entries.We observe that Kalman's rank condition rank(D, AD, . . ., A N −1 D) = N (7) is necessary for this uniqueness of solution (see Theorem 7).However, since a matrix D satisfying Kalman's rank condition (7) is not invertible in general, the partial observation (6) cannot imply the nullity of the full observation: so the uniqueness of solution to the over-determined system (5)-( 6) cannot be obtained by the standard Carleman's theorem of uniqueness of continuation (see [3,4]).Since only Kalman's rank condition is not sufficient, some additional conditions should be required for the uniqueness of continuation.
Definition 1.Let λ 1 , . . ., λ m denote the distinct eigenvalues of A. The matrix A satisfies theclosing condition if there exists a number a such that Definition 2. The operator L satisfies the c-gap condition if there exists a number c > 0, such that holds true for all distinct eigenvalues α n = α n of L.
Definition 3. The pair (L, γ) is observable if there exists a constant c > 0, independent of β ∈ R and f ∈ H , such that the observability inequality holds for any given solution φ to the following over-determined scalar problem Let us recall the following generalized rank condition of Kalman's type (see [8]).Now we can give the main result on the uniqueness of continuation.
Theorem 5. Assume that the pair (A, D) satisfies Kalman's rank condition (7).Then, the overdetermined system (5)-( 6) has only the trivial solution Φ ≡ 0 in any one of the following situations: (ii) The matrix A satisfies the -closing condition (9) with > 0 small enough, the operator L satisfies the c-gap condition (10), and the over-determined scalar problem has only the trivial solution φ ≡ 0. (iii) The matrix A satisfies the -closing condition (9) with > 0 small enough and the pair (L, γ) is observable.
On the other hand, if we replace A by A + bI , and β 2 by β 2 + b for any given b > 0 in (5), it will not modify anything in Theorem 5. So, without loss of generality, we may assume that the eigenvalues of A are strictly positive.
Proof.Now we give the proof of Theorem 5.
Case (i).From ( 6) and ( 14) we have Then, applying D T to (5) and noting the first formula of (4), it follows that We write (5) as Since L is self-adjoint, the eigenspaces associated with the different eigenvalues λ l are mutually orthogonal.Then it follows from ( 16) that Since for each 1 l m, the eigenvalue λ l = 0 and the vectors d µ l −1 +1 , . . ., d µ l are linearly independent, it follows that Case (ii).Assume that there exist two integers l , k with 1 l , k m and l = k, such that φ (i ) ≡ 0 for some i with µ l −1 + 1 i µ l , and φ (i ) ≡ 0 for some i with µ k−1 + 1 i µ k .Then φ (i ) and φ (i ) will be eigenfunctions of L, so there exist the corresponding eigenvalues α n l and α n k such that However, because of the -closing condition ( 9) and the c-gap condition (10), the above equality cannot be satisfied for > 0 small enough.Therefore, there exists at most one integer k with 1 k m, such that Then ( 6) reduces to Since the vectors d µ k−1 +1 , . . ., d µ k are linearly independent, it follows from (17) that Then the uniqueness of continuation of the scalar problem (15) implies that which, combined with (19), leads to for 1 i N and 1 j M , we have and On the other hand, by the definition of G in (4), condition (6) leads to Then, taking the j -th component of ( 21) and ( 23), we get with the additional condition If β 2 − a 0, multiplying (24) by w j , we get If β 2 − a > 0, then w j satisfies the scalar problem (12).Since (L, γ) is observable, we get again (26).
By the orthogonality of the eigenspaces (18), it follows from (22) that By the -closing condition (9), we get Thus, it follows from (26) that provided that < c; namely, we have Since d µ l −1 +1 , . . ., d µ l are linearly independent, we obtain that namely, Φ ≡ 0. The proof is then complete.
The above theorem can be read as "under Kalman's rank condition on the coupling matrices A and D, the observability of a scalar equation implies the uniqueness of solution to a complex system".Thus, it provides a simple and efficient approach to solve a seemingly difficult problem of uniqueness for a complex system.
Case (i) of global observation is similar to the finite-dimensional case.In this case, without any additional conditions on the matrix A or on the operator L, only Kalman's rank condition is sufficient for the uniqueness of solution to the over-determined system (see Theorem 16).In case (ii), thanks to the c-gap condition, the unique continuation of a scalar problem implies the uniqueness of solution to the over-determined system (see Theorem 14).In the general case (iii), the observability inequality is required and can be established under suitable geometrical control conditions (see Theorem 15).Moreover, the necessity of -closing condition is shown by an example in [8].
In the previous discussion, we have assumed that the matrix A is symmetric, which is actually essential for the stability of the evolution systems that we will study in the next section.A generalization of Theorem 5 can be found in the complete version [6].

Asymptotic synchronization by groups
Recall that the operators L and γ satisfy the conditions (1)- (3).In what follows, we assume furthermore that the operator γ is compact from V into V .
Let A and D be symmetric and positive semi-definite matrices.Consider the following second order evolution system with variable U = (u (1) , . . ., u (N ) ) T : associated with the initial data t = 0 : Defining the linear operator A by we formulate (27) into a first-order evolution system Clearly (see [11]), the operator A generates a semi-group of contractions in the space (V × H ) N .Definition 6.The system (27) is asymptotically (strongly) stable if for any given initial data (U 0 ,U 1 ) ∈ (V × H ) N , the corresponding solution U to problem (27)-( 28) satisfies Theorem 7. If the system (27) is asymptotically stable in (V × H ) N , then Kalman's rank condition (7) holds.Inversely, assume that the pair (A, D) satisfies Kalman's rank condition (7), then the system (27) is asymptotically stable in any one of the three situations described in Theorem 5.
Proof.Assume that the rank condition (7) fails.By Proposition 4, Ker(D) contains at least an eigenvector E ∈ R N associated with an eigenvalue λ ∈ R: Then, applying E to (27) and setting φ = (E ,U ), we get which is conservative, so never asymptotically stable.Inversely, because the resolvent of A is compact in the space (V × H ) N , by the classic theory of semi-groups (see [1,2]), the system (27) is asymptotically stable if and only if A has no pure imaginary eigenvalues.

It follows that
Since L + A is positive definite, we have β = 0. Since L + A and DG are symmetric, equation ( 32) is equivalent to the over-determined system ( 5)-( 6), which has only the trivial solution Φ ≡ 0 by virtue of Theorem 5.

Definition 8. The system (27) is asymptotically synchronizable by p-groups if for any given initial data
for all n r −1 + 1 k, l n r and 1 r p.
Let S r be the full row-rank matrix of order (n r − n r −1 − 1) × (n r − n r −1 ): Define the (N − p) × N matrix C p of synchronization by p-groups as Then the asymptotic synchronization by p-groups (33) can be equivalently rewritten as Since the above asymptotic synchronization by p-groups investigates the behaviour of the solutions on the infinite horizon, the notion of synchronizable state by p-groups is no longer available as for the synchronization on a finite interval considered in [7][8][9][10]; therefore the corresponding asymptotic synchronization by p-groups certainly proposes interesting questions and needs new effective methods.This is the second topic to be developed in this paper.
Before starting the study on the asymptotic synchronization, we first give some algebraic preliminaries.

Definition 9. The matrix A satisfies the condition of C p -compatibility if
or equivalently, there exists a positive semi-definite matrix A p of order (N − p), such that

Applications to wave equations
We will give some classic examples to illustrate possible applications of the abstract theory mentioned above.However, the approach is quite flexible and can easily be applied to other types of wave equations with variable density, or outside a star-shaped domain (cf.[5,12,13]).

Case with the gap condition
We first consider a specific situation on a rectangular domain where a > 0 is a parameter.
Theorem 14.Let a 2 be a rational.Assume that the pair (A, D) satisfies the rank condition (7) and that A satisfies the -closing condition with > 0 small enough.Then the system (44) is asymptotically stable.
Proof.Consider the following eigensystem: By Carleman's uniqueness theorem (see [3,4]), under the additional condition φ = 0 on Γ 1 , the above system has only the trivial solution.Moreover, a straightforward computation gives the eigenvalues and the associated eigenvectors as follows: Since a 2 is a rational, the eigenvalues satisfy the c-gap condition: for all α k,l > α p,q .Then, by Theorem 7, the system (44) is asymptotically stable, provided that A satisfies the -closing condition with > 0 small enough.If a 2 is an irrational, the c-gap condition is no longer valid.In this case, we don't know if the system (44) is asymptotically stable or not, even though the coupling matrix A satisfies theclosing condition with > 0 small enough.

Case with the observability inequality
In this subsection, we assume that there exists x 0 ∈R n , such that setting m = x − x 0 , we have (m • ν) 0 on Γ 0 .Then there exists a constant c > 0 independent of β and f , such that any given solution φ to the over-determined scalar problem In other words, the pair (L, γ) defined by ( 46) is observable (see [6] for details).Noting that γ is compact from H 1 Γ 0 (Ω) into (H 1 Γ 0 (Ω)) , by Theorem 12 we have the following Theorem 15.Assume that the pair (A, D) satisfies the rank condition (43) and that A satisfies the -closing condition (9) with > 0 small enough.Assume furthermore that A satisfies the condition of C p -compatibility (36) and that D is given by (38).Then the system (44) is asymptotically synchronizable by p-groups.

Case of global damping
Finally, we investigate the asymptotic stability of a system of wave equations with globally distributed damping: U t t − ∆U + AU + DU t = 0 in R + × Ω, As in the finite-dimensional case of ordinary differential equations, without any additional conditions, only Kalman's rank condition is sufficient for the asymptotic stability of wave equations.By Theorem 7, we have the following Theorem 16.Let the pair (A, D) satisfy the rank condition (7).Then the system (49) is asymptotically stable.

CProposition 4 .
. R. Mathématique, 2020, 358, n 3, 285-295 The rank condition rank(D, AD, . . ., A N −1 D) = N − d (13) holds for one integer d 0 if and only if d is the largest dimension of the subspaces which are invariant for A and contained in Ker(D T ).

) Definition 10 .
The matrix D satisfies the condition of strong C p -compatibility ifKer(C p ) ⊆ Ker(D), (37)or equivalently, there exists a positive semi-definite matrix R of order (N − p), such thatD = C T p RC p .(38)