APPROXIMATE INTERNAL CONTROLLABILITY AND SYNCHRONIZATION OF A COUPLED SYSTEM

. Based on the uniqueness of solution to a coupled system of wave equations associated with incomplete internal observations, we establish the approximate internal synchronization by groups, the induced internal synchronization and the approximate internal synchronization in the pinning sense.


Introduction
Let A be a matrix of order N and D be a full column-rank matrix of order N × M .Both A and D are composed of constant entries.Let Ω ⊂ R n be a bounded domain with a C 1 boundary Γ.We consider the following problem for the variable U = (u (1) , • • • , u (N ) ) T : and where H = (h (1) , • • • , h (M ) ) T denotes the internal controls and χ ω is the characteristic function of the subdomain ω ⊂ Ω.The approximate internal controllability was well studied in [22].In this paper, we will further develop the theory of approximate internal synchronization and clarify the relation between the internal synchronization in the pinning sense and that in the consensus sense.
The exact internal controllability of wave equations was abundantly studied in the literatures.We quote [6,10,30] for a single 1-d wave equation with locally distributed control in any fixed nonempty subinterval of a bounded interval.In higher dimensional case, the exact controllability was established by HUM method in [23,32,33] with control distributed in an ϵ-neighbourhood ω of the boundary Γ satisfying the usual multiplier control condition.
In the case of fewer internal controls, namely, when rank(D) < N , system (1.1) is not exactly controllable (see [22], Thm.10).In order to relax the restrictions on the domain ω and on the control matrix D, we studied in [22] the approximate controllability of system (1.1), which is equivalent to the uniqueness of solution to the adjoint system for the variable Φ = (ϕ (1) , • • • , ϕ (N ) ) T : associated with the initial data and the internal observation By classic theory of semigroups (see [27]), system (1.3) forms a C 0 -semigroup in the space (L 2 (Ω) × H −1 (Ω)) N .Moreover, it is easy to see that Kalman's rank condition: is necessary for the uniqueness.However, when rank(D) < N , the incomplete observation (1.5) cannot imply the nullity of all the components: So, the uniqueness of solution to problems (1.3)-(1.4)cannot be considered as a standard continuation theorem of Holmgren's type.However, because of the commutation of the internal observation with d'Alembert operator: it turns out that Kalman's rank condition (1.6) plays the same role as in the case of ODEs [9].We have stablished the following result.
The above theorem is valid without any restriction on the matrix A, nor on the damping domain ω.Moreover, the observation time T > 2d(Ω) is independent of rank(D) and of the order N of A. This is essentially different from the case of boundary observation, where the observation time T depends on the rank of the matrix D and the number N of the equations (see [14,31]).Now let us comment the related literature.The situation is much complicated for the uniqueness of problems (1.3)-(1.4)associated with boundary observation: (1.9) Our basic idea is to combine the uniform observability of a scalar system and Kalman's rank condition (1.6).
The first attempt for realizing this idea was carried out in [13,14] for a system of wave equations with Dirichlet boundary condition by incomplete Neumann observations.Later, this idea was used in [11,19] for Neumann and Robin boundary conditions, and further developed in [18,21] for an elliptic system with Neumann boundary conditions observed by incomplete Dirichlet observations.Additionally, the coupling matrix A should be nilpotent (see [2,14,16]), or symmetric and close to a scalar matrix etc. (see [19,31]).Unlike the hyperbolic system, which was less studied and the obtained results are of different natures (see [1,7,26,29]), the parabolic problem has been abundantly investigated in the literature.We only quote [3,4] and the references therein for the internal control of coupled systems of heat equations with the same diffusion coefficients and constant or timedependent coupling terms by means of Carleman estimates.We also mention [24] for the internal controllability of a system of heat equations with analytic nonlocal coupling terms and [25] for the internal observability of some parabolic equations with constant or time-dependent coupling terms using Lebeau-Robbiano strategy.
The paper is organized as follows.In Section 2, we complete the work in [22] by a charaterization of the approximate internal controllability, then we first show the equivalence of the approximate internal controllability of system (1.1) and the D-observability of adjoint system (1.3).In Section 3, we show the equivalence of the approximate internal controllability of system (1.1) and Kalman's rank condition (1.6).We then recall the relation between the approximate controllability and the exact controllability.In Section 4, we clarify the notion of indirect control by successive projections on the null space of controls.In Section 5, we study the approximate internal synchronization by p-groups of system (1.1) under the condition of C p -compatibility.In Section 6, we establish the necessity of the condition of C p -compatibility under the minimal rank condition.In Section 7, we further give the additional information of synchronization in the general situation.In the last section, we summarize the materials obtained for approximate internal and boundary controllabilities, and indicate some problems to be examined in the future.

Approximate internal controllability and D-observability
Let us first recall the following standard well-posedness result (see [5], [6] and [27]).Proposition 2.1.Let Ω ⊂ R n be a bounded domain with a smooth boundary Γ of class C 1 .For any given 2) admits a unique weak solution U in the space Moreover, the mapping is linear and continuous for the corresponding topologies.
Let C be the set of all the initial states (V (0), V ′ (0)) given by the backward problem: as the internal control H runs through the space (L 2 (0, T ; L 2 (Ω))) M .Similarly to the approximate boundary controllability in [17], we can show the following result.

Approximate internal controllability
As a direct consequence of Theorems 1.1 and 2.5, we have the following result.Let Ω ⊂ R n be a bounded domain with a C 2 boundary Γ.For any given x 0 ∈ R n , we define where ν(x) is the unit outer normal vector on Γ.For a good understanding of the conception of the approximate controllability, we first recall the following result on the exact controllability.
Assume furthermore that D is invertible.Then system (1.1) is exactly controllable at any given time T > T 0 in the space (H 1 0 (Ω) × L 2 (Ω)) N .In the case of fewer internal controls, we have the following negative result.This is the motivation for considering the approximate controllability.Proposition 3.3.( [22], Thm.10) Assume that rank (D) < N .Then no matter how large the time T > 0 is, system (1.1) is not exactly null controllable in the space (H 1 0 (Ω) × L 2 (Ω)) N .Noting that the rank of D in equation (1.6) may be much smaller than N , this is the advantage to consider the approximate internal controllability.However, the following result shows that the sequence of controls and only if it is approximately null controllable at the time T in the space (H 1 0 (Ω) × L 2 (Ω)) N by means of a bounded sequence {H n } of internal controls in (L 2 (0, T ; L 2 (Ω))) N .

Notion on indirect controls
In [30], D. Russell introduced the notion of indirect stabilization.It concerns if the dissipation induced by one of the equations can be sufficiently transmitted to the other ones in order to realize the stability of the overall system.The effectiveness of the indirect damping depends in a very complex way on all of the involved factors such as the nature of the coupling term, the order of dissipation, the hidden regularity of undamped equations, the accordance of boundary conditions and many others.We refer [8,28] and the reference therein for the recent progress.
In this section, we try to explain the meaning of indirect controls and the mechanism of their roles for approximate controllability.The basic idea is to project system (1.1) to Ker(D T ) for getting a family of systems with homogeneous boundary condition.We first show the idea by a simple example: We have Then, applying the row-vectors (1, 0, 0) and (0, 1, 0) in Ker(D T ) to system (4.1),we get 2) The reduced system (4.2) is for the variables u 1 and u 2 , so at the first step, the variable h (1) = −u 3 can be formally regarded as an internal control appearing in system (4.2).However, the value of h (1) cannot be freely chosen, then we call it as an indirect internal control.Next let We have Then, applying the row-vector (1, 0) in Ker(D T 1 ) to the reduced system (4.2), at the second step we get This is a system for the variable u 1 , in which the variable h (2) = −u 2 can be regarded as an indirect internal control.Finally, let Since Ker(D T 2 ) = (0), we stop the projection.By this way, we decompose the original system (4.1)into two sub-systems (4.2) and (4.3).Consequently, besides the direct boundary control h acting on the boundary and appearing in the original system (4.1),we find two indirect internal controls h (1) and h (2) , which are hidden in the sub-systems (4.2) and (4.3), respectively.Now we present the general procedure.Let A 0 be a matrix of order N 0 , and D 0 be a matrix of order In particular, we have Noting that there exists a matrix A l of order N l and a matrix D l of order N l × M , such that Then, noting (4.7), we have and Since K l−1 is of full column-rank, then Ker(K l−1 ) = {0}.It follows from equation (4.10) that We may continue the procedure of projection until D L = 0 or Ker(D T L ) = {0}.Now let and applying K T 0 to equation (1.1) and noting equations (4.7) and (4.9) for l = 1, we get Assume that U (l−1) are determined for l ⩾ 2 by For l ⩾ 2, applying K T l−1 to equation (4.15) and noting equations (4.7) and (4.9), we get The terms H (l) (1 ⩽ l ⩽ L) can be formally regarded as internal controls in the reduced systems (4.17).Thus, the original system (1.1) is directly controlled by H (0) , and indirectly controlled by H (1) , • • • , H (L) , which are hidden in subsystems (4.17) and intervene into the system at different steps of the reduction.
Accordingly, let rank(D l ) denote the number of indirect controls H (l) for 1 ⩽ l ⩽ L.Then, the following formula (see [20], Prop.3.3) shows that rank(D, AD, is the total number of (direct and indirect) controls.However, the control H (l) is implicitly given via U (l−1) for 1 ⩽ l ⩽ L, so, its value cannot be independently chosen, then H (l) (1 ⩽ l ⩽ L) will be called the indirect internal controls.Nevertheless, the following result reveals the role of the indirect internal controls H (l) (1 ⩽ l ⩽ L) for the approximate internal null controllability.
Proposition 4.1.If system (1.1) is approximately null controllable, then the subsystems (4.17) are approximately null controllable for all l with 1 ⩽ l ⩽ L.
Proof.Noting that the approximate internal controllability of system (1.1) implies well that of subsystems (4.17).Moreover, by [20], Proposition 3.4, we have for all l with 0 ⩽ l ⩽ L.
Consider the following example: In this case, K 0 can be taken as Then, applying equations (5.4)-(5.5)with l = 1, a straightforward computation gives and Similarly, applying equations (5.4)-(5.5)with l = 2 to we get Since Ker(D T 2 ) = {0}, we stop the projection.All the pairs (A l , D l ) satisfy Kalman's rank condition (4.20) with System (1.1) as well as subsystems (4.17) for l = 1, 2 are approximately null controllable.

Approximate internal synchronization by groups
When the rank of Kalman's matrix is fewer than N , the approximate internal controllability fails.As in the case of boundary controls studied in [14], we will consider the corresponding synchronization by groups.
Let p ⩾ 1 be an integer such that We re-arrange the components of the state variable U into p groups (u (1) , Definition 5.1.System (1.1) is approximately synchronizable by p-groups at the time T in the space ( as n → +∞ for all n r−1 + 1 ⩽ k, l ⩽ n r and 1 ⩽ r ⩽ p. Let S r be a 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 where Then, (5.1) is equivalent to as n → +∞.The convergence (5.1) or (5.4) will be called the approximate internal synchronization by p-groups in the consensus sense.if and only if there exists a matrix A p of order (N − p), such that Under the condition of C p -compatibility (5.6), applying C p to system (1.1) and setting W p = C p U and D p = C p D, we get the following reduced system Thus, the approximate internal synchronization by p-groups of system (1.1) is transformed into the approximate internal null controllability of the reduced system (5.7), which can be treated by Theorem 3.1.).Since C p D = C p D 1 , the matrix D 0 does not play any role for the reduced system (5.7).Without loss of generality, we may assume that D 0 = 0 so that Im(D) ⊆ Im(C T p ).

Condition of C p -compatibility
The condition of C p -compatibility equation (5.5) is a key ingredient for the synchronization of system (1.1).However, its necessity is a delicate question.In this section, we will first discuss this question under the minimum total number of controls, and further develop the study in the next section.
Lemma 6.1.Let d ⩾ 0 be an integer.The rank condition rank(D, AD, holds if and only if d is the largest dimension of the subspaces which are contained in Ker(D T ) and invariant for A T .Proposition 6.2.Assume that system (1.1) is approximately synchronizable by p-groups.Then for any given subspace V invariant for A T and contained in Ker(D T ), we have We have For any given initial data ( U 0 , U 1 ) ∈ (H 1 0 (Ω) × L 2 (Ω)) N , let {H n } be a sequence of internal controls, which realizes the approximate internal synchronization by p-groups for system (1.1).We denote by {U n } the sequence of the corresponding solutions.Applying E r to system (1.1) and noting Let E be a unit vector in V ∩ Im(C T p ).Since E ∈ Im(C T p ), there exists x ∈ R N −p , such that E = C T p x.It follows from (5.4) that On the other hand, since E ∈ V , there exist coefficients a 1 , • • • , a p not all zero such that E = d r=1 a r E r , then we have a r u r .(6.6) The homogeneous system (6.4) is independent of internal control, therefore, independent of n.It follows from equations (6.5) and (6.6) that d r=1 a r u r (T ) = 0. (6.7) By the time-invertibility of system (6.4), the mapping Then we deduce from equation (6.7) a contradiction Proposition 6.3.Assume that system (1.1) is approximately synchronizable by p-groups.Then we necessarily have (6.9) By Lemma 6.1, there exists a subspace V of dimension q, which is invariant for A T and contained in Ker(D T ).Since then Im(C T p ) ∩ V ̸ = {0}, which contradicts Proposition 6.2 Theorem 6.4.Assume that system (1.1) is approximately synchronizable by p-groups under the minimal rank condition Then, we have (a) Ker(C p ) admits a complement which is invariant for A.
(b) There exist u 1 , • • • , u p independent of applied internal controls, such that as n → +∞.(c) A satisfies the condition of C p -compatibility equation (5.5).
Remark 6.5.The convergence in equation (6.11) will be called the approximate internal synchronization by pgroups in the pinning sense, and the vector-valued function (u 1 , • • • , u p ) T will be called the synchronizable state by p-groups.Theorem (6.4) means that under the minimum rank condition (6.10), the approximate internal synchronization by p-groups takes place in the pinning sense.
Proof.(a) By Lemma 6.1 with d = p, there exists a subspace V = Span{E 1 , • • • , E p } which is invariant for A T and contained in Ker(D T ).By Proposition 6.2, we have It follows that On the other hand, since then Ker(C p ) admits V ⊥ as a complement, which is invariant for A.
(b) Let Q p be a matrix of order N × (N − p) such that Im(Q p ) = V ⊥ .By equation (6.12), Im(C T p ) ∩ (V ⊥ ) ⊥ = {0}.Then, by [17], Proposition 2.5 Im(C T p ) and V ⊥ are bi-orthonormal.Without loss of generality, we may assume that C p Q p = I N −p .Similarly, Ker(C p ) and V are bi-orthonormal, we can chose be a sequence of internal controls, which realizes the approximate internal synchronization by p-groups for system (1.1).We denote by {U n } the sequence of the corresponding solutions.Since V ⊥ is a complement of Ker(C p ) and Im(Q p ) = V ⊥ , we have Ker(C p )⊕ Im(Q p ) = R N .Then, a straightforward computation gives where for r = 1, • • • , p, u r = E T r U n are determined by equation (6.4) with d = p.Using equation (5.4), we get thus equation (6.11).
(c) Applying C p to the equations in (1.1), we get Using equation (6.11) and passing to the limit as n → +∞, we get where u 1 , • • • , u p are given by equation (6.4).Since the mapping equation (6.8) is surjective from (

Induced internal synchronization
In this section, we will further examine the convergence (5.4).For this purpose, we introduce the following notion.
Applying C p to the equations in system (1.1), and noting (5.4), we have Then, applying C p A to the equations in system (1.1), we can successively get for all integers k ⩾ 0, which implies that for 0 ⩽ i ⩽ N − 1 as n → +∞.Now applying (A i ) T x to system (1.1) and using Cayley-Hamilton's Theorem, for 0 ⩽ i ⩽ N − 1, there exist coefficients α ij such that Since the homogeneous system (7.8) is independent of n, it follows from equation (7.7) that By the time-invertibility of system (7.8),we get for all 0 ⩽ i ⩽ N − 1 and ( U 0 , U 1 ) ∈ (H 1 0 (Ω) × L 2 (Ω)) N .In particular, we have for all ( U 0 , U 1 ) ∈ (H 1 0 (Ω) × L 2 (Ω)) N .We thus get a contradiction x = 0. Theorem 7.3.Assume that system (1.1) is approximately synchronizable by p-groups.Then, it is induced synchronizable by C p.
Proof.Since formula (5.10) is still valid for the (N − p) × N full row-rank matrix C p, Theorem 5.3 applies under the condition of C p-compatibility equation (7.3) and the rank condition (7.4).We get thus the convergence (7.1) with C r = C p.
Corollary 7.4.Assume that system (1.1) is approximately synchronizable by p-groups under the minimum rank condition (6.10).Then A satisfies the condition of C p -compatibility equation (5.5).
Proof.This result was already proved in Theorem 6.4.Here we give another proof, which further clarifies the necessity of the extension described by equation (7.2).
By equation (7.4), we have Now we consider the case that A satisfies the condition of C p -compatibility equation (5.5).Let D p denote the set of matrices D, which realize the approximate internal synchronization by p-groups for system (1.1).By Proposition 6.By Theorem 6.4, when the equality in equation (7.13) is true, Ker(C p ) admits a complement, which is invariant for A. In this situation, system (1.1) is approximately synchronizable by p-groups in the pinning sense and the synchronizable state by p-groups is independent of applied internal controls.
In the general case, we will extend C p to a suitable matrix C * q , and bring the problem to the optimal situation considered in Theorem 6.4.Proposition 7.5.( [17], Prop.2.19) Assume that A satisfies the condition of C p -compatibility equation (5.5) and that Im(C T p ) is A T -marked (there exists a Jordan basis of A T in Im(C T p ), which can be extended by adding new vectors to a Jordan basis of A T in C N ).Then there exists an (N − q) × N (0 ⩽ q < p) full row-rank matrix C * q such that Ker(C * q ) is the largest subspace satisfying the following two requirements.(a) Ker(C * q ) is contained in Ker(C T p ), (b) Ker(C * q ) is invariant for A (then A satisfies the condition of C * q -compatibility) and admits a supplement which is also invariant for A. We necessarily have the rank condition So, when we make the reduction by using C p for transforming system (1.1) to system (5.7), the rank of the matrices (D, AD, . This process losses (p − q) controls.However, when we make the reduction by C * q , the matrices (D, AD, ) have the same rank, therefore, there is no loss of controls in this process.In other words, the (p − q) controls lost in Ker(C p ) is recovered in the reduction by C * q .
Theorem 7.7.Assume that system (1.1) is approximately synchronizable by p-groups under the minimal rank condition (7.15).Then it is induced synchronizable by C * q .
Proof.Obviously, C * q satisfies (a) and (b) in Definition 7.1 with C r = C * q .In particular, A satisfies the condition of C * q -compatibility, by Proposition 5.2, there exists a matrix A * q such that C * q A = A * q C * q .Applying C * q to system (1.1) and setting W q = C * q U and D * q = C * q D, we get .17) By Proposition 7.6, condition (7.16) holds under the minimum rank condition (7.15).Then, applying Theorem 5.3 to the reduced system (7.17),we deduce the convergence (7.1), in which C r is replaced by C * q .We get thus the induced synchronization by C * q .
The above theorem shows that the recovered (p − q) controls will provide the convergence of (p − q) more components of U than the convergence (5.4).As a result, we can realize the approximate internal controllability of (N − q) equations by means of (N − q) controls.This is the best thing that we can do.
Corollary 7.8.Assume that system (1.1) is approximately synchronizable by p-groups under the minimal rank condition (7.15).Then it is approximately synchronizable by p-groups in the pinning sense.

Proof. Let Span{e
For any given initial data ( U 0 , U 1 ) ∈ (H 1 0 (Ω) × L 2 (Ω)) N , let {H n } be a sequence of internal controls, which realizes the approximate internal synchronization by p-groups for system (1.1).We denote by {U n } the sequence of the corresponding solutions.Under the minimal rank condition (7.15), by Theorems 7.7 and 6.4, there exist as n → +∞.Since Ker(C * q ) ⊂ Ker(C p ), there exist some coefficients α sr such that as n → +∞.In other words, the primary system (1.1) is approximately synchronizable by p-groups in the pinning sense.However, since q < p, u 1 , • • • , u p given by equation (7.21) are linearly dependent.

Synthesis and open problems
There are some commun points for the approximate boundary and internal controllabilities and synchronizations.
The advantages of the approximate internal controllability and synchronization are as follows.1. Kalman's rank condition (1.6) is also sufficient for the approximate internal controllability (Thm.1.1), but insufficient in general for the approximate boundary controllability (see [17], Thms.8.11 and 16.12).
3. The controllability time can be uniquely determined by the geodesic diameter of Ω and independent of both rank(D) and N for the internal controllability (Thm.1.1).However, it depends on both rank(D) and N for the approximate boundary controllability (see [17], Thms.8. 25 and 16.15 and [31]).
Finally, we give some open problems.
1.The study can be similarly carried up for a system of wave equations with Neumann boundary condition We easily check that Kalman's rank condition (1.6) with D = (D 1 , D 2 ) is still necessary for the approximate controllability, the sufficiency will be investigated in a forthcoming work.
3. System of wave equations with coupled Robin condition which is uncontrollable.As in [19], "No invariant subspace of A T and B T is contained in Ker(D T )" is still a necessary condition for the approximate controllability.But the sufficiency is largely open.

. 3 )
Let V = Span{E 1 , • • • , E d }be a subspace which is invariant for A T and B T , and contained in Ker(D T ) with D = (D 1 , D 2 ).Then, there exist coefficients α rs and β rs such thatA T E r = d s=1 α rs E s , B T E r = d s=1 β rs E s , r = 1, • • • , d.(8.4)ApplyingE r to system (8.3) and setting u r = (E r , U ), we get a homogeneous system for r = 1, • • • , d:u ′′ r − ∆u r + d s=1 α rs u s = 0 in R + × Ω, ∂ ν u r + d s=1 β rs u s = 0 on R + × Γ,(8.5) .12) It follows that p ⩽ p.Noting that p ⩾ p, we get p = p.Then, the extension matrix C p of C p is actually C p itself.From equation (7.2), we deduce that A T Im(C T p ) ⊆ Im(C T p ), namely, AKer(C p ) ⊆ Ker(C p ).