Xing LUTatsien LI

Abstract This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM method for wave equations, however, by transforming the system into a first order hyperbolic system, the HUM method for 1-D first order hyperbolic systems, established by Li-Lu (2022) and Lu-Li (2022), can be applied to get the corresponding results.

Keywords Exact boundary controllability, Exact boundary synchronization, Coupled system of wave equations, HUM method

1 Introduction

The synchronization is a widespread natural phenomenon (see [5, 23]) which has been intensively studied in PDEs case in recent years (see [16] and the references therein, and [1–3]).The study of synchronization for the coupled system of wave equations

in a bounded smooth domain with various boundary conditions, in whichU=U(t,x) =(u(1),···,u(n))Tis the state variable, ?is the Laplacian operator, andAis a coupling matrix with constant elements, has been carried out in [4, 7–19, 22], etc.

However, for the system of wave equations coupled with velocities

the situation is quite different: Its exact boundary controllability can not be obtained by usual HUM method.In fact,since(1.2)does not possess the property of energy conservation,one can not establish the corresponding observability inequalities for the adjoint system by the energy estimate and the multiplier method directly from (1.2).On the other hand, if we regardAUtas a perturbation term, unlikeAU, which is not a compact one, the compact perturbation method given in [16] does not work.Thus, the exact boundary controllability and then the exact boundary synchronization for system (1.2) is still an open problem up to now.

In this paper, we will specially consider the corresponding problem in the 1-D case, namely,we will consider the following 1-D system of wave equations coupled with velocities

whereU=U(t,x)=(u(1),···,u(n))Tis the state variable, andAis a coupling matrix of ordern.

We give the following Dirichlet boundary condition onx=0:

While, onx=Lwe take any one of boundary conditions of Dirichlet type, Neumann type and coupled dissipative type:

where the boundary control matrixDis ann×M(M≤n) full column-rank matrix, andBis a boundary coupling matrix of ordern.AllA,BandDhave real constant elements, andH=(h(1),···,h(M))Tdenotes the boundary control.

The initial data is given by

The basic idea is to transform system (1.3)–(1.5) into a first order hyperbolic system, then the characteristic method can be applied to establish the corresponding observability inequalities for the corresponding adjoint system, so that the HUM method still works (see [21]), in other words,by means of the general result given in [6], we can get the desired exact boundary synchronization by groups for system (1.3)–(1.5).

For this purpose, we first transform system (1.3)–(1.5) into a first order hyperbolic system without zero eigenvalues.Let

It is easy to see thatVsatisfies

where

in whichInis the identity matrix of ordern.

By (1.4)–(1.5), onx=0 we have

and, assuming that ?1 is not an eigenvalue ofBin case (1.5c), onx=Lwe have any one of the following boundary conditions:

where

Moreover, by (1.6)–(1.7), the initial data is given by

In 1-D case, instead of discussing separately system (1.3)–(1.4) with different boundary conditions (1.5a), (1.5b) and (1.5c), respectively, by transforming system (1.3)–(1.5) into a first order hyperbolic system(1.8)and(1.10)–(1.11)with different parameters on the boundary conditions onx=L, we can use the theory of controllability and synchronization for first order hyperbolic systems obtained in [6] to uniformly get the boundary controllability and the boundary synchronization for system (1.3)–(1.5).

We first present the well-posedness of system(1.3)–(1.5)in Section 2,then the exact boundary synchronization by groups,corresponding exactly synchronizable states by groups and some necessary conditions will be studied in Sections 3–6, respectively.In Section 7 we give remarks for a more general coupled system.

2 Well-Posedness

Let

In what follows, we always assume that the following conditions ofC0compatibility at the points (t,x)=(0,0) and (0,L) are satisfied, respectively:

Applying[20,Theorem 3.1]to the first order hyperbolic system(1.8)and(1.10)–(1.11), for any givenwe have the following lemma.

Lemma 2.1For any given T >0, for any given initial data V0∈(L2(0,L))2nand any given boundary function H∈H, satisfying the conditions of C0compatibility(2.2)at the points(t,x)=(0,0)and(0,L), respectively, the mixed problem(1.8),(1.10)–(1.11)and(1.13)admits a unique weak solution V=V(t,x)∈(L2(0,T;L2(0,L)))2n, satisfying

and

here and hereafter, c denotes a positive constant.

By Lemma 2.1, noting (1.13), we have the following theorem.

Theorem 2.1Assume that?1is not an eigenvalue of B in case(1.5c).For any givenT >0, for any given initial data∈(H1(0,L)×L2(0,L))nand any given boundary function H∈H, satisfying the conditions of C0compatibility(2.2)at the points(t,x) =(0,0)and(0,L), respectively, problem(1.3)–(1.6)admits a unique weak solution(U,Ut) ∈(L2(0,T;H1(0,L)×L2(0,L)))n, satisfying

3 Exact Boundary Synchronization by p-Groups

We now take a look to the exact boundary synchronization byp-groups for system (1.3)–(1.5).Letp≥1 be an integer,ni(≥2,i= 1,···,p) be any given positive integers, and letLet the state variableU=U(t,x) in system (1.3)–(1.5) be divided intopgroups,and fori= 1,···,p, theith group consists ofnicomponents,Assume that for any given initial data∈(H1(0,L)×L2(0,L))n, there exists aT >0 such that by boundary controlH∈H given by (2.1), the exact synchronization is realized in each group at the timet=T(>0):

Correspondingly,letCpbe the following(n?p)×nfull row-rank matrix of synchronization

an (ni?1)×nifull row-rank matrix fori=1,···,p.We have

and

If system (1.3)–(1.5) is exactly synchronizable byp-groups at the timet=T, then

where?i(i=1,···,p) are given by (3.4), or, equivalently,

Let

and let H be given by (2.1).Applying [6, Lemma 2.7] to system (1.8) and (1.10)–(1.11), for any given∈(H1(0,L)×L2(0,L))nwe have the following lemma.

Lemma 3.1Assume that?1is not an eigenvalue of B in case(1.11c).Let T≥2L.If M= rank(D) =n, then for any given initial data V0(x) ∈(L2(0,L))2ngiven by(1.13), there exists a boundary control H(t)∈H, satisfying

such that system(1.8)and(1.10)–(1.11)is exactly null controllable.

Remark 3.1In case (1.11a), applying [6, Lemma 2.7] to system (1.8)–(1.10) and (1.11a),we can findH′∈(L2(0,T))M, satisfying

such that the system is exactly null controllable.Then,noting the condition ofC0compatibility at the point (0,L) in (2.2), (3.8) still holds.

Noting (1.4) and (1.7), it is easy to get the following lemma.

Lemma 3.2The exact boundary(null)controllability for system(1.8)and(1.10)–(1.11)is equivalent to that for system(1.3)–(1.5).

By Lemmas 3.1–3.2, we immediately get the following theorem.

Theorem 3.1Assume that?1is not an eigenvalue of B in case(1.5c).Let T≥2L.If M=rank(D)=n, then system(1.3)–(1.5)is exactly null controllable for any given initial data∈(H1(0,L)×L2(0,L))n, and the boundary control H(t)∈Hsatisfies(3.8).

Remark 3.2By [6, Lemma 2.7], system (1.8) and (1.10)–(1.11) is in fact exactly controllable at the timet=TifM= rank(D) =nforT≥2L.Thus, by Lemma 3.2, system(1.3)–(1.5) is also exactly controllable under the conditions of Theorem 3.1.

However,the exact boundary null controllability and the exact boundary controllability for system (1.3)–(1.5) is not always equivalent.By applying [20, Remark 3.4] to system (1.8) and(1.10)–(1.11), we have that system(1.3)–(1.4)with boundary condition of Dirichlet type (1.5a)or Neumann type (1.5b) are time reversible; moreover, assuming that 1 is not an eigenvalue ofB, thenG1given by (1.12) is invertible, and system (1.3)–(1.4) with dissipative boundary condition (1.5c) is also time reversible.Thus by [20, Theorem 4.1], we have the following corollary.

Corollary 3.1For system(1.3)–(1.4)with(1.5a)or(1.5b), the exact boundary null controllability and the exact boundary controllability are equivalent.Moreover, if1is not an eigenvalue of B, then we also have the same result for system(1.3)–(1.4)and(1.5c).

Remark 3.3The exact boundary (null) controllability is important for getting the exact boundary synchronization by groups for system (1.3)–(1.5).It is usually complicated to establish the exact boundary (null) controllability especially in higher-dimensional space.In this paper we do it in the one-dimensional space with the aid of the controllability results on first order hyperbolic systems given in [6].It is challenging to deal with higher-dimensional case,system with (1.5c) will be more difficult because of the coupling on the boundary.

Once the exact boundary (null) controllability is established, the exact boundary synchronization byp-groups and corresponding exactly synchronizable states byp-groups can be discussed under the following conditions ofCp-compatibility for the coupling matrices.

Definition 3.1Let ?i(i= 1,···,p)be given by(3.4).Matrix A satisfies the condition of Cp-compatibility if

in which Apis a matrix of order(n?p), and αij(i,j=1,···,p)are constants.

Matrix B satisfies the condition of Cp-compatibility if

in which Bpis matrix of order(n?p), and βij(i,j=1,···,p)are constants.

Theorem 3.2Assume that A satisfies the condition of Cp-compatibility(3.9).Assume furthermore that?1is not an eigenvalue of B, and B satisfies the condition of Cp-compatibility(3.10)in case(1.5c).Ifrank(CpD) =n?p, then there exists a boundary control H(t) ∈H,satisfying

such that system(1.3)–(1.5)is exactly synchronizable by p-groups, whereHis given by(2.1).

ProofUnder the condition ofCp-compatibility (3.9) forAand (3.10) forBin case (1.5c),letW=CpU, whereUsatisfies system (1.3)–(1.5) with (1.6).We have the following reduced system ofW:

and any one of

whereandare given by (3.9) and (3.10), respectively.By Theorem 3.1, the reduced system (3.12)–(3.14) is exactly null controllability when rank(CpD) =n?p.Noting that the exact boundary null controllability of the reduced system (3.12)–(3.14) is equivalent to the exact boundary synchronization byp-groups of the original system(1.3)–(1.5), we immediately get Theorem 3.2.

Remark 3.4Under the conditions ofCp-compatibility for the coupling matrices, Theorem 3.2 and the following results on exactly synchronizable states byp-groups are discussed directly from the viewpoint of wave equations as in [16].

On the other hand, these results can be also built as an application of the results for first order hyperbolic systems obtained in [6] by transforming the exact boundary synchronization byp-groups for system(1.3)–(1.5)into the exact boundary synchronization for system(1.8)and(1.10)–(1.11) with respect to the matrix of synchronizationThe perspective of first order hyperbolic system is practical since no matter whatpis for the exact boundary synchronization byp-groups for system(1.3)–(1.5),it is always exact boundary synchronization for system (1.8) and (1.10)–(1.11) but with different size of C1.

However,for a system of wave equations coupled with velocities,since there is a lack of compactness, the conditions ofCp-compatibility can not be directly derived from both viewpoints.

4 Exactly Synchronizable States by p-Groups

Under the conditions ofCp-compatibility forAandB, by inserting (3.5)into (1.3)–(1.5), it is easy to get the system satisfied by the exactly synchronizable state byp-groups,and similarly to [6, Theorem 4.2, Lemma 4.3], if the system of exactly synchronizable states byp-groups is time reversible, then the attainable set of exactly synchronizable statesat the timet=Tis the whole space (H1(0,L)×L2(0,L))p.

Theorem 4.1Assume that A satisfies the condition of Cp-compatibility(3.9).Assume furthermore that?1is not an eigenvalue of B, and B satisfies the condition of Cp-compatibility(3.10)in case(1.5c).If system(1.3)–(1.5)is exactly synchronizable by p-groups at the time t=T, then, as t≥T, the exactly synchronizable state by p-groupssatisfies

and any one of

for i=1,···,p, where αijand βij(i,j=1,···,p)are given by(3.9)and(3.10), respectively.

Moreover, the attainable set of exactly synchronizable statesat thetime t=T is the whole space(H1(0,L)×L2(0,L))pfor cases(1.5a)and(1.5b).Assume furthermore thatKer(G1)∩Ker(Cp) = {0}, then this result is also true for case(1.5c), where G1is given by(1.12).

Remark 4.1By [16, Proposition 2.21], in order to have Ker(G1)∩Ker(Cp)={0}, we can assume that 1 is not an eigenvalue ofB.

In order to further determine corresponding exactly synchronizable states, let

satisfy that Span{ε1,···,εp}and Ker(Cp)=Span{?1,···,?p}are bi-orthonormal,where?i(i=1,···,p) are given by (3.4).

Theorem 4.2Assume that A satisfies the condition of Cp-compatibility(3.9).Assume furthermore that?1is not an eigenvalue of B, and B satisfies the condition of Cp-compatibility(3.10)in case(1.5c).Define D byKer(DT)=Span{ε1,···,εp}.Then we have

and system(1.3)–(1.5)is exactly synchronizable by p-groups.

Moreover, ifSpan{ε1,···,εp}is an invariant subspace of AT, then the exactly synchroniz-able state by p-groupsof system(1.3)–(1.4)with(1.5a) (resp.(1.5b))is independent of applied boundary controls.IfSpan{ε1,···,εp}is a common invariant subspace of ATand BT, then the exactly synchronizable state by p-groupsof system(1.3)–(1.4)with(1.5c)is also independent of applied boundary controls.

ProofSince Ker(DT) = Span{ε1,···,εp}, noting Span{ε1,···,εp} and Ker(Cp) are biorthonormal, by [16, Propositions 2.5, 2.11], we immediately get (4.5), then, by Theorem 3.2,system (1.3)–(1.5) is exactly synchronizable byp-groups.

Assume that Span{ε1,···,εp} is an invariant subspace ofATandBT, respectively, noting(3.9)–(3.10) and that Span{ε1,···,εp} and Ker(Cp) = Span{?1,···,?p} are bi-orthonormal,it is easy to check thatwhereαijandβij(i,j= 1,···,p) are given by (3.9) and (3.10), respectively.Letφi= (U,εi)(i= 1,···,p),whereU=U(t,x) is the solution to system (1.3)–(1.5), which realizes the exact boundary synchronization byp-groups at the timet=T.Then, multiplyingεi(i= 1,···,p) on system(1.3)–(1.5), fori=1,···,pwe have

and any one of

with the initial data

Noting (3.5), since Span{ε1,···,εp} and Ker(Cp) = Span{?1,···,?p} are bi-orthonormal, we have

Thus the exactly synchronizable state byp-groupsof system(1.3)–(1.5)is entirely determined by the solution to problem (4.6)–(4.9), which is independent of applied boundary controls.

Inversely to Theorem 4.2, we have the following theorem.

Theorem 4.3Assume that A satisfies the condition of Cp-compatibility(3.9).Assume furthermore that?1is not an eigenvalue of B, and B satisfies the condition of Cp-compatibility(3.10)in case(1.5c).Assume finally that system(1.3)–(1.5)is exactly synchronizable by pgroups under the conditionrank(CpD) =n?p.Let U be the solution to problem(1.3)–(1.6), which realizes the exact boundary synchronization by p-groups at the time t=T.If(U,εi)with εi(i= 1,···,p)given by(4.4)are independent of applied boundary controls, thenSpan{ε1,···,εp}is an invariant subspace of ATfor system(1.3)–(1.4)with(1.5a) (resp.(1.5b)), whileSpan{ε1,···,εp}is a common invariant subspace of ATand BTfor system(1.3)–(1.4)with(1.5c).Moreover, we have εi∈Ker(DT)(i= 1,···,p).In particular, if D satisfies(4.5), then we haveKer(DT)=Span{ε1,···,εp}.

ProofWe only give a sketch of the proof, which is similar to that of [6, Theorem 4.8].

LetU=U(t,x) be the solution to system (1.3)–(1.5), which realizes the exact boundary synchronization byp-groups at the timet=T.Takingby Theorem 2.1, the linear mappingF:H→(U,Ut) is continuous from H to ((0,+∞;H1(0,L)×L2(0,L)))n,where H is given by (2.1).By linearity, the Frchet derivative

satisfies also system (1.3)–(1.5) withSince (U,εi)(i= 1,···,p) are independent of applied boundary controls, we have

for any givenH∈H.

Since (ε1,···,εp,CTp) constitutes a basis in Rn, we have

whereaijandbij(i,j= 1,···,p) are constants,PiandQi∈Rn?p(i= 1,···,p).Then,multiplyingεi(i=1,···,p) on system (1.3) of, it follows from (4.12)–(4.13) that

Theorem 4.4Assume that A satisfies the condition of Cp-compatibility(3.9).Assume furthermore that?1is not an eigenvalue of B, and B satisfies the condition of Cp-compatibility(3.10)in case(1.5c).Assume finally that system(1.3)–(1.5)is exactly synchronizable by pgroups under conditionrank(CpD) =n?p with H(t)satisfying(3.11).Then the exactly synchronizable state by p-groupssatisfies the following estimate:

for i=1,···,p, where φi(i=1,···,p)satisfy problem(4.6)–(4.9).

ProofThe proof is similar to that of [9, Theorem 8.4], we only give a sketch here.LetUbe the solution to problem (1.3)–(1.6), which realizes the exact boundary synchronization byp-groups at the timet=T, and letzi= (εi,U)(i= 1,···,p) withεi(i= 1,···,p) given by (4.4).It is easy to prove thatfori= 1,···,p, whereαijandβij(i,j= 1,···,p) are given by (3.9) and (3.10), respectively.Then there existPiandQi∈Rn?p(i= 1,···,p), such thatandfori=1,···,p.Thus we have

and

Multiplyingεi(i= 1,···,p) on both sides of problem (1.3)–(1.6), noting (4.17)–(4.18), fori=1,···,p, we have

and any one of

with

Letyi=zi?φi(i= 1,···,p), whereφi(t,x)(i= 1,···,p) is the solution to problem(4.6)–(4.9).Fori=1,···,p, we have

and any one of

with

According to the theory of first order hyperbolic systems, by [20, Theorem 3.3], with H given by (2.1) we have the following estimate fori=1,···,p:

in cases (4.25a) and (4.25b); while

By Theorem 2.1, noting (3.11), we have

and

Therefore, it follows from (3.11), (4.27)–(4.28) and (4.30)–(4.31) that

On the other hand,noting(3.5)and that Span{ε1,···,εp}and Ker(Cp)=Span{?1,···,?p}are bi-orthonormal, we have

Substituting (4.33) into (4.32), we get (4.16).

Remark 4.2[4] and [18] discussed the exact boundary synchronization by groups for the coupled system of 1-D wave equations(1.1)with various types of boundary conditions but in the framework of classical solutions.It was proved that whenDin (1.5) is the identity matrix, we can find(n?p)boundary controls so that system(1.1)and(1.4)–(1.5)is exactly synchronization byp-groups.In this paper we extend the corresponding result to system (1.3)–(1.5) in the framework of weak solutions, for which not only corresponding exactly synchronizable states byp-groups are further determined and estimated, but also the necessary rank conditions of Kalman type can be obtained in Section 6.

5 Necessity of the Conditions of Cp-Compatibility

Assume that system(1.3)–(1.5)is exactly synchronizable byp-groups,namely,we have(3.5),then, multiplyingCpon (1.3), we have

where?i(i=1,···,p)are given by(3.4).IfCpA?i=0(i=1,···,p),then,noting(3.4),we have the condition ofCp-compatibility (3.9) forA.Otherwise,,···,are linearly dependent,without loss of generality, we may assume that

whereδi(i=1,···,p?1) are constants.

Let

be an invertible matrix, and let

and

On the other hand, noting (1.7) and (5.3), let

Noting (5.4)–(5.6) and (5.8), ast≥Twe have

namely, by an invertible linear transformation, one group of the components of system (1.8)and(1.10)–(1.11)is in fact exactly null controllable,while any one of the other groups is exactly synchronizable.From the perspective of first order hyperbolic systems, we should exclude this situation (see [6, Theorem 6.1]).Hence, we have the following theorem.

Theorem 5.1If system(1.3)–(1.5)is exactly synchronizable by p-groups,but the derivatives of the exactly synchronizable states with respect to t,are not linearly dependent,then A satisfies the condition of Cp-compatibility(3.9).

In particular, whenp=1, by [6, Theorems 6.1–6.2] we have the following corollary.

Corollary 5.1If system(1.3)–(1.5)is exactly synchronizable but not exactly null controllable, then A must satisfy the condition of C1-compatibility(3.9)with p=1, moreover, B must satisfy the condition of C1-compatibility(3.10)with p=1in case(1.5c), where C1is given by(3.2).

Remark 5.1The discussions on the exact boundary synchronization byp-groups and corresponding exactly synchronizable states byp-groups for system (1.1) and for system (1.3) are quite similar.In the study of the necessity of the conditions ofCp-compatibility for the coupling matrixAin Theorem 5.1, noting the form of couplingAUtin system (1.3), we need to check the linear independence ofIn comparison, for system (1.1) with couplingAU, we need to check the linear independence of, instead.Similarly for that ofBon the boundary.

Specifically,for system(1.1),the coupling of displacementsAUcan be regarded as a compact perturbation.Because of this compactness,we can prove thatis linearly independent in the domain or on the boundary,then we get the necessity of the conditions ofCp-compatibility for the coupling matrices (see more details in [22]).However, the coupling of velocitiesAUtin system (1.3) is not compact, we can not prove the linear independence ofin the same way, the necessity of the conditions ofCp-compatibility for bothAandBfor the exact boundary synchronization byp-groups for system (1.3)–(1.5) is still an open problem.

6 Kalman’s Criterion

In this section, we give some necessary conditions for the coupling matrices.

Theorem 6.1If system(1.3)–(1.4)with(1.5a) (resp.(1.5b))is exactly null controllable,then we necessarily have

ProofThe result can be proved by applying [6, Theorem 7.1] to system (1.8), (1.10) and(1.11a)(resp.(1.11b)).We only need to point out that the rank condition(6.1)holds if and only ifATdoesn’t have any invariant subspace that is contained in Ker(DT) (see [16, Proposition 2.12]).

Corollary 6.1Assume that A satisfies the condition of Cp-compatibility(3.9).If system(1.3)–(1.4)with(1.5a) (resp.(1.5b))is exactly synchronizable by p-groups, then we necessarily have

ProofUnder the condition ofCp-compatibility (3.9) forA, the exact boundary synchronization byp-groups for system(1.3)–(1.4)with(1.5a)(resp.(1.5b))can be equivalently transformed into the exact boundary null controllability for the reduced system (3.12)–(3.13) with(3.14a)(resp.(3.14b)).Then,applying Theorem 6.1 to system(3.12)–(3.13)with(3.14a)(resp.(3.14b)), we have

Noting (3.9), (6.2) follows from (6.3).

For system (1.3)–(1.4) with (1.5c), the Kalman’s criterion, which is similar to that in [17],is different because there is another coupling matrixBon the boundary.Let

be ann×Mmatrix for any given non-negative integersp,q,···,r,s≥0, and let

be an enlarged matrix by the matrices R(p,q,···,r,s)for all possible (p,q,···,r,s).

Lemma 6.1(see [17, Lemma 2.1]) Ker(RT)is the largest subspace of all the subspaces which are contained inKer(DT)and invariant for ATand BT.

Lemma 6.2Assume that?1is not an eigenvalue of the coupling matrix B.For any given k×n matrix C, there exists a matrix of order k, such that

if and only if there exists a matrix of order k, such that

where G1is given by(1.12).

ProofAssume first that (6.5) holds, then we have

Since ?1 is not an eigenvalue ofB, by[16,Proposition 2.21],?1 is not an eigenvalue ofB,then it follows from (6.7) that

Inversely,assume that(6.6)holds.We claim that ?1 is not an eigenvalue ofG1.Otherwise,noting (1.12), there exists a non-trivialξ∈Rn, such that

then we have (In?B)ξ= ?(In+B)ξ,which leads toξ= ?ξ, a contradiction.Thus, by [16,Proposition 2.21], ?1 is not an eigenvalue of.Then, similarly to (6.8), we have

Noting (1.12), we have

Theorem 6.2Assume that both?1and1are not eigenvalues of B.If system(1.3)–(1.4)with(1.5c)is exactly null controllable, then we necessarily haverank(R)=n.

ProofAssume by contradiction that rank(R)=n?dwithd>0, then dim Ker(RT)=d.By Lemma 6.1,Ker(RT)is contained in Ker(DT)and invariant forATandBT.Since Ker(RT)is invariant forBT, by Lemma 6.2, Ker(RT) is also invariant forGT1.Noting that 1 is not an eigenvalue ofB,G1is invertible.Moreover, since Ker(RT) is contained in Ker(DT), it is easy to see that Ker(RT)is contained in Ker(DT(In+B)?T).Thus the desired result can be proved by applying [6, Theorem 7.1] to system (1.8), (1.10) and (1.11c).

Corollary 6.2Assume that A and B satisfy the conditions of Cp-compatibility(3.9)and(3.10), respectively.Assume furthermore that?1is not an eigenvalue of B, and1is not an eigenvalue of given by(3.10).If system(1.3)–(1.4)with(1.5c)is exactly synchronizable by p-groups, then we necessarily have

ProofUnder the conditions ofCp-compatibility forAandB, the exact boundary synchronization byp-groups for system(1.3)–(1.4)with(1.5c)can be equivalently transformed into the exact boundary null controllability for the reduced system (3.12)–(3.13) with (3.14c).Then,applying Theorem 6.2 to this reduced system(3.12)–(3.13)with(3.14c)and noting(3.9)–(3.10),we have (6.13).

Remark 6.1By [16, Proposition 2.21], if 1 is not an eigenvalue ofB, then 1 is not an eigenvalue of

7 Remar k

The preceding method can be used to consider the corresponding problems for the following 1-D coupled system of wave equations

with the same boundary conditions (1.4) and (1.5) and the same initial data (1.6), where bothAandare matrices of ordernwith real constant elements.

LetVbe defined by (1.7).We have still system (1.8) and (1.10)–(1.11) with

Suppose furthermore that the matrixalso satisfies the condition ofCp-compatibility

in whichis a matrix of order(n?p), and(i,j=1,···,p)are constants, we can similarly get all the results mentioned above, in which the exactly synchronizable state byp-groupssatisfies

with(4.2)–(4.3)fori=1,···,p,and the exactly synchronizable state byp-groupssatisfies the estimate (4.16), whereφi(i=1,···,p) satisfy the system

with (4.7)–(4.9), whereαij,βijand(i,j= 1,···,p) are given by (3.9), (3.10) and (7.3),respectively.