Exact Boundary Controllability for a Coupled System of Wave Equations with Neumann Boundary Controls?

Tatsien LI Bopeng RAO

(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

1 Introduction

Since the pioneering work of Russell[25],and the celebrated paper of Lions[19]in which a systematic approach,the so-called Hilbert Uniqueness Method,was developed,the control of wave equations has undergone a significant progress.In the last decades,the control of systems has become a very challenging issue.The aim of this paper is to investigate the exact boundary controllability and the non-exact boundary controllability for the following coupled system of wave equations with Neumann boundary controls:

where??Rnis a bounded domain with smooth boundaryΓ=Γ1∪Γ0such that?νdenotes the outward normal derivative on the boundary,A is a matrix of order N with constant elements,D is a full column-rank matrix of order N×M(M≤N)with constant elements,denote the state variables and the boundary controls,respectively.Here,the introduction of the boundary control matrix D makes the discussion on the control problem more flexible.

Let us denote

whereis the subspace of H1(?)composed of functions with null trace on Γ0.We will show the exact boundary controllability of(1.1)for any given initial datavia the HUM approach.To this end,we first establish the following theorem.

Theorem 1.1There exist positive constants T>0 and C>0 independent of initial data,such that the following observability inequality

holds for all solutionsto the corresponding adjoint problem:

wheredenotes the adjoint variables,H?1is the dual space of H1,and the initial data(Φ0,Φ1)is taken in a subspace F ? H0×H?1.

Recall that without any assumption on the coupling matrix A,the usual multiplier method can not be applied directly.The absorption of coupling lower terms is a delicate issue even for a single wave equation(see[9,13]).In order to deal with the lower order terms,we propose a method based on the compactness-uniqueness argument that we formulate in the following lemma.

Lemma 1.1Let F be a Hilbert space endowed with the p-norm.Assume that

where⊕denotes the direct sum and L is a finite co-dimensional closed subspace in F.Assume that q is another norm in F such that the projection from F into N is continuous with respect to the q-norm.Assume furthermore that

Then there exists a positive constant C>0 such that

Following the above lemma,we have to first show the observability inequality for the initial data with higher frequencies in L.In order to extend it to the whole space F,it is sufficient to verify the continuity of the projection from F into N for the q-norm.In many situations,it often occurs that the subspaces N and L are mutually orthogonal with respect to the q-inner product,and this is true in the present case.This new approach turns out to be particularly simple and efficient for getting the observability of some distributed systems with lower order terms.

As for the Dirichlet boundary problem(see[15–16]),we show the non-exact boundary controllability for the Neumann boundary problem(1.1)in the case of fewer boundary controls,i.e.,M

Let us comment the related literatures.The exact boundary controllability and the approximate boundary controllability for a coupled system of wave equations with Dirichlet boundary controls were established by Li and Rao in[14–16].Moreover,in the case of fewer boundary controls,the authors also obtained various results on the exact boundary synchronization and the approximate boundary synchronization for the same system.Using Carleman estimates,Duyckaerts,Zhang and Zuazua studied in[7]the observability inequality for a coupled system of wave equations with Dirichlet boundary condition by means of internal or boundary observation.The optimality of the observability inequality was proved in even space dimensions.In a similar work[27],Zhang and Zuazua established the sharp observability inequality for Kirchhoff plate systems in any space dimensions.In a recent work[6],Dehman,Le Rousseau and Léautaud established the controllability of two coupled wave equations on a compact manifold with only one local distributed control.The optimal time of controllability and the controllable spaces were given in the cases that the waves propagate with the same speed or with different speeds.Using the Riemannian geometry method,Yao established in[26]the controllability of wave equation with variable coefficients for Dirichlet or Neumann boundary condition.We finally mention the work of Hu,Ji and Wang[8]for the exact boundary controllability of one space-dimensional quasilinear system of wave equations with various boundary controls.

The non-exact boundary controllability for a coupled system of wave equations depends on the level of energy and the property of controllability.In fact,if the components of initial data are allowed to have different levels of energy,then the exact boundary controllability for a system of two wave equations was established by means of only one boundary control in[1,21,24],or more recently,for a cascade system of N wave equations by means of only one boundary control in[2].In contrast with the exact boundary controllability,the approximate boundary controllability for a coupled system of wave equations is more flexible with respect to the number of boundary controls.It was recently shown in[17]that for Dirichlet boundary controls,this property could be characterized by means of the famous Kalman’s criterion on the rank of an enlarged matrix composed of the coupling matrix A and the boundary control matrix D.

Differently from the hyperbolic systems,the exact boundary null controllability of coupled systems of parabolic equations can be realized in the case of fewer controls for the initial data with the same level of energy.There are a number of works on this topic.We only quote[3]and the references therein for the null controllability of coupled systems of parabolic equations with a local distributed control or with a boundary control.

The paper is organized as follows.In Section 2,we give the proof of the basic lemma of compact perturbation.Section 3 is devoted to the proof of Theorem 1.1.In order to clarify the idea,we divide the proof into several propositions.In Section 4,we prove the exact boundary controllability(see Theorem 4.1)and the non-exact boundary controllability(see Theorem 4.2)for a system of wave equations with Neumann boundary condition.

2 Proof of Lemma 1.1

Assume that(1.7)fails,then there exists a sequence zn∈F such that

Using(1.5),we write zn=xn+ynwith xn∈N and yn∈L.Since the projection from F into N is continuous with respect to the q-norm,there exists a positive constant c>0 such that

Since N is of finite dimension,we may assume that there exists x∈N such thatin N.Then,since the second relation of(2.1)means thatin F,we deduce thatin L for the p-norm.Therefore,we get x∈L∩N,which leads to x=0.Then,we have

Then using(1.6),we get a contradiction:

The proof is then complete.

Remark 2.1Noting that L is not necessarily closed with respect to the weaker q-norm,so,a priori,the projectionis not continuous with respect to the q-norm(see[5]).

3 Proof of Theorem 1.1

Theorem 1.1 will be proved at the end of this section.We first start with some useful preliminary results.

Let? ? Rnbe a bounded open set with smooth boundaryΓ.Letbe a partition ofΓsuch thatThroughout this paper,we assume that? satisfies the usual geometric control condition(see[4]).More precisely,assume that there exists x0∈Rn,such that setting m=x?x0,we have

where(·,·)denotes the inner product in Rn.

Let

We consider the following homogenous adjoint problem:

Let

We define the linear unbounded operator?? in H0by

Clearly,?? is a densely defined self-adjoint and coercive operator with a compact resolvent in H0.Then we can define the power operatorfor any given s∈R(see[18]).Moreover,the domainendowed with the normbeing the norm of H0,is a Hilbert space,and its dual space with respect to the pivot spaceIn particular,we have

Then we formulate(3.3)into an abstract evolution problem:

Clearly,the problem(3.7)generates a C0-semigroup in the space Hs× Hs?1.Moreover,we have the following result(see[18,Chapter III-8]and[23,Chapter III]).

Proposition 3.1For any given initial datawith s∈ R,the problem(3.7)admits a unique weak solution in the sense of C0-semigroups,such that

Now let embe the normalized eigenfunction defined by

where the sequence of positive terms{μm}m≥1is increasing so thatClearly,{em}m≥1is a Hilbert basis in L2(?).

For each m≥1,we define the subspace Zmby

It is clear that the subspaces Zm(m≥1)are invariant with respect to AT.Moreover,for any given integersand any given vectors α,β ∈ RN,we have

Then the subspaces Zm(m≥1)are mutually orthogonal in the Hilbert space Hswith any given s∈R and in particular,we have

Proposition 3.2LetΦ be the solution to the problem(3.7)with the initial data(Φ0,Φ1)∈H1×H0,which satisfies an additional condition:

for T>0 large enough.Then,we haveΦ0≡Φ1≡0,i.e.,Φ≡0.

ProofBy Schur’s theorem,we may assume that A is an upper triangular matrix so that the problem(3.7)with the additional condition(3.13)can be rewritten as

for k=1,···,N.Then using Holmgren’s uniqueness theorem(see[20,Chapter I-8]),there exists a positive constant T>0 large enough and independent of the initial datasuch that?(1)≡ 0.Then,we get successively ?(k)≡ 0 for k=1,···,N.The proof is then complete.

Letσdenote the Euclidian norm of the matrix A.Then we have the following energy estimates:

ProofFirst,a straightforward computation yields

Then,using(3.12),we get

It follows that

Therefore,the functionis increasing,while,the functionis decreasing,which implies(3.16).The proof is then complete.

Proposition 3.4There exist an integer m0≥1 and positive constants T>0 and C>0 independent of initial data,such that the following observability inequality:

ProofFirst we write(3.7)as

for=1,2,···,N.Then,multiplying the k-th equation of(3.21)by

and integrating by parts over the domain[0,T]×?,we get easily the following identities(see[9,Chapter III],[20,Chapter III]):

Noting the geometrical control condition(3.1),we have

then,it follows from(3.23)that

Taking the summation of(3.25)with respect to k=1,···,N,we get

where M is the vector composed of

Next,we estimate the last two terms on the right-hand side of(3.26).First,it follows from(3.22)that

where R= ∥m∥∞is the diameter of? and

Similarly,we have

Thus,setting

and noting(3.16),we get

Inserting(3.29)and(3.32)into(3.26)gives

Thus,we have

provided that m0is so large that

Now,integrating the inequality on the left-hand side of(3.16)over[0,T],we get

then,noting(3.31),we get

Thus,it follows from(3.34)and(3.37)that

which is guaranteed by the following choice(see(3.31),(3.35)and(3.39)):

The proof is then complete.

Proposition 3.5There exist an integer m0≥1 and positive constants T>0 and C>0 independent of initial data,such that the following observability inequality:

ProofNoting that Ker(??+AT)is of finite dimension,there exists an integer m0≥ 1 so large that

Let

We have

Next letΨ be the solution of the problem(3.7)with the initial data(Ψ0,Ψ1)given by(3.45).We have

By the well-posedness,we get

Then,applying(3.20)toΨ,we get

Thus,using(3.46)and(3.48),we get immediately(3.41).The proof is finished.

Finally,we give the proof of Theorem 1.1.

We next define

with

Clearly,N is a finite-dimensional subspace and L is a closed subspace in F.In particular,the observability inequality(3.41)can be extended to all initial data(Φ0,Φ1)in the whole subspace L.

Now we introduce the q-norm by

By(3.11),the subspaces(Zm×Zm)are mutually orthogonal infor all m≥1,then the subspace N is an orthogonal complement of L in H0×H?1.In particular,the projection from F into N is continuous with respect to the q-norm.On the other hand,since the observability inequality(3.41)holds for all initial data(Φ0,Φ1)in the subspace L,the condition(1.6)is verified.Then,applying Lemma 1.1,we get the inequality(1.7),which precisely means that(3.41)can be extended to all(Φ0,Φ1)∈ F,we then get(1.3).The proof of Theorem 1.1 is now completed.

Remark 3.1Without any additional assumptions on the coupling matrix A,the adjoint system(3.7)is not conservative and the usual multiplier method can not be applied directly.However,since each subspace Zmis invariant with respect to the matrix AT,for any given initial data(Φ0,Φ1)∈ Zm×Zm,the corresponding solution Φ of(3.7)still lies in the subspace Zm.Then because of the identity(3.12),the coupling termis negligible comparing withTherefore,we first expect the observability inequality(3.15)only for the initial data⊕(Φ0,Φ1)with higher frequencies lying in the sub-linear hullwith an integer m0≥1 large enough.We next extend it to the closure of the whole linear hullby an argument of compact perturbation as shown in Lemma 1.1.

Remark 3.2The compactness-uniqueness arguments are frequently used in the study of the observability of distributed parameter systems.It turns out that this method is particularly simple and efficient for dealing with some systems with lower order terms.A natural formulation is to consider the problem as a compact perturbation of a skew-adjoint operator(see[10,22]).This approach requests that the eigen-system of the underlying system forms a Riesz basis in the energy space.Since the Riesz basis is not stable even for the compact perturbation,this may cause serious problems in the application.By contrast,the method proposed here does not require any spectral conditions on the underlying system.In particular,instead of Riesz basis property,we assume only that the projection from F into N is continuous with respect to the q-norm.Moreover,it often occurs that the subspaces N and L are mutually orthogonal with respect to the q-inner product,hence,the continuity of the projection from F into N is much easier to be checked than the Riesz basis property.

Remark 3.3The present method can be generalized to a larger class of problems,for example,to the coupled system of wave equations with different speeds of propagation,which will be considered in a forthcoming work.

Remark 3.4As to the optimality of controllability time T,we refer to[6–7,27]for some related discussions.

4 Exact Boundary Controllability with Neumann Boundary Controls

Let D be a boundary control matrix of order N×M(M≤N)and denote

We consider the following inhomogeneous problem:

We will first show the exact boundary controllability of(4.2)by a standard application of the HUM method of Lions[20].We next show the non-exact boundary controllability in the case of fewer boundary controls(M

Obviously,we have

On the other hand,by(1.3)and the trace embeddingfor all,we get the following continuous embedding:

Multiplying the equation in(4.2)by a solutionΦof the adjoint problem(3.7)and integrating by parts,we get

Taking H0as the pivot space and noting(4.4),(4.5)can be written as

Definition 4.1U is a weak solution to the problem(4.2),if

such that the variational equation(4.6)holds for any given

Proposition 4.1For any givenand any givenwiththe problem(4.2)admits a unique weak solution U.Moreover,the linear map

is continuous with respect to the corresponding topologies.

ProofDefine the linear form

By the definition(3.51)of the p-norm and the continuous embedding(4.4),the linear form Ltis bounded infor any given t ≥ 0.Let Stbe the semi-group associated to the homogeneous problem(3.7)on the Hilbert spacewhich is an isomorphism onThe composed linear formis bounded inThen,by Riesz-Frêchet’s representation theorem,there exists a unique elementsuch that

for any given.Since

we get(4.6)for any givenMoreover,we have

Then,by a classic argument of density,we get the regularity(4.7).The proof is thus complete.

Definition 4.2The problem(4.2)is exactly null controllable at the time T in the space H0×H1,if for any given(U1,U0)∈ H0×H1,there exists a boundary control H ∈ L2(0,T;L2(Γ1))Msuch that the problem(4.2)admits a unique weak solution U satisfying the final condition

and the continuous dependance

Theorem 4.1Assume that M=N.Then there exists a positive constant T>0 such that the problem(4.2)is exactly null controllable at the time T for any given initial data(U1,U0)∈H0×H1.

ProofLetΦ be the solution to the adjoint problem(3.7)inwith.Let

Because of the first inclusion in(4.4),we have H ∈ L2(0,T;L2(Γ1))N.Then by Proposition 4.1,the corresponding backward problem

admits a unique weak solution U with(4.7).Accordingly,we define the linear map

Clearly,Λis a continuous map from

Next,using(4.6),it follows that

whereΨ is the solution to the problem(3.7)with the initial data(Ψ0,Ψ1).It follows that

By definition,is dense in F,then the linear form

can be continuously extended to F,so thatMoreover,we have

Once again,by the density of Hs×Hs?1in F,the linear map Λ can be continuously extended to F,so thatΛ becomes a continuous linear map from F to F′.Therefore,the symmetric bilinear formis continuous and coercive in the product space F × F.By Lax and Milgram’s lemma,Λ is an isomorphism from F onto F′.Then for any giventhere exists an element(Φ0,Φ1)∈ F,such that

This is precisely the exact boundary null controllability of the problem(4.2)for any given initial data(U1,?U0)∈ F′,in particular,for any given initial data(U1,?U0)∈ H0×H1? F′,because of the second inclusion in(4.4).

Finally,from(4.13)and(4.16),we deduce the continuous dependance

The proof is thus complete.

In the case of fewer boundary controls,we have the following negative result.

Theorem 4.2Assume that M0.

ProofSince M

whereθ∈ D(?)is arbitrarily given.If the problem(4.2)is exactly null controllable,we can find a boundary control H with the least norm such that

Then,by Proposition 4.1,we have

Now,taking the inner product of e with(4.2)and noting?=(e,U),we get

Noting(4.3),by the well-posedness,we get

for any givenθ∈ D(?).Choosingwe have 2?s>1.This gives a contradiction.The proof is then complete.

Remark 4.1As shown in the proof of Theorem 4.1,a weaker regularity such as(U′,U)∈C0([0,T];H?1×H0)is sufficient to make sense to the value(U′(0),U(0)),therefore,sufficient for proving the exact boundary controllability.At this stage,it is not necessary to pay much attention to the regularity of the weak solution with respect to the space variable.However,in order to establish the non-exact boundary controllability in Theorem 4.2,this regularity becomes indispensable for the argument of compact perturbation.In the case of Dirichlet boundary controls,the weak solution has the same smoothness as the controllable initial data.This regularity yields the non-exact boundary controllability in the case of fewer boundary controls(see[14–16]).But for Neumann boundary controls,the direct inequality is much weaker than the inverse inequality.For example,in Proposition 4.1,we can get only(U′,U)∈for anywhile,the controllable initial data(U1,U0)lies in the space H0×H1.Even though this regularity is not sharp in general(see[11,Theorem 1.1]and[12,Main Theorems 1.2–1.3]),it is already sufficient for the proof of the non-exact boundary controllability of the problem(4.2).

Remark 4.2As for the case of Dirichlet boundary controls discussed in[14–16],we have shown in Theorems 4.1–4.2 that for Neumann boundary controls the problem(4.2)is exactly null controllable if and only if the boundary controls have the same number as the state variables or the wave equations.Of course,the non-exact boundary controllability is valid only in the framework that all the components of the initial data are in the same energy space.In fact,if the components of the initial data are allowed to have different levels of finite energy,then we can realize the exact boundary controllability by means of only one boundary control for a system of two wave equations(see[1,21]),or more generally,for a cascade system of N wave equations(see[2]).On the other hand,in contrast with the exact boundary controllability,the approximate boundary controllability is more flexible with respect to the number of boundary controls,and is closely related to the so-called Kalman’s criterion on the rank of an enlarged matrix composed of the coupling matrix A and the boundary control matrix D(see[15,17]).

AcknowledgementsPart of the work was done during the visit of the second author at the Laboratoire International Associé Sino-Fran?cais de Mathématiques Appliquées(LIASFMA)and the School of Mathematical Sciences of Fudan University during June–August 2014.He would like to thank their hospitality and support.

The authors are very grateful to Professor Xu Zhang for bringing their attention to the references[11,12]and for the valuable discussions on several occasions,and would like also to thank the referees for their very valuable comments and remarks,which were greatly appreciated to improve the presentation of the paper.

[1]Alabau-Boussouira,F.,A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems,SIAM J.Control Optim.,42,2003,871–904.

[2]Alabau-Boussouira,F.,A hierarchic multi-level energy method for the control of bidiagonal and mixed ncoupled cascade systems of PDEs by a reduced number of controls,Adv.Diff.Equ.,18,2013,1005–1072.

[3]Ammar Khodja,F.,Benabdallah,A.,González-Burgos,M.and de Teresa,L.,Recent results on the controllability of linear coupled parabolic problems:A survey,Math.Control Relat.Fields,1,2011,267–306.

[4]Bardos,C.,Lebeau,G.and Rauch,J.,Sharp sufficient conditions for the observation,control,and stabilization of waves from the boundary,SIAM J.Control Optim.,30,1992,1024–1064.

[5]Brezis,H.,Functional Analysis,Sobolev Spaces and Partial Differential Equations,Springer-Verlag,New York,2011.

[6]Dehman,B.,Le Rousseau,J.and Léautaud,M.,Controllability of two coupled wave equations on a compact manifold,Arch.Ration.Mech.Anal.,211,2014,113–187.

[7]Duyckaerts,T.,Zhang,X.and Zuazua,E.,On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials,Ann.Inst.H.Poincaré Anal.Non Linéaire,25,2008,1–41.

[8]Hu,L.,Ji,F.Q.and Wang,K.,Exact boundary controllability and exact boundary observability for a coupled system of quasilinear wave equations,Chin.Ann.Math.,Ser.B,34(4),2013,479–490.

[9]Komornik,V.,Exact Controllability and Stabilization,The Multiplier Method,Masson,Paris,1994.

[10]Komornik,V.and Loreti,P.,Observability of compactly perturbed systems,J.Math.Anal.Appl.,243,2000,409–428.

[11]Lasiecka,I.and Triggiani,R.,Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and data supported away from the boundary,J.Math.Anal.Appl.,141,1989,49–71.

[12]Lasiecka,I.and Triggiani,R.,Sharp regularity for mixed second-order hyperbolic equations of Neumann type,I.L2nonhomogeneous data,Ann.Mat.Pura Appl.,157,1990,285–367.

[13]Lasiecka,I.,Triggiani,R.and Zhang,X.,Nonconservative wave equations with unobserved Neumann B.C.:Global uniqueness and observability in one shot,Differential geometric methods in the control of partial differential equations,Contemp.Math.,268,A.M.S.,Providence,RI,2000,227–325.

[14]Li,T.-T.and Rao,B.P.,Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls,Chin.Ann.Math.,Ser.B,34(1),2013,139–160.

[15]Li,T.-T.and Rao,B.P.,Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls,Asym.Anal.,86,2014,199–226.

[16]Li,T.-T.and Rao,B.P.,A note on the exact synchronization by groups for a coupled system of wave equations,Math.Methods Appl.Sci.,38,2015,241–246.

[17]Li,T.-T.and Rao,B.P.,Criteria of Kalman’s type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls,C.R.Acad.Sci.Paris,Ser.I,353,2015,63–68.

[18]Lions,J.-L.and Magenes,E.,Problèmes aux Limites Non Homogènes et Applications,Vol.1,Dunod,Paris,1968.

[19]Lions,J.-L.,Exact controllability,stabilization and perturbations for distributed systems,SIAM Rev.,30,1988,1–68.

[20]Lions,J.-L.,Contr?labilité Exacte,Perturbations et Stabilisation de Systes Distribués,Vol.1,Masson,Paris,1988.

[21]Liu,Z.and Rao,B.P.,A spectral approach to the indirect boundary control of a system of weakly coupled wave equations,Discrete Contin.Dyn.Syst.,23,2009,399–414.

[22]Mehrenberger,M.,Observability of coupled systems,Acta Math.Hungar.,103,2004,321–348.

[23]Pazy,A.,Semigroups of Linear Operators and Applications to Partial Differential Equations,Springer-Verlag,New York,1983.

[24]Rosier,L.and de Teresa,L.,Exact controllability of a cascade system of conservative equations,C.R.Math.Acad.Sci.,Paris,349,2011,291–296.

[25]Russell,D.L.,Controllability and stabilization theory for linear partial differential equations:Recent progress and open questions,SIAM Rev.,20,1978,639–739.

[26]Yao,P.F.,On the observability inequalities for exact controllability of wave equations with variable coefficients,SIAM J.Control Optim.,37,1999,1568–1599.

[27]Zhang,X.and Zuazua,E.,A sharp observability inequality for Kirchhoff plate systems with potentials,Comput.Appl.Math.,25,2006,353–373.