Global Exact Boundary Controllability for General First-Order Quasilinear Hyperbolic Systems?

Cunming LIU Peng QU

1 Introduction

Consider the following first-order quasilinear hyperbolic system with one space variable:

whereu=(u1,···,un)T∈Uis the unknown vector function of(t,x),taking values in a bounded and connected domainU?(for convenience,all equations hold foru∈throughout this paper,unless otherwise indicated),andA(u)=(aij(u))is aC2smoothn×nmatrix function.By hyperbolicity,the coefficient matrixA(u)possessesnreal eigenvaluesλi(u)(i=1,···,n)a complete set of left eigenvectors

satisfying

and

and a complete set of right eigenvectors

satisfying

and

Without loss of generality,one may assume that

whereδijis the Kronecker symbol.Suppose thatλi(u),li(u)andri(u)(i=1,···,n)are alsoC2smooth with respect tou.Since,generally speaking,first-order quasilinear hyperbolic systems with zero eigenvalues do not have exact boundary controllability(see[12]),we assume that the system possesses no zero eigenvalue,namely,

Besides,for the simplicity of analyzing simple wave solutions,we assume the following condition for each eigenvalue:

Remark 1.1For each eigenvalueλi(u)of the system,if either it is linearly degenerate,namely,

or it is genuinely nonlinear,namely,

then the assumption(1.8)obviously holds.

We consider the mixed initial-boundary value problem for the system(1.1)with the initial condition

and the following boundary conditions:

whereHi(i=1,···,n)areC1functions and satisfy the following solvability conditions:

andhi(t)∈C1(i=1,···,n)will be taken as boundary controls.

This mixed initial-boundary value problem(1.1)and(1.9)–(1.11)admits a unique localC1solutionu=u(t,x)for any given initial data satisfying suitableC1compatibility conditions(see[13]).Many papers,including the present one,consider the following problem of exact boundary controllability:For any given initial dataφ(x)and final dataψ(x)∈C1([0,L];U),under certain reasonable assumptions,it is asked to find a controlling timeT0>0 andnboundary controls

such that the mixed initial-boundary value problem(1.1)and(1.9)–(1.11)admits a uniqueC1solutionu=u(t,x)on the domain[0,T0]×[0,L],which verifies the following final condition:

For linear hyperbolic systems,the exact boundary controllability has a complete theory(see[5,14]).For semilinear hyperbolic systems,one can refer to[4]and[19–20].For quasilinear hyperbolic systems,the theory of local exact boundary controllability has been established(see[1,7,9–10]).Roughly speaking,if the initial dataφ(x)and the final dataψ(x)are both smallC1perturbations around one constant equilibrium of the system,then one can find a sharp controlling timeT0and use some or all of the boundary functionshi(t)(i=1,···,n)as controls to achieve the exact boundary controllability in the framework ofC1classical solutions.Since the whole controlling process is achieved around one point in the phase space,this result is called to be the local controllability.

Now it is natural to ask that for general initial data and final data,whether or not one has the global controllability.Since,in the general situation,the classical solutions may blowup in a finite time,the general global exact boundary controllability is hard to be built in the framework of classical solutions.However,for linearly degenerate hyperbolic systems of the diagonal form,since the blowup can be prevented for classical solutions,[18]presents the corresponding results on the global exact boundary controllability.

On the other hand,we may consider the global exact boundary controllability in a slightly different sense as follows.If the initial data and the final data are smallC1perturbations of two distinct constant equilibriau?andu??of the system,namely,

we wish to get the corresponding exact boundary controllability.For the results in this aspect,one may refer to[2,8,11,15]for hyperbolic systems of the diagonal form and wave equations,and[3]for the Saint-Venant equations with slope and friction,which is a special hyperbolic system of the diagonal form with source terms.The main method of these works is successively using exact boundary controllable neighborhoods of finitely many constant equilibria of the system to cover a curve connectingu?andu??composed of equilibria of the system,and then using the local exact boundary controllability to move the solution step by step from the controllable neighborhood ofu?to the one ofu??.In this way,the total controlling time of the global exact boundary controllability might be quite long.We point out that up to now,all the known results on the global exact boundary controllability are only restricted to quasilinear hyperbolic systems of the diagonal form,but not on the general quasilinear hyperbolic systems.

In this paper,the general quasilinear hyperbolic system(1.1)(not necessarily of the diagonal form)is concerned with the initial dataφ(x)and the final dataψ(x)asC1perturbations of two distinct constant equilibriau?andu??,respectively.Besides,we hope to reduce the controlling time of the global exact boundary controllability.Our strategy is listed as follows.The first step is connecting those two constant equilibria with a set of characteristic trajectories of the system,and along these characteristic trajectories we can construct the simple wave solutions of the system which take values on them,respectively.By this analysis,a special solutionu=to system(1.1)with

can be constructed with a possibly largeC1norm,which gives the exact boundary controllability connecting those two equilibria.Based on this,the desired global exact boundary controllability can then be achieved by applying the local exact boundary controllability near those two equilibria,respectively.Since the simple wave solutions used in this process may possess a largeC1norm,the values taken by this solution may change rapidly in the phase space,which overcomes the long-time consumption of the original pointwise extension control method.By means of characteristic trajectories,we are able to develop this simple wave method for the general quasilinear hyperbolic systems.

Inspecial solutions with monotone initial data are analyzed for the transport equation.Based on this,in§3,simple wave solutions to system(1.1)are constructed along the corresponding characteristic trajectories and are combined to form a special solution to system(1.1)on the domain[0,T0]×[0,L]withu?as its initial data andu??as its final data.This special solution,together with the local exact boundary controllability,then leads to the global exact boundary controllability.In§4,the quasilinear hyperbolic system of the diagonal form is analyzed as an example to show the reduction of the controlling time when using this new method.

2 Special Solutions to the Transport Equation

In this section,we consider the following transport equation:

wherezis the unknown function of(t,x),taking its values on a closed intervalI?andλ(z)∈C1satisfies

Consider the Cauchy problem of(2.1)with the following initial condition:

From Theorem 1.1 of[6],we have the following lemma.

Lemma 2.1Under the hypothesis(2.2),if

then the Cauchy problem(2.1)and(2.3)admits a unique global C1solution

on t≥0.

Using this result,someC1solutions with specific properties to the equation(2.1)can be constructed as follows.

Lemma 2.2Under the hypothesis(2.2)and

if z?and z??are two given points on I,then for

and any given δ>0,there exists a C1solution

to the equation(2.1),satisfying

and

ProofSet

We constructz(t,x)according to the following different cases.

Case 1(Forward rarefaction waves)Ifλ(z)>0 andz??≤z?,one may choose a monotone function?∈C1(R;I)satisfying

Then by Lemma 2.1,the Cauchy problem(2.1)and(2.3)admits a uniqueC1solutionz=z(t,x)ont≥0.Moreover,by the method of characteristics,one has

which imply thatz=z(t,x)satisfies all the requirements of Lemma 2.2 on the domain[0,T1+δ]×[0,L](see Fig.1).

Figure 1 Forward rarefaction waves in the case λ(z)>0 and z?? ≤ z?

Case 2(Backward rarefaction waves)Ifλ(z)<0 andz??≥z?,one may choose a monotone function?∈C1satisfying

Then by Lemma 2.1,the Cauchy problem(2.1)and(2.3)admits a uniqueC1solutionz=z(t,x)ont≥0.Moreover,by the method of characteristics,it is easy to show thatz=z(t,x)fulfills all the requirements of Lemma 2.2 on the domain[0,T1+δ]×[0,L](see Fig.2).

Figure 2 Backward rarefaction waves in the case λ(z)<0 and z?? ≥ z?

Case 3(Compression waves)Ifλ(z)>0 andz??>z?,orλ(z)<0 andz??

to the equation(2.1),satisfying

and

Setting

it is easy to show thatz=z(t,x)is aC1solution to the equation(2.1)on the domain[0,T1+δ]×[0,L],which fulfills all the requirements of Lemma 2.2(see Fig.3).

Figure 3 Compression waves

3 SimpleWave Solutions to Quasilinear Hyperbolic Systems and Global Exact Boundary Controllability

In this section,simple wave solutions to the quasilinear hyperbolic system(1.1),suitably constructed based on the result of§2,are combined on the stripex∈[0,L]to form a desired special solution,and then the global exact boundary controllability can be realized.

Fori=1,···,n,the curveu=u(i)(s,uB)defined inUby the following initial value problem of ODE:

is called theith characteristic trajectory passing through the pointu=uB(see[6]).By hyperbolicity of the system,there arendistinct characteristic trajectories passing through any given pointuB∈U,and they form a set of local curved coordinates in the neighborhood ofuB.Therefore,for any given pointsu?andu??inone can find a set of(finitely many)characteristic trajectories to connect them successively.In other words,there existKcharacteristic trajectoriesu(ik)andKreal numberssk(k=1,···,K),such that

and

Obviously,the selection of these characteristic trajectories is not unique.In applications,we may choose the one with the least number of characteristic trajectories.

On each given characteristic trajectory,we have the following lemma.

Lemma 3.1Suppose that u=u(i)(s,uB)is the ith characteristic trajectory of the system(1.1)passing through u=uB,i=1,···,n,and z=z(t,x)is a C1solution to the following transport equation:

on the domain[0,T]×[0,L].Then u=u(i)(z(t,x),uB)is a C1solution to the system(1.1)on the domain[0,T]×[0,L].

ProofDue to the smoothness assumption of the right eigenvectorri(u),u=u(i)(z(t,x),uB)is aC1function on the domain[0,T]×[0,L].Substituting it into the equation(1.1)and noting(3.1)and(1.4),we have

Note that the solutionu=u(i)(z(t,x),uB)given above depends on one scalar functionz=z(t,x),which means it is a simple wave solution.Moreover,it is easy to show that fori=1,···,n,if(1.8)holds for theith eigenvalueλi(u)of the system(1.1),then the functionλ(z)=λi(u(i)(z,uB))satisfies(2.2).Then,using Lemmas 2.2 and 3.1,we get the following corollary.

Corollary 3.1Under hypotheses(1.7)–(1.8),if uE∈U is located on the ith characteristic trajectory passing through uB∈U,namely,

for sE∈then for

and any given δ>0,there exists a C1solution u=u(t,x)∈C1([0,Ti,?+δ]×[0,L];U)to the system(1.1),satisfying

and

By this result,fork=1,···,K,on each characteristic trajectoryu(ik)a simple wave solution can be constructed to get aC1solution developing fromtoAfter suitable translations with respect tot,these solutions can be combined to get the following result.

Proposition 3.1Under hypotheses(1.7)–(1.8),for any given points u?and u??in U,if they can be connected by K characteristic trajectories of the system(1.1),then for

and any given T0>KT?,there exists a C1solution u=u(t,x)∈C1([0,T0]×[0,L];U)to the system(1.1)on the domain[0,T0]×[0,L],satisfying

and

Remark 3.1If we choose?(x)∈Ck(k≥1)in the above process,may have higher regularity:(t,x)∈Ck([0,T0]×[0,L];U).

Now the global exact boundary controllability can be precisely presented and proved by means of Proposition 3.1 and the local exact boundary controllability.

Theorem 3.1Under hypotheses(1.7)–(1.8)and(1.12)–(1.13),for any given initial data φ(x)and final data ψ(x)satisfying(1.15)–(1.16),where u?and u??are constant equilibria of the system(1.1),being connected by K characteristic trajectories,let T?be defined by(3.7).Then for any given T0>(K+2)T?,there exist C1controls hi(t)(i=1,···,n)on[0,T0],such that the mixed initial-boundary value problem(1.1)and(1.9)–(1.11)admits a unique C1solution u=u(t,x)on the domain[0,T0]×[0,L],which verifies exactly the final condition(1.14).

ProofBy the local exact boundary controllability given in[7,10],a local control can be performed to get aC1solutionu=u(t,x)on the domain[0,T?+δ]×[0,L]to the system(1.1),satisfying

Then by Proposition 3.1,aC1solutionu=u(t,x)to the system(1.1)can be constructed on the domain[T?+δ,(K+1)(T?+δ)]×[0,L],satisfying

Finally,another local control can be performed to get aC1solutionu=u(t,x)to the system(1.1)on the domain[(K+1)(T?+δ),(K+2)(T?+δ)]×[0,L],satisfying

Now a special solutionu(t,x)∈C1([0,T0]×[0,L];U)to the system(1.1),which possesses the initial data(1.9)and the final data(1.14),can be constructed just by combining the above three solutions.Substituting this solution into boundary conditions(1.10)–(1.11),one can obtain theC1boundary controls

Obviously,for these boundary controls,the aforementioned solutionu=u(t,x)is the uniqueC1solution to the corresponding mixed initial-boundary value problem(1.1)and(1.9)–(1.11).Moreover,u=u(t,x)satisfies the final condition(1.14).

4 Further Discussions

First,we take the quasilinear hyperbolic system of the diagonal form as an example to show the validity of our methods as well as the reduction of the controlling time.

Consider the following quasilinear hyperbolic system of the diagonal form:

whereλi(u)(i=1,···,n)are smooth and bounded,satisfying

and

For the following boundary conditions given atx=0 andx=L:

we hope to find suitable boundary controlshi(t)(i=1,···,n)such that there exists aC1solutionu=u(t,x)which develops from the initial data

in a neighborhood of one given constant equilibrium to the final data

in a neighborhood of another given constant equilibrium.Since theith characteristic trajectory of(4.1)passing through any given point is a straight line parallel to theuiaxis,it is possible to give a much clearer estimate on the controlling time.Actually,from Theorem 3.1 we have the following result.

Corollary 4.1Under hypotheses(4.2)–(4.3),for any given points u?and u??inany given controlling time

and any given initial data φ(x)and final data ψ(x)satisfying

there exist boundary controls hi(t)∈C1[0,T0](i=1,···,n)such that the mixed initial-boundary value problem(4.1)and(4.4)–(4.6)admits a unique C1solution u=u(t,x)on the domain[0,T0]×[0,L],which verifies exactly the final data(4.7).Here U can be chosen as any rectangular domain containing u?and u??.

Roughly speaking,the global controllability time given in Corollary 4.1 is(n+2)times of the time of two-sided local exact boundary controllability(see[7]),which,in general,is much shorter than that consumed by the method of pointwise extension control.

Compared with the method given in previous results,the process shown in this paper has the following advantages:First,the global exact boundary controllability is established for general hyperbolic systems,but not just for systems of the diagonal form.Secondly,the controlling time can be significantly reduced.However,the result given in this paper is only feasible to homogeneous systems,since the constant equilibria of an inhomogeneous system generally form a complicated manifold in the phase space,and there may not exist a set of finite characteristic trajectories,composed of equilibria,that connect any two given equilibria.Moreover,the global exact boundary controllability is established only with two-sided boundary controls.In general,since the simple waves do not satisfy the uncontrolled boundary conditions,the global one-sided exact boundary controllability or the two-sided one with less controls(see[7]for the corresponding theories of local controllability)can not be achieved using the above method of simple waves.

AcknowledgementThe authors would like to thank Prof.Ta-Tsien Li of Fudan University for his patient guidance,precious suggestions and productive discussions.

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