Control and Stabilization of High-Order KdV Equation Posed on the Periodic Domain

ZHAO Xiangqingand BAI Meng

1DepartmentofMathematics,Zhejiang OceanUniversity,Zhoushan 316000,China.

2Key Laboratory of Oceanographic Big Data Mining&Application of Zhejiang Province,Zhoushan 316000,China.

3School of Mathematics and Statistics Sciences,Zhaoqing University,Zhaoqing 526061,China.

1 Introduction

In this paper,we will investigate the following higher-order dispersive equation posed on the periodic domain T(a unit circle in the plane)from the control point of view：

wherefis the control input supported in a given open setω?T.The assumption on the periodic domain is equivalent to impose the periodic boundary conditions over the interval(0,2π)：

The following two fundamental control theory problems will be discussed：

Exact controllability:For the given initial state u0and terminal state u1belong in a certain space,can one find an appropriate control input f such that equation(1.1)admits a solution u which satisfies

Feedback stabilization:Is there a feedback control law:f=Ku such that the resulting closedloop system

is exponentially stable as t→∞?

Since for the solution of(1.1)satis fi es

the mass will be conserved provided that

For the purpose of mass conservation,the control input as follows is chosen(see[5])：

whereg(x)is a given nonnegative smooth function such that{g＞0}=ω?T and

Withhas a new control input,the resulting control system turns to be

We state the main results as follows：

Theorem 1.1(Exact controllability).Let T＞0and s≥s0(see Lemma3.2)be given.Then there exists a δ＞0such that for any u0,u1∈Hs(T)with[u0]=[u1]and

one can fi nd a control function h∈L2([0,T];Hs(T))such that the system(1.3)has a solution u∈C([0,T];Hs(T))satisfying

Theorem 1.2(Feedback Stabilizability).Let s≥s0and λ＞ 0be given.If one chooses the Slemrod’s feedback control Kλ(see Section2)in the system(1.3),then the resulting close-loop system

is locally exponentially stable in the space Hs(T):there exists δ＞0such that for any u0∈Hs(T)with‖u0‖Hs(T)＜δ,the corresponding solution u of(1.4)satisfies

for any t＞0.

Whenl=1,the dispersive equation

became the famous KdV equation,therefor people often call(1.5)as high-order KdV equation[1,2].More information on higher-order dispersive equations can be found in[3].The periodic exact controllability and exponentially stability of KdV equation and Kawahara equation are investigated by[4-6]and[7,8]respectively.

The Cauchy problem of the general dispersive equation(1.5)has been shown well posed recently by Bourgain method(see[1,2,9]).The subtle Bourgain smoothing effect established in[1,2,9]will play an indispensable role in the proofs of exactly controllable and exponentially stability in this paper.Results on initial-boundary value problem for(1.5)see[10-14].

Since it is time-reversible for dispersive equation,we assumet＞0 in the following sections.

The paper is organized as follows：In Section 2,we study the associated linearized system.We obtain the controllability of the linear open loop system in the spaceHs(T)for anys∈R through solving a moment problem.Then the exponentially stabilizable with arbitrarily large decay rate is proved when the Slemrod’s feedback control input is chosen.In Section 3,aided by Bourgain smoothing properties of linear equation,we show the nonlinear system is locally exactly controllable in the spaceHs(T)for anys≥s0by Banach fixed point theorem.In Section 4,the nonlinear feedback systemisfirst shown to be globally well-posed in the spaceHs(T)for anys≥s0and then it is shown to locally exponentially stabilizable with arbitrarily large decay rate.

2 Linear system

Consideration is first given to the associated linear open loop control system

where the operatorGand control inputh=h(x,t)are defined in Section 1.

LetAdenote the operator

with the domainD(A)=H2l+1(T).It generates a strongly continuous groupW(t)on the spaceL2(T)and its eigenfunctions are simply the orthonormal Fourier basis functions inL2(T),

The corresponding eigenvalue ofφkis

For anyl∈Z,let

Thenm(l)≤2l+1 for anylandm(l)=1 iflis large enough.Moreover,

The solutionvof the system(2.1)can be expressed in the form

wherev0,kandGk[h]are the Fourier coef fi cients ofv0andG[h],respectively,

fork=0,±1,±2,···.Furthermore,for givens∈R,ifv0∈Hs(T)andh∈L2(0,T;Hs(T)),the function given by(2.2)belongs to the spaceC([0,T];Hs(T)).

We have the following exact controllability result for the open loop control system(2.1).

Theorem 2.1.Let T＞0and s∈Rbe given.There exists a bounded linear operator

such that for any v0,v1∈Hs(T),if one chooses h=Φ(v0,vT)in(2.1),then the system(2.1)admits a solution v∈C([0,T];Hs(T))satisfying

In the sequelwe will denotebyCnumerical constant which maybe different from line to line.Moreover,let

Proof.For givev0,v1∈Hs(T),we need to fi ndh∈L2(0,T;Hs(T))such

which is equivalent to the moment equation：

fork=±1,±2,···.

If we definepk=eλkt,thenP≡{pk|-∞＜k＜∞}will form a Riesz basis for its closed spanPTinL2(0,T).We letQ≡{qk|-∞＜k＜∞}be the unique dual Riesz basis forPin PTsuch that

We take the controlhin(2.3)to have the form

where the coefficientshjare to be determined so that,among other things,the series(2.5)is appropriately convergent.Substituting(2.5)into(2.3)yields,using the biorthogonality(2.4),that

for-∞＜k＜∞.AsGis a self-adjoint operator inL2(T),

We have

It is easy to see thatβ0=0 andβk/=0 ifk/=0.Moreover,the familiar Lebesgue lemma together with the second identity above shows that

It follows that there is aδ＞0 such that

Settingh0=0 and

It remains to show thathdefined by(2.5)and(2.7)is inL2([0,T];Hs(T))provided thatv0,v1∈Hs(T).To this end,let us write

where

Thus

and

where the constantCcomes from the Riesz basis property ofQinPT.However

where

Hence

and

Thus,in the case ofs≥0,

We have,according to(2.7),that

In the case ofs＜0,as for any-∞＜k,j＜∞,

and therefore

Now we turn to consider feedback stabilization problem of the linear system(2.1).According to[4,7],it is possible to establish the exponential stability with decay rate as large as one desires for the resulting closed-loop system if the Selmord feedback law is chosen.For anyλ＞0,define

for anyφ∈Hs(T).Then,we have

Lemma 2.1.For any s≥0,the operator Lλis an isomorphism from Hs(T)onto Hs(T)for all s≥0.

Proof.See Lemma 2.4 in[4].

According to Lemma 2.1,Lλhas bounded inverse inHs(T).Taking the control func-we obtain the following closed-loop system：

with the feedback control law

Proposition 2.1.Let s≥0and λ＞0be given.Then for any v0∈Hs(T),the system(2.9)admits a unique solution v∈C([0,T];Hs(T)).Moreover there exist positive constants Msdepending only on s such that

for any t＞0.

Proof.The existence of the solutionvfollows from the standard semigroup theory[15].

The decay estimate(2.10)can be proved by interpolation：

The case ofs=0 follows from[16].

Fors=2l+1,letw=vt.Thenwsolves

where

for anyt≥0.It then follows from

that

for anyt≥0.

The case of 0＜s＜2l+1 follows by interpolation.The other cases ofscan be proved similarly.

3 Exact controllability

In this section,we study the exact controllability for the open loop nonlinear control system：

To apply the Bourgain smoothing effect,some technical preparations are needed.For givenb,s∈R,and a functionu：T×R→R,define the quantities

wheredenotes the Fourier transform ofuwith respect to the space variablexand the time variabletandp(k)=-k2l+1.Then Bourgain spaceXb,s(resp.Yb,s)associated to the higher-orderKdV equation onT is the completion of the spaceS(T×R)under the norm ‖u‖Xb,s(resp.‖u‖Yb,s).

For givenb,s∈R,let

be endowed with the norm

For a given intervalI,letXb,s(I)(resp.Zb,s(I))be the restriction space ofXb,sto the intervalIwith the norm

Forsimplicity,we denoteXb,s(I)(resp.Zb,s(I))byifI=(0,T).Inaddition,let

There are a series of smoothing estimates：

Lemma 3.1.Let b,s∈Rand T＞0be given.There exists a constant C＞0such that

(i)for any φ∈Hs(T),

(ii)for any

Proof.See[2,9].

Lemma 3.2.For

There exist a constant C such that the following bilinear estimate

holds.

Proof.See[17,18]and[9].

Proof of Theorem 1.1.Rewrite the system(3.1)in its equivalent integral equation form：

define

According to Theorem 2.1,for givenu0,u1∈(T),if one chooses

in the equation(3.2),then

and

This leads us to consider the map

If we can prove that Γ is a contraction mapping in an appropriate space,then its fi xed pointuis a solution of(3.2)withh=Φ(u0,u1+ω(T,u))and satis fi esu|t=T=u1.

Applying Lemma 3.1-3.2 yields that

Notice that

Consequently,

ForR＞0,letBRbe a bounded subset of

Then,for anyu∈BR

We chooseδ＞0 andR＞0 such that

Then,

which means that Γ mapBRinto itself.

Similarly,for anyu,v∈BR,we deduce that

which implies that Γ is an contracting map onBR.

By the Banach fi xed point theorem,there is unique solution to the integral equation(3.2)which is the desired solutionof(3.1). □

4 Exponential stabilizability

For the linearized system

Its solution can be written as

whereWλis theC0-semigroup associated to the linearized system.

Lemma 4.1.Let s∈Rand T＞0be given.There exists a constant C＞0such that

(i)

for any φ∈(T);

(ii)

for any

Proof.For givenφ∈(T)andlet

Thenusolves

Consequently,

and for any 0＜T′≤T,

whereC＞0 depends only onsandT.As

for someν＞0 andC1depending only onsandT.Thus ifT′is chosen small enough,we have

It then follows from the semigroup property of the system(4.1)that

The proof is complete.

We first show the closed loop system

is well-posedness in the spaceHs(T)for anys≥0.

Proposition 4.1.Let T＞0and s≥s0be given.Then there exists a δ＞0such that for any u0∈Hs(T)with

the system(4.2)admits a unique solution Moreover,the corresponding solution map is Lipschitz continuous.

Proof.Rewrite the system(4.2)in its equivalent integral equation form：

Then define the map

Applying Lemma 4.1,Lemmas 3.1-3.2,we obtain

ForR＞0,letBRbe a bounded subset of

Then,for anyu∈BR,we have

We chooseδ＞0 andR＞0 such that

Then,which implies that Γ mapsBRinto itself.In addition,for anyu,v∈BR,we have

which implies that Γ is an contracting mapping onBR.By Banach’s contracting mapping principle,Γ has unique fixed point which is the desired solution of the system(4.2).

Remark 4.1.The local well-posedness result presented in Proposition 4.2 can be restated as follows：

Lets≥s0andr＞0 be given.There exists aT＞0 such that for anyu0∈Hs(T)with‖u0‖s≤r,the system(4.2)admits a unique solutionu∈

Next we show that the system(4.2)is globally well-posed in the spaceHs(T)for anys≥0.

Theorem 4.1.Let s≥s0and T＞0be given.For any u0∈Hs(T),the system(4.2)admits a unique solution u∈Furthermore,the following estimate holds

where αT,s：R+→R+is a nondecreasing continuous function depending only on T and s.

Proof.The proof is very much similar to that of Theorem 4.7 in[5]and is therefore omitted.

Proof of Theorem 1.2.For givens≥0 andλ＞0,by Proposition 2.3,there exists positive constantCsuch that

For any given 0＜λ′＜λ,pickT＞0 such that

We seek a solutionuto the integral equation(4.3)as a fi xed point of the map

in some closed ballBR(0)in the function spaceThis will be done provided that‖u0|s≤δwhereδis a small number to be determined.Furthermore,to ensure the exponential stability with the claimed decay rate,the numbersδandRwill be chosen in such a way that

By Lemma 4.1,there exist some positive constantC1,C2(independent ofδandR)such that

On the other hand,we have for some constantC＞0 and allu∈BR(0)

Pickδ=C4R2,whereC4andRare chosen so that

Then we have

Therefore,Γ is a contractioninBR(0).Furthermore,its unique fi xedpointu∈BR(0)ful fi lls

Assume now that 0＜‖u0‖0)＜δ.Changingδintoδ′≡‖u0‖sandRintoR′≡(δ′/δ)12R,we infer that

and an obvious induction yields

for anyn≥0.We infer by the semigroup property that there exists some positive constantC＞0 such that

if‖u0‖s≤δ.

Acknowledgement

The fi rst author is fi nancially supported by the Natural Science Foundation of Zhejiang Province(#LY18A010024,#LQ16A010003),the China National Natural Science Foundation(#11505154,#11605156)and the Open Foundation from Marine Sciences in the Most Important Subjects of Zhejiang(#20160101).The second author is fi nancially supported by Foundation for Distinguished Young Teacher in Higher Education of Guangdong,China(YQ2015167),Foundation for Characteristic Innovation in Higher Education of Guangdong,China(Analysis of some kinds of models of cell division and the spread of epidemics),NSF of Guangdong Province(2015A030313707).The authors are greatly in debt to the anonymous referee for his/her valuable comments and suggestions on modifying this manuscript.

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