Sufficiency of Kalman’s Rank Condition for the Approximate Boundary Controllability on an Annular Domain

Chengxia ZU

Abstract In this paper the author establishes the sufficiency of Kalman’s rank condition on the approximate boundary controllability at a finite time for diagonalizable systems on an annular domain in higher dimensional case.

Keywords Kalman’s rank condition, Approximate boundary controllability, Diagonalizable systems of wave equations, Annular domain

1 Introduction

Let ? be a bounded domain in Rdwith smooth boundary Γ=Γ1∪Γ0such thatand mes(Γ1)>0.Let H0=L2(?), H1=H10(?), L=(0,+∞;L2(Γ1))and H?1=H?1(?)denotes the dual of H1.

Let U = (u(1),··· ,u(N))T.Consider the following coupled system of wave equations with Dirichlet boundary controls:

with the initial condition

where “′” stands for the time derivative;is the Laplacian operator; the coupling matrix A=(aij)is of order N and the boundary control matrix D =(dpq)is a full column-rank matrix of order N×M (M ≤N), both with constant elements; H =(h(1),··· ,h(M))Tdenotes the boundary controls.

From the approximate boundary null controllability of system (1.1)introduced by Li and Rao in [5–6], we have the following definition.

Definition 1.1System(1.1)is approximately boundary null controllable at the timeT >0,if for any given initial datathere exists a sequence{Hn}ofboundary controls inLMwith compact support in[0,T], such that the corresponding sequence{Un}of solutions to problem(1.1)–(1.2)satisfies

Let Φ=(φ(1),··· ,φ(N))T.The adjoint system of system (1.1)is given by

with the initial data

Definition 1.2(see [8–9])The adjoint system(1.4)is D-observable on the interval[0,T],if the following partial Neumann observation

?νbeing the outward normal derivative, implies that(Φ0,Φ1)=(0,0), thenΦ ≡0.

The relationship between the approximate boundary null controllability of system(1.1)and the D-observability of the adjoint system(1.4)was also given by Li and Rao in [5–6]as follows.

Theorem 1.1System(1.1)is approximately null controllable at the timeT >0if and only if the adjoint system(1.4)is D-observable on the interval[0,T].

The necessity of Kalman’s rank condition to the D-observability of the adjoint system(1.4),proved by Li and Rao in [5–6], can be written as the following theorem.

Theorem 1.2If the adjoint system(1.4)is D-observable, then we necessarily have the following Kalman’s rank condition:

Kalman’s rank condition (1.7)is not sufficient for the approximate boundary null controllability of system (1.1)in general.Otherwise, noting that Kalman’s rank condition (1.7)is independent of the control (and observation)time T >0, if system (1.1)is approximately null controllable at the time T >0, then the approximate boundary null controllability can be realized almost immediately, which contradicts the finite speed of wave propagation.However, in some special cases,Kalman’s rank condition(1.7)is sufficient for the approximate boundary null controllability of system (1.1)on a finite time interval [0,T], when T > 0 is large enough (see[5–6, 9]).This paper as a continuation of [9] is to investigate the sufficiency of Kalman’s rank condition (1.7)for diagonalizable systems on an annular domain ? = {x : a < |x| < 1} ?Rdwith Γ0={x:|x|=a} and Γ1={x:|x|=1}, where a is a positive constant with a<1.

In Section 2, we will investigate the eigenfunctions and eigenvalues of ??on the annular domain ?={x:a<|x|<1}based on the coordinate transformation,and give some properties of the eigenvalues.The uniqueness result for non-harmonic series on this annular domain will be established in Section 3.The sufficiency of Kalman’s rank condition (1.7)for diagonalizable systems on an annular domain will be given in Section 4 by a way similar to the one-spacedimensional case and to [9].

2 Preliminaries

In this section, we will give the eigenfunctions and eigenvalues of ??on an annular domain?={x:a<|x|<1} with 00 large enough for diagonalizable systems.For this purpose, we consider the eigenvalue problem

in spherical coordinates.

Let e(x)= R(r)Y(θ).Similarly to the spherical domain in [9], we get the corresponding eigenvalue problems for Y(θ)and R(r), respectively.

For Y(θ), we have

where m ∈N and ?θis the Laplacian operator on the unit sphere Sd?1with d ≥2 (see [1]).

For R(r), we have

Atkinson and Han introduced the eigenfunctions and eigenvalues of problem (2.2)in [1].In what follows, N and N+denote the set of natural numbers and the set of positive integers,respectively.

Lemma 2.1(see[1])Let?θbe the Laplacian operator on the unit sphereSd?1withd ≥2.Then, we have

(i)for any givenm ∈N,{Ym,j}1≤j≤jmare the eigenfunctions of??θ, corresponding to the eigenvaluem(m+d ?2), i.e., we have

wherejm, the multiplicity of the eigenvaluem(m+d ?2), is given by

(ii){Ym,j}m∈N,1≤j≤jmare orthonormal inL2(Sd?1), i.e., we have

withm,m′∈N,1 ≤j ≤jmand1 ≤j′≤jm′, whereδm,m′stands for the Kronecker symbol;

(iii){Ym,j}m∈N,1≤j≤jmare complete inL2(Sd?1).

Next, we consider problem(2.3)as a Sturm-Liouville problem(2.7)below.Some properties of the eigenfunctions of the Sturm-Liouville problem was introduced by Bagrov and Belov in[2] as follows.

Lemma 2.2Consider the following Sturm-Liouville problem

Assume thatφ′(r),q(r)andρ(r)are continuous, andφ(r)> 0,ρ(r)> 0andq(r)≥0on the interval[a,1].Then, we have

(i)there exists a sequence of eigenvalues{λk}k∈N+and the corresponding sequence of eigenfunctions{xk(r)}k∈N+for Sturm-Liouville problem(2.7), and all eigenvalues can be ordered so that

(ii)every eigenvalue corresponds to, up to a multiplier constant, only one eigenfunction;

(iii)the eigenfunctions of Sturm-Liouville problem(2.7)corresponding to different eigenvalues are pairwisely orthogonal on the interval(a,1)with the weight functionρ(t), i.e., fork,k′∈N+, we have

wherexk(r)andxk′(r)are eigenfunctions of(2.7)corresponding to the eigenvaluesλkandλk′withλk≠λk′, respectively;

(iv)(Steklov’s expansion theorem)if a functionf(r)is twice continuously differentiable on[a,1]and satisfies the boundary conditions in(2.7), it can be expanded in a series of the eigenfunctionsxk(r)of Sturm-Liouville problem(2.7)absolutely and uniformly converging on[a,1].

Remark 2.1Lemma 2.2 is also valid when the interval [a,1] is replaced by [r1,r2].

By Wu [8], we have the following lemma.

Lemma 2.3For any fixedm ∈N, let

where(r)and(r)are the（m+?1）-th Bessel function and Neumann function,respectively, andμm,kis thek-th positive root of

We have

By Lemma 2.2(iii)and Lemma 2.3, for

we have the following proposition.

Proposition 2.1(i)For any givenm ∈N,{Rm,k(r)}k∈N+is a sequence of orthogonal functions with the weightrd?1inL2(a,1).

(ii)(see [5])Ifd=2andm ∈N+, or ifd ≥3andm ∈N, we have

while, ford=2andm=0, we have

Proof(i)Since (2.10)is a Sturm-Liouville problem(2.7)with ρ(r)=r, by Lemma 2.2(iii),for any fixed m ∈N, we have

where k,k′∈N+.Then, noting (2.13), for any fixed m ∈N, we have

Namely, for any given m ∈N, {Rm,k(r)}k∈N+is a sequence of orthogonal functions with the weight rd?1in L2(a,1).

(ii)By [4, Lemma 4(iv)], a direct computation gives (2.14)and (2.15).

Remark 2.2(i)Let

Then, for any given m ∈N, {cm,kRm,k(r)}k∈N+is a sequence of orthonormal functions with the weight rd?1in L2(a,1).

(ii)Furthermore, let

Then, by Proposition 2.1(ii), we have

The inequality (2.18)will be useful to guarantee the convergence of the infinite series given in Section 4.We now give the eigenfunctions and eigenvalues of ??on ? = {x: a< |x| < 1}as follows.

Lemma 2.4Let?={x:a<|x|<1}with0

in whichRm,k(r)is given by(2.13),cm,kis given by(2.16), andYm,j(θ)is given by Lemma2.1(ii).We have

(i)for any givenm ∈Nandk ∈N+,em,k,j(x)(1 ≤j ≤jm)are all the eigenfunctions of??, corresponding to the eigenvalue;

(ii){em,k,j(x)}m∈N;k∈N+;1≤j≤jmis an orthonormal sequence inL2(?);

(iii)?νem,k,j(x)|Γ1= ?2π?1cm,kYm,j(θ), where?νdenotes the outward normal derivative on the boundary.

Proof(i)By the above discussion, it is easy to get (i).

(ii)Let m,m′∈N, k,k′∈N+, 1 ≤j ≤jmand 1 ≤j′≤jm′.By Lemma 2.1(ii)–(iii)and Proposition 2.1, we have

then we get (ii).

(iii)Since

by the boundary condition in (2.10), we have

in which we used the fact that Jσ(x)(x)?Nσ(x)(x)=(see [2]).By (2.20), we have

Now, we introduce some properties of the eigenvaluesof ??on ?={x:a<|x|<1}.Let ασ,kdenote the k-th zero point of the cross-product of the σ-th order Bessel function and Neumann function:

where K is a positive constant with K > 1.The property of ασ,kis given in [3–4].By (2.12)and taking K =a?1, we get a·μm,k=ασ,kwith σ =m+?1.Then, we have the following peroposition.

Proposition 2.2Letμm,kbe thek-th positive root of(2.12)withm ∈Nandk ∈N+.

(i)Ford=2, we have

while, for any fixedd>2, we have

(ii)For any fixedd ≥2, whenk →+∞, we have

(iii)For any fixedd ≥2, we have

(iv)For any fixedd ≥2, we have

3 A Uniquess Result

Let Z?denote the set of all nonzero integers.We now give the following uniqueness result introduced by Zu,Li and Rao in[9],which will be useful for proving the sufficiency of Kalman’s rank condition (1.7)on the annular domain ?={x:a<|x|<1}.

Lemma 3.1Assume that

for any fixedm ∈N.Assume furthermore that for any givenm ∈N, there exist positive constantsγm,cmandτmsuch that

and

for all1 ≤l ≤sand allk ∈Z?with|k|large enough.

Assume finally that

with

andT >2πD+, where

in whichis the upper density of the sequenceThen, we have

When ? is an annular domain, by Proposition 2.2(i), for d > 2 and d = 2, the uniform gap condition of sequence {μm,k}k∈N+starts from k = 2 for any given m ∈N and m ∈N+,respectively, which is different from the case that ? is a spherical domain.Hence, in order to use Lemma 3.1, we should add condition (3.10)below and rearrange 2s elements of sequenceto guarantee condition (3.1).

Corollary 3.1Assume that

For any fixedm ∈N, we define

whereμm,kis thek-th positive root of(2.12)for any fixedm ∈N.Then, for?>0small enough and

the sequencesatisfies(3.1)–(3.3)and(3.6)for any givenm ∈N.

ProofFor any given m ∈N and k ∈N+, by the definition of μm,kand Proposition 2.2(ii),we have

For d=2, by Proposition 2.2(ii), we have

Let

For any fixed d ≥2, by Proposition 2.2(i), we have

On the other hand,for any fixed m ∈N,by Proposition 2.2(ii),a direct computation similar to that on a spherical domain gives that

and

Then it is easy to see that the sequencesatisfies (3.2)–(3.3)with

Next, we will prove that the sequencesatisfies (3.6)in a way similar to the proof of that on a spherical domain.

For any fixed m ∈N and k ∈N+, iffor 1 ≤l ≤s ?1, then we have

By Proposition 2.2(ii), for each m ∈N, we have

Thus, by (3.13)–(3.16), we get

Since

4 The Sufficiency of Kalman’s Rank Condition on an Annular Domain

On the annular domain ?={x:a<|x|<1}, we consider the following system:

with the initial condition

The adjoint system of (4.1)is given by

with the initial data

In this section,we will prove the sufficiency of Kalman’s rank condition(1.7)for T >0 large enough to the approximate boundary null controllability of system(4.1)on the annular domain?.By Theorem 1.1, it is sufficient to prove the sufficiency of Kalman’s rank condition (1.7)to the D-observability of the corresponding adjoint system (4.3)on ?.The following necessary and sufficient condition of Kalman’s rank condition (1.7)given by Li and Rao in [5–6] is very useful to prove the sufficiency of Kalman’s rank condition (1.7)in this case.

Lemma 4.1Assume thatk ≥0is an integer,Ais a matrix of orderNandDis a full column-rank matrix of orderN ×MwithM ≤N.Then Kalman’s rank condition

holds if and only if the largest dimension of invariant subspaces ofAT, contained inKer(DT),is equal tok.

Theorem 4.1Let

with0 < a < 1and let the sequencebe defined by(3.9).Assume that thecoupling matrixAis diagonalizable with the real eigenvalues given by

Assume furthermore that?> 0is small enough and(3.10)holds so that for eachm ∈N, thesequencesatisfies(3.1)–(3.3)and(3.6).

Then Kalman’s rank condition(1.7)is sufficient for the approximate boundary null controllability of system(4.1), provided thatT >2s(1 ?a).

ProofBy Theorem 1.1, it is sufficient to prove the sufficiency of Kalman’s rank condition(1.7)to the D-observability of the corresponding adjoint system (4.3).The proof is similar to that on a spherical domain in higher dimension case (see [9]).In this paper, we just give the essential differences.

with

Furthermore, let

in which we define em,?k,j= em,k,jfor all m ∈N,k ∈N+and 1 ≤j ≤jm.Then, by [9,Theorem 3],forms a Riesz basis of ((?))N×(L2(?))N.

Thus, for any given initial datathere existssuch that

with

Then the corresponding solution to problem (4.3)–(4.4)is given by

where em,k,j(x)are given by (2.19).

Using Lemma 2.4(iii), the observation (1.6)becomes

Noting D =(dpq), we define

and μm,?k=μm,kfor all m ∈N and k ∈N+.

Then, for any fixed q with 1 ≤q ≤M, the observation (1.6)can be rewritten as

The difference from the proof of that on a spherical domain is the verification of

for any fixed q with 1 ≤q ≤M.

Using (2.18)and (4.8), we have

In the present situation, by Proposition 2.2(iv), for any fixed l with 1 ≤l ≤s, we have

Hence, there exists a positive constant c1such that

By (4.12)and Cauchy-Schwartz inequality, we have

Let

By (4.10)and (4.13), for q =1,2,··· ,M, we have

Applying Lemma 3.1 to each line of (4.9), we get

By Lemma 4.1, it follows from Kalman’s rank condition (1.7)that Ker(DT)does not contain any non-trivial invariant subspace of AT, then we have

Since {ω(l,μ)}1≤μ≤μlare linearly independent, noting that by (2.16), cm,k≠ 0 for any given m ∈N and any given k ∈N+, we have

namely, Φ ≡0.The proof is complete.

Example 4.1Let δ1,δ2be positive constants with δ1<δ2, and k1,k2be positive integers.Consider the adjoint system (4.3)with A =diag(δ1,δ2)and D = (1,?1)T.We will show that Kalman’s rank condition is not sufficient for the D-observability of adjoint system (4.3)at the infinite horizon for ?∈N, where

For ?>0 with ?∈N, there exist m and k1,k2with 1 ≤k1

where α>0.Let

where Ym,1is given in Lemma 2.1 with j =1; Rm,k1(r)and Rm,k2(r)are given by (2.13)with k=k1and k =k2,respectively.Then by Lemma 2.4,Φ is a non-trivial solution of system(4.3)and satisfies the observation.

Example 4.1 shows that Kalman’s rank condition (1.7)is not sufficient in general for the approximate boundary null controllability of system (1.2)even at the infinite horizon.Hence,it is essential to add condition (3.10)to guarantee the sufficiency of Kalman’s rank condition(1.7).

We now indicate the relationship between the controllability time T and the rank of D.

Theorem 4.2Let? = {x : a < |x| < 1}with0 < a < 1.Assume thatrank(D)= N ?kwith0 ≤k ≤N ?1and the coupling matrixAis diagonalizable with the real eigenvalues given by(4.6).Then, Kalman’s rank condition(1.7)is sufficient for the approximate boundary null controllability of system(4.1)on the interval[0,T], provided thatT >2(k+1)(1 ?a)and?>0is small enough.

ProofThe proof is same as that of Theorem 4 given by Zu, Li and Rao in [9].

Remark 4.1Let ? = {x : r1< |x| < r2} with 0 < r1< r2.Assume that the coupling matrix A is diagonalizable with the real eigenvalues given by (4.6).Assume furthermore that?>0 is so small that for each m ∈N, the sequencedefined by (3.9)satisfies(3.1)–(3.3)and (3.6).

Then, Kalman’s rank condition (1.7)is sufficient for the approximate boundary null controllability of system (4.1)on the interval [0,T], provided that T >2s(r2?r1).

ProofWhen ? ={x: a<|x| <1} is changed to {x:r1<|x| < r2}, a, μm,kand D+are replaced byand r2D+, respectively.Thus, by Lemma 3.1, the controllability time T is replaced by r2T, i.e., 2s(r2?r1).

Similarly, we have the following remark.

Remark 4.2Let ?={x:r1<|x| 0 is small enough and condition (3.10)holds.Then,Kalman’s rank condition (1.7)is sufficient for the approximate boundary null controllability of system (4.1)on the interval [0,T], provided that T >2(k+1)(r2?r1).

Remark 4.3The controllability time T given by Theorems 4.1 or 4.2 is not optimal.

AcknowledgementThe author would like to thank Professors Tatsien Li and Bopeng Rao for their valuable advices.