# Controllability of  Second-OrderEquations inL2(Ω)

Authors: Hugo Leiva; Nelson Merentes
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert spaceL2(Ω): ÿ+Ay=1ωu(t), t∈(0,τ], y(0)=y0, ẏ(0)=y1, where Ω is a domain in ℝN(N≥1), y0,y1∈L2(Ω), ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0,τ;L2(Ω)), and A:D(A)⊂L2(Ω)→L2(Ω) is an unbounded linear operator with the following spectral decomposition: Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k, with the eigenvalues λj given by the following formula: λj=j2mπ2m, j=1,2,3,… and m≥1 is a fixed integer number, multiplicity γj is equal to the dimension of the corresponding eigenspace, and {ϕj,k} is a complete orthonormal set of eigenvectors (eigenfunctions) of A. Specifically, we prove the following statement: if for an open nonempty set ω⊂Ω the restrictions ϕj,kω=ϕj,k|ω of ϕj,k to ω are linearly independent functions on ω, then for all τ≥2/πm-1 the system is approximately controllable on [0,τ].  As an application, we prove the controllability of the 1D wave equation.


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## Body

## 1. Introduction

This paper has been motivated by the work in [1] and the articles [2, 3], where a new technique is used to prove the interior approximate controllability of some diffusion process. Particularly in [3], where the authors prove the interior approximate controllability of the following broad class of reaction diffusion equations in the Hilbert space Z=L2(Ω) given by (1.1)z′=-Az+1ωu(t),t∈[0,τ],
where Ω is a domain in ℝn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u∈L2(0,τ;L2(Ω)) and A:D(A)⊂Z→Z is an unbounded linear operator with the spectral decomposition(1.2)Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k.
The eigenvalues 0<λ1<λ2<⋯<⋯λn→∞ of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace, and {ϕj,k} is a complete orthonormal set of eigenvectors of A. The operator -A generates a strongly continuous semigroup {TA(t)}t≥0 given by (1.3)TA(t)z=∑j=1∞e-λjt∑k=1γj〈z,ϕj,k〉ϕj,k.
As a consequence of this result, the controllability of the following heat equation follows trivially by putting A=-Δ(1.4)zt=Δz+1ωu(t,x),in(0,τ)×Ω,z=0,on(0,τ)×∂Ω,z(0,x)=z0(x),inΩ.
Following [1–3], in this paper, we study the interior approximate controllability of the following broad class of second-order equations in the Hilbert space L2(Ω):(1.5)ÿ+Ay=1ωu(t),t∈(0,τ],y(0)=y0,ẏ(0)=y1,
where the eigenvalues λj of the operator of A are given by the following formula:(1.6)λj=j2mπ2m,j=1,2,3,…,m≥1is  a  fixed  integer  number.
Specifically, we prove the following statement: if for an open nonempty set ω⊂Ω the restrictions ϕj,kω=ϕj,k|ω of ϕj,k to ω are linearly independent functions on ω, then for all τ≥2/πm-1 the system (1.5) is approximately controllable on [0,τ]. Moreover, we can exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time (see Theorem 2.8).This result implies the interior controllability of the following well-known examples of partial differential equations.Example 1.1.
The 1D Wave Equation(1.7)ytt-Δy=1ωu(t,x),in(0,τ]×Ω,y=0,on(0,τ)×∂Ω,y(0,x)=y0(x),yt(0,x)=y1(x),x∈Ω,
where ω is an open nonempty subset of Ω=[0,1], 1ω denotes the characteristic function of the set ω, and the distributed control u∈L2(0,τ;L2(Ω)).Example 1.2.
The Model of Vibrating String Equation(1.8)wtt+Δ2w=1ωu(t,x),in(0,τ]×Ω,w=Δw=0,on(0,τ)×∂Ω,w(0,x)=ϕ0(x),wt(0,x)=ψ0(x),x∈Ω,
where Ω=[0,1], ω is an open nonempty subset of Ω, u∈L2([0,τ];L2(Ω)), ϕ0,ψ0∈L2(Ω).


## 2. Main Results

In this section, we will prove the main result of this work; to this end, we consider byX=U=L2(Ω) and the linear unbounded operator A:D(A)⊂X→X can be written as follows. (a)
For allx∈D(A), we have
(2.1)Ax=∑j=1∞λjEjξ,
where
(2.2)Ejx=∑k=1γj〈ξ,ϕj,k〉ϕj,k.
So, {Ej} is a family of complete orthogonal projections in X and x=∑j=1∞Ejx,x∈X.(b)
The semigroup{TA(t)} generated by -A can be written as follows:(2.3)TA(t)x=∑j=1∞e-λjtEjx.(c)
The fractional powered spacesXr are given by
(2.4)Xr=D(Ar)={x∈X:∑n=1∞λn2r‖Enx‖2<∞},r≥0,
with the norm
(2.5)‖x‖r=‖Arx‖={∑n=1∞λn2r‖Enx‖2}1/2,x∈Xr,Arx=∑n=1∞λnrEnx.
Also, for r≥0, we define Zr=Xr×X, which is a Hilbert space endowed with the norm given by
(2.6)‖[yv]‖Zr2=‖y‖r2+‖v‖2.Proposition 2.1.
The operatorPj:Zr→Zr, j≥0, defined by
(2.7)pj=[Ej00Ej],j≥1,
is a continuous (bounded) orthogonal projections in the Hilbert space Zr.Proof.
First we will show thatPj(Zr)⊂Zr, which is equivalent to show that Ej(Xr)⊂Xr. In fact, let x be in Xr and consider Ejx. Then,
(2.8)∑n=1∞λn2r‖EnEjx‖2=λj2r‖Ejx‖2<∞.
Therefore, Ejx∈Xr,  for all x∈Xr.
Now, we will prove that this projection is bounded. In fact, from the continuous inclusionXr⊂X, there exists a constant k>0 such that
(2.9)‖x‖≤k‖x‖r,∀x∈Xr.
Then, for all x∈Xr, we have the following estimate
(2.10)‖Ejx‖r2=∑n=1∞λn2r‖EnEjx‖2=λj2r‖Ejx‖2≤λj2r‖x‖2≤λj2rk2‖x‖r2.
Hence, ∥Ejx∥≤λjrk∥x∥r, which implies the continuity of Ej:Xr→Xr. So, Pj is a continuous projection on Zr.Hence, with the change of variabley′=v, the system (1.5) can be written as a first-order system of ordinary differential equations in the Hilbert space Z1/2=X1/2×X as follows: (2.11)z'=Az+Bωu,z(0)=z0,z∈Z1/2,t∈(0,τ],
where(2.12)z=[yv],Bω=[01ω],A=[0IX-A0]
is an unbounded linear operator with domain D(𝒜)=D(A)×D(A1/2).The proof of the following theorem follows in the same way as [4, Theorem  3.1], by putting c=0 and d=1 or directly from [5, lemma  2.1] or [6, Lemma  3.1].Theorem 2.2.
The operator𝒜 given by (2.12) is the infinitesimal generator of a strongly continuous semigroup  {T(t)}t∈ℝ given by
(2.13)T(t)z=∑j=1∞eAjtPjz,z∈Z,t≥0,
where {Pj}j≥1 is a complete family of orthogonal projections in the Hilbert space Z1/2 given by
(2.14)Pj=diag⁡[Ej,Ej],j≥1,Aj=RjPj,Rj=[01-λj0].
Also,(2.15)Aj*=Rj*Pj,Rj*=[0-1λj0].
Moreover, eAjs=eRjsPj and the eigenvalues of Rj are λji and -λji.Now, before proving the main theorem, we will give the definition of approximate controllability for this system. To this end, for allz0∈Z1/2 and u∈L2(0,τ;U), the initial value problem(2.16)z′=Az+Bωu(t),z∈Z,t∈(0,τ],z(0)=z0,
where the control function u belongs to L2(0,τ;U), admits only one mild solution given by(2.17)z(t)=T(t)z0+∫0tT(t-s)Bωu(s)ds,t∈[0,τ].Definition 2.3 (Approximate Controllability).
The system (2.16) is said to be approximately controllable on[0,τ]if for everyz0,z1∈Z1/2,ɛ>0there existsu∈L2(0,τ;U)such that the solutionz(t)of (2.17) corresponding touverifies(2.18)z(0)=z0,‖z(τ)-z1‖<ɛ.Consider the following bounded linear operator:(2.19)G:L2(0,τ;Z)⟶Z,Gu=∫0τT(τ-s)Bωu(s)ds,
whose adjoint operator G*:Z→L2(0,τ;Z) is given by(2.20)(G*z)(s)=Bω*T*(τ-s)z,∀s∈[0,τ],∀z∈Z.
The following lemma is trivialLemma 2.4.
The equation (2.16) is approximately controllable on [0,τ] if, and only if, Rang(G)¯=Z.The following result is well known from linear operator theory.Lemma 2.5.
LetW and Z be Hilbert spaces and G*∈L(Z,W) the adjoint operator of the linear operator G∈L(W,Z). Then,
(2.21)Rang⁡(G)¯=Z⟺Ker⁡(G*)={0}.As a consequence of the foregoing Lemma, one can prove the following result.Lemma 2.6.
LetW and Z be Hilbert spaces and G*∈L(Z,W) the adjoint operator of the linear operator G∈L(W,Z). Then, Rang(G)¯=Z if, and only if, one of the following statements holds:  (a)
Ker(G*)={0},(b)
〈GG*z,z〉>0, z≠0 in Z,(c)
lim∝→0+α(αI+GG*)-1z=0,(d)
supα>0∥α(αI+GG*)-1∥≤1.The following theorem follows directly from (2.20) and Lemmas 2.4 and 2.6.Theorem 2.7.
The equation (2.16) is approximately controllable on  [0,τ] iff
(2.22)Bω*T*(t)z=0,∀t∈[0,τ],⟹z=0.Now, we are ready to formulate and prove the main theorem of this work.Theorem 2.8 (Main Theorem).
If for an open nonempty setω⊂Ω the restrictions  ϕj,kω=ϕj,k|ω of  ϕj,k to  ω are linearly independent functions on  ω, then for all  τ≥2/πm-1 the system (2.16) is approximately controllable on  [0,τ]. Moreover, a sequence of controls steering the system (2.16) from initial state  z0 to an  ϵ neighborhood of the final state  z1 at time  τ>0 is given by
(2.23)uα(t)=Bω*T*(τ-t)(αI+GG*)-1(z1-T(τ)z0),
and the error of this approximation Eα is given by
(2.24)Eα=α(αI+GG*)-1(z1-T(τ)z0).Proof.
We will apply Theorem2.7 to prove the controllability of system (2.16). To this end, we observe that the adjoint of operator Bω is by
(2.25)Bω*=[01ω],T*(t)z=∑j=1∞eAj*tPjz,z∈Z,t≥0.
Therefore,
(2.26)Bω*T*(t)z=∑j=1∞Bω*eRj*tPjz.
On the other hand, we have that
(2.27)eRj*t=[cos⁡⁡λjt-1λjsin⁡λjtλjsin⁡λjtcos⁡⁡λjt]=[cos⁡⁡jmπmt-1jmπmsin⁡jmπmtjmπmsin⁡jmπmtcos⁡⁡jmπmt].
Suppose for all z∈Z1/2 that
(2.28)Bω*T*(t)z=∑j=1∞{jmπmsin⁡(jmπmt)(1ωEjz1)+cos⁡⁡(jmπmt)(1ωEjz2)}=0,∀t∈[0,τ].
Then, if we make the change of variable s=πmt, we obtain that
(2.29)∑j=1∞{jmπmsin⁡(jms)(1ωEjz1)+cos⁡⁡(jms)(1ωEjz2)}=0,∀s∈[0,πkτ].
Since τ≥2/πm-1 we get that
(2.30)∑j=1∞{jmπmsin⁡(jms)(1ωEjz1)+cos⁡⁡(jms)(1ωEjz2)}=0,∀s∈[0,2π].
On the order hand, it is well known that {1,cos(ns),sin(ns):n=1,2,3,…} is an orthogonal base of L2[0,2π], which implies that {cos(jms),sin(jms):j=1,2,3,…} is an orthogonal set in L2[0,2π], and therefore
(2.31)(1ωEjz1)(x)=0,(1ωEjz2)(x)=0,∀x∈Ω,j=1,2,3,…,
that is,
(2.32)(Ejz1)(x)=0,(Ejz2)(x)=0,∀x∈ω,j=1,2,3,…,
that is,
(2.33)∑k=1γj〈z1,ϕj,k〉ϕj,k(x)=0,∑k=1γj〈z2,ϕj,k〉ϕj,k(x)=0,∀x∈ω,j=1,2,3,….
Since the restrictions ϕj,kω=ϕj,k|ω of ϕj,k to ω are linearly independent functions on ω, we get that
(2.34)〈z1,ϕj,k〉=0,〈z2,ϕj,k〉=0,j=1,2,3,…;k=1,2,…,γj.
Therefore,
(2.35)Pj(z)=[Ej(z1)Ej(z2)]=[00].
Hence, z=∑j=1∞Pjz=0, and the proof of the approximate controllability of the system (2.16) is completed.
Now, given the initial and the final statesz0 and z1, we consider the sequence of controls
(2.36)uα(⋅)=Bω*T*(τ-⋅)(αI+GG*)-1(z1-T(τ)z0)=G*(αI+GG*)-1(z1-T(τ)z0),α>0.
Then,
(2.37)Guα=GG*(αI+GG*)-1(z1-T(τ)z0)=(αI+GG*-αI)(αI+GG*)-1(z1-T(τ)z0)=z1-T(τ)z0-α(αI+GG*)-1(z1-T(τ)z0).
From part (c) of Lemma 2.6, we know that
(2.38)lim⁡α→0+⁡α(αI+GG*)-1(z1-T(τ)z0)=0.
Therefore,
(2.39)lim⁡α→0+⁡Guα=z1-T(τ)z0,
that is,
(2.40)lim⁡α→0+⁡{T(τ)z0+∫0τT(τ-s)Bωuα(s)ds}=z1.
This completes the proof of the theorem.The following basic theorem will be used to prove an important consequence of the foregoing theorem.Theorem 2.9 (see [7,  Theorem  1.23,  page  20]).
SupposeΩ⊂ℝn is open, nonempty, and connected set, and  f is real analytic function in  Ω with  f=0 on a nonempty open subset  ω of  Ω. Then,  f=0 in  Ω.Corollary 2.10.
Ifϕj,k are analytic functions on  Ω, then for all open nonempty set  ω⊂Ω and all  τ≥2/πm-1 the system (2.16) is approximately controllable on  [0,τ].Proof.
It is enough to prove that, for all open nonempty setω⊂Ω the restrictions ϕj,kω=ϕj,k|ω of ϕj,k to ω are linearly independent functions on ω, which follows directly from Theorem 2.9.


## 3. Applications

For the applications, we will use Corollary2.10 and the following fact.Theorem 3.1 (see [3]).
The eigenfunctions of the operator-Δ with Dirichlet boundary conditions on  Ω are real analytic functions in  Ω.In this section, we will prove the approximate controllability of  (1.7) and (1.8). Specifically, we will prove the following theorem.Theorem 3.2.
For all open nonempty setω⊂Ω=[0,1], we have the following statements.  (a)
For allτ≥2 the system (1.7) is approximately controllable on [0,τ].(b)
For allτ≥2/π the system (1.8) is approximately controllable on [0,τ].Proof.
LetX=L2(Ω) and consider the linear unbounded operator -Δ:D(-Δ)⊂X→X with D(-Δ)=H01(Ω)∩H2(Ω). In this case, the eigenvalues and the eigenfunctions of A=-Δ are given, respectively, by
(3.1)λj=j2π2,ϕj(x)=2sin⁡(jπx),j=1,2,3,…,Ax=-Δx=∑j=1∞λj〈x,ϕj〉ϕj=∑j=1∞j2π2〈x,ϕj〉ϕj.
Then, m=1 and τ≥2. So, (a) follows from Corollary 2.10.
To prove (b), we consider the operator(3.2)Ax=(-Δ)2x=∑j=1∞λj2〈x,ϕj〉ϕj=∑j=1∞j4π4〈x,ϕj〉ϕj.
Then, m=2 and τ≥2/π. So, (b) follows from Corollary 2.10.


## 4. Final Remark

The result presented in this paper can be formulated in a more general setting. Indeed, we can consider the following second-order evolution equation in a general Hilbert spaceX:(4.1)ÿ+Ay=Cu(t),t∈(0,τ],y(0)=y0,ẏ(0)=y1,
where, A:D(A)⊂X→X is an unbounded linear operator in X with the spectral decomposition given by (1.2), the control u∈L2(0,τ;X) and C:X→X is a linear and bounded operator (linear and continuous). In this case, the characteristic function set is a particular operator C, and the following theorem is a generalization of Theorem 2.8.Theorem 4.1.
If the vectorsC*ϕj,k are linearly independent in  X, then for all  τ≥2/πm-1 the system (4.1) is approximately controllable on  [0,τ]. Moreover, a sequence of controls steering the system (2.16) from initial state  z0 to an  ϵ neighborhood of the final state  z1 at time  τ>0 is given by
(4.2)uα(t)=B*T*(τ-t)(αI+GG*)-1(z1-T(τ)z0),
and the error of this approximation Eα is given by
(4.3)Eα=α(αI+GG*)-1(z1-T(τ)z0),
where the operator B is given by B=[0C].The novelty of this result is based on the fact that, it is general, rigorous, applicable, and easily comprehensible by those young mathematician who are located in places away from majors research center.



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