DOI:10.1051/cocv/2013057 www.esaim-cocv.org
A SINGULAR CONTROLLABILITY PROBLEM WITH VANISHING VISCOSITY
Ioan Florin Bugariu
1and Sorin Micu
1Abstract. The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional powerαof the Dirichlet Laplace operator. Through the parameterαwe may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption thatα∈[0,1)\1
2
. It follows that, under this assumption, our starting question has a positive answer.
Mathematics Subject Classification. 93B05, 30E05, 35E20.
Received June 6, 2012. Revised February 27, 2013.
Published online December 10, 2013.
1. Introduction
ForT >0, we consider the one-dimensional linear wave equation with “lumped” control
⎧⎪
⎨
⎪⎩
wtt(t, x)−∂xx2 w(t, x) =v(t)f(x) (t, x)∈(0, T)×(0, π) w(t,0) =w(t, π) = 0 t∈(0, T)
w(0, x) =w0(x), wt(0, x) =w1(x)x∈(0, π),
(1.1)
where the profile f ∈L2(0, π) is given and verifies fn = 0 for every n≥1. Here and in the sequel, given any functiong∈L2(0, π), we denote bygn then−th Fourier coefficient ofg,
gn=
π
0 g(x) sin(nx)dx (n≥1).
Equation (1.1) is said to benull-controllable in timeT >0 if, for every initial data (w0, w1)∈ H0⊂H01(0, π)× L2(0, π), there exists a controlv∈L2(0, T) such that the corresponding solution of (1.1) verifies
w(T,·) =wt(T,·) = 0, (1.2)
Keywords and phrases.Wave equation, null-controllability, vanishing viscosity, moment problem, biorthogonals.
1 Facultatea de Stiinte Exacte, Universitatea din Craiova, 200585, Romania.
florin bugariu 86@yahoo.com; sd micu@yahoo.com
Article published by EDP Sciences c EDP Sciences, SMAI 2013
A SINGULAR CONTROLLABILITY PROBLEM WITH VANISHING VISCOSITY
where the spaceH0is defined as follows H0=
⎧⎪
⎨
⎪⎩(w0, w1)∈H01(0, π)×L2(0, π)
n≥1
n2wn02+wn12 fn2
<∞
⎫⎪
⎬
⎪⎭. (1.3)
The goal is to drive the initial data (w0, w1) to rest by using a controlv(t), depending only on time and acting on the system through a given shape function in spacef(x). Such types of controls are often used and sometimes called “lumped” or “bilinear” (see, for instance, [1,12,21]).
The controllability properties of (1.1) are by now well-known (see, for instance, the monographs [7,34]).
One of the oldest methods used to study such controllability problems consists in reducing them to a moment problem whose solution is given in terms of an explicit biorthogonal sequence to a family Λ of exponential functions. For instance, this method was used by Fattorini and Russell in the pioneering articles [12,13] to prove the controllability of the one dimensional heat equation. In their case, the family Λ has only real exponential functions. On the contrary, when equation (1.1) comes into discussion, the family Λ is given by (eμnt)n∈Z∗, where μn =in, n ∈ Z∗, are the eigenvalues of the wave operator
0 −I
−∂2xx 0
and are purely imaginary. It follows easily that, (1.1) is null-controllable in timeT if, and only if, for every initial data (w0, w1)∈ H0, there exists a solutionv∈L2(0, T) of the following moment problem:
T2
−T2
v
t+T 2
eμntdt=−e−T2μn f|n|
w1|n|+μnw|0n|
(n∈Z∗). (1.4)
In order to fix some ideas and to illustrate the method used in this paper, let us briefly show how do we obtain a solution of (1.4). We begin by defining the function
Ψm(z) =sin(π(z+m))
π(z+m) , (1.5)
which is an entire function of exponential type πsuch that
R
Ψm(x)2dx <∞.It results from Paley–Wiener Theorem that the Fourier transform ofΨm,
θm(t) = 1
2π RΨm(x)e−ixtdx (m∈Z∗), (1.6)
belongs to L2(−π, π). Moreover, from the inversion formula, it follows that θm
m∈Z∗ forms a biorthogonal sequence to the familyΛ= (eμnt)n∈Z∗,i.e.verify
π
−π
θm(t)eμntdt=δmn (m, n∈Z∗). (1.7)
From the above properties, we deduce that a formal solution of the moment problem (1.4) is given by v(t) =−
m∈Z∗
e−πμm f|m|
w|1m|+μmw0|m|
θm(t−π) (t∈(0,2π)). (1.8)
In fact (1.8) gives a true solution of (1.4) if the right hand side of (1.8) converges in L2(0,2π). For each (w0, w1)∈ H0, the convergence of this series follows from the existence of a constantC >0 such that
θmL2(−π,π)≤C (m∈Z∗), (1.9)
I.F. BUGARIU AND S. MICU
which is a consequence of the uniform boundedness (inm) of theL2(R)−norms of (Ψm)m∈Z∗ and Plancherel’s Theorem. Hence, for any initial data (w0, w1)∈ H0, the moment problem has at least a solutionv∈L2(0,2π), given by (1.8), and the controllability of (1.1) in timeT = 2πfollows.
In many applications it is of interest to study the uniform controllability properties of (1.1) when a viscous term is introduced in the equation. Indeed, the mechanism of vanishing viscosity is a common tool in the study of Cauchy problems or in improving convergence of numerical schemes for hyperbolic conservation laws and shocks. For instance, in [16,17], it is proved that, by adding an extra numerical viscosity term, the dispersive properties of the finite difference scheme for the nonlinear Schr¨odinger equation become uniform when the mesh- size tends to zero. This scheme reproduces at the discrete level the properties of the continuous Schr¨odinger equation by dissipating the high frequency numerical spurious solutions. On the other hand, a viscosity term is introduced in [10] to prove the existence of solutions of hyperbolic equations. In both examples the viscosity is devoted to tend to zero in order to recover the original system. Thus, a legitimate question is related to the behavior and the sensitivity of the controls during this process. For instance, given T >0 andε∈ (0,1), one could consider the perturbed wave equation
⎧⎨
⎩
utt(t, x)−∂xx2 u(t, x) + 2ε(−∂xx2 )αut(t, x) =vε(t)f(x) (t, x)∈(0, T)×(0, π)
u(t,0) =u(t, π) = 0 t∈(0, T)
u(0, x) =u0(x), ut(0, x) =u1(x) x∈(0, π)
(1.10)
and study the possibility of obtaining a controlvof (1.1) as limit of controlsvε∈L2(0, T) of (1.10). Here and in what follows (−∂2xx)αdenotes the fractional power of orderα≥0 of the Dirichlet Laplace operator in (0, π).
More precisely,
(−∂xx2 )α:D((−∂xx2 )α)⊂L2(0, π)→L2(0, π), D((−∂xx2 )α) =
⎧⎨
⎩u∈L2(0, π) : u=
n≥1
ansin(nx) and
n≥1
|an|2n4α<∞
⎫⎬
⎭, u(x) =
n≥1
ansin(nx) −→ (−∂xx2 )αu(x) =
n≥1
ann2αsin(nx).
(1.11)
Equation (1.10) is dissipative and it can be easily checked that, iff = 0, d
dt
u(t)2H1
0 +ut(t)2L2
=−2ε
π 0
(−∂xx2 )α2ut(t, x)2dt≤0. (1.12) Hence, 2ε(−∂xx2 )αut(t, x) represents an added viscous term devoted to vanish as εtends to zero. However, the controllability properties of (1.10) are poor. Indeed, the family of exponential functions corresponding to this case is given by Λ= (eνnt)n∈Z∗, whereνn=ε|n|2α+ sgn (n)
|n|4α−n2. Ifα > 12, we have that
n→−∞lim νn= 0,
which implies that the family Λis not minimal. Consequently, equation (1.10) is not spectrally controllable if α > 12 (for more details in the caseα= 1, see [30]).
Since we want to allow stronger dissipative terms which correspond to the caseα > 12, we perturb the wave equation (1.1) in the following slightly different way
⎧⎪
⎨
⎪⎩
utt(t, x)−∂xx2 u(t, x) + 2ε(−∂2xx)αut(t, x) +ε2(−∂xx2 )2αu(t, x) =vε(t)f(x) (t, x)∈(0, T)×(0, π)
u(t,0) =u(t, π) = 0 t∈(0, T)
u(0, x) =u0(x), ut(0, x) =u1(x) x∈(0, π).
(1.13)
A SINGULAR CONTROLLABILITY PROBLEM WITH VANISHING VISCOSITY
Equation (1.13) is still dissipative. Indeed, iff = 0, we have that d
dt
u(t)2H1
0 +ε2(−∂xx)αu(t)2L2+ut(t)2L2
=−2ε π
0
(−∂xx2 )α2ut(t, x)2dt≤0. (1.14) Note that, if α ≤ 12, the norm
u2H1
0+ε2(−∂xx)αu2L2+ut2L2 is equivalent to u2H1
0 +ut2L2 and the controllability properties of (1.10) and (1.13) are similar. However, in the case α > 12, the controllability properties of (1.13) are better than those of (1.10). Hence, the term ε2(−∂xx)2αu(t, x) allows us to consider stronger dissipation and also to simplify some of our estimates.
The aim of this paper is to study the controllability properties of (1.13) and their relation with the ones of (1.1). The controllability of (1.13) is defined in a similar way as for (1.1). More precisely, given T >0 and f ∈L2(0, π) withfn= 0 forn≥1, equation (1.13) isnull-controllable in timeT if, for any (u0, u1)∈ H0, there exists a controlvε∈L2(0, T) such that the corresponding solution (u, ut) of (1.13) verifies
u(T,·) =ut(T,·) = 0. (1.15)
The null-controllability problem is equivalent to find, for every initial data (u0, u1)∈ H0, a solutionvε∈L2(0, T) of the following moment problem:
T2
−T2 vε
t+T
2
eλntdt=−e−λnT2 f|n|
u1|n|+λnu0|n|
(n∈Z∗), (1.16)
where λn =in+ε|n|2α,are the eigenvalues of the operator
0 −I
−∂xx2 +ε2(−∂xx2 )2α 2ε(−∂2xx)α
corresponding to the “adjoint” problem of (1.13).
As in (1.4), if we have at our disposal a biorthogonal sequence to the family eλnt
n∈Z∗, denoted by (θm)m∈Z∗, we can give immediately a formal solution of (1.16),
vε(t) =−
m∈Z∗
e−λmT2 f|m|
u1|m|+λmu0|m|
θm
t−T
2
(t∈(0, T)). (1.17) This time the family
eλnt
n∈Z∗ has no longer purely imaginarily exponents like in (1.1). Thus, it is not so easy as for (1.1) to give explicit entire functions (Ψm)m∈Z∗ whose Fourier transforms define a biorthogonal sequence (θm)m∈Z∗ to
eλnt
n∈Z∗. Moreover, we cannot guarantee anymore the boundedness of the sequence (θm)m∈Z∗
and (1.9) will be replaced by an estimate of the form
||θm||L2≤Ceβ|(λm)| (m∈Z∗). (1.18) Note that||θm||L2 may become exponentially large as m goes to infinity. By taking into account the damping mechanism introduced in equation (1.13), this growth estimate guarantees the convergence of series (1.17) for each initial data (u0, u1)∈ H0, ifT is large enough. However, in order show that a control timeT independent ofεcan be chosen and to prove the boundedness of the family of controls (vε)ε∈(0,1)inL2(0, T), the dependence in ε of the constants C and β from (1.18) is required. This represents one of the most difficult tasks of our work. We shall prove that C andβ from (1.18) can be chosen independent ofε, fact that ensures the uniform boundedness of the sequence (vε)ε∈(0,1) and the possibility to pass to the limit asεtends to zero in (1.13). The main result of this paper reads as follows.
Theorem 1.1. Letα∈[0,1)\1
2
andf ∈L2(0, π)be a function such thatfn = 0for everyn≥1. There exists a time T >0 with the property that, for any (u0, u1)∈ H0 andε∈(0,1), there exists a control vε ∈L2(0, T) of (1.13)such that the family(vε)ε∈(0,1)is uniformly bounded inL2(0, T)and any weak limit vof it, asεtends to zero, is a control in timeT for equation (1.1).
I.F. BUGARIU AND S. MICU
The controllability problem studied in this paper belongs to the interface between parabolic and hyperbolic equations. From this point of view, it is related to [8,14,23], where the controllability of the transport equation is addressed after the introduction of a vanishing viscosity term. In [8] Carleman estimates are used to obtain an uniform bound for the family of controls. The same result is shown in [14], improving the control time, by means of nonharmonic Fourier analysis and biorthogonal technique. The recent article [23] deals with a nonlinear scalar conservation law perturbed by a small viscosity term and proves the uniform boundedness of the boundary controls. Related problems in which controls for an equation are obtained as limits of controls of equations of different type may be also found in [24,26,28,31,37].
In order to justify the damping mechanism introduced in (1.13), which involves the fractional powerα of the Laplace operator, let us point out that sometimes it may be useful to control the amount of dissipation introduced in the system not only by means of the vanishing parameterε but also by an adequate choice of the differential operator. For instance, the convergence rates in some perturbed problems can be improved by choosing a viscosity operator of lower order (see, for instance, [18] in the context of Hamilton−Jacobi equations).
In (1.13) this is achieved through the parameter α. The caseα = 1 has been studied, for a slightly different problem, in [25], where a uniform controllability result with respect to the viscosity is proved. Theorem 1.1 shows that a similar result holds for any α∈[0,1)\1
2
. Note that, ifα∈ 0,12
, the imaginary parts of the eigenvalues λn dominate the real ones and problem (1.13) has the same hyperbolic character as in the limit caseε= 0. On the contrary, ifα∈1
2,1
, (1.13) has a parabolic type. In this case we are dealing with a truly singular control problem and the pass to the limit is sensibly more difficult. Finally, let us remark thatα= 12 is a singular case in which the basic controllability properties (such as spectral controllability) of (1.13) do not hold.
Forα∈[0,1)\ {12}, the construction from the proof of Theorem 1.1implies that the following Ingham-type inequality (see [20]) holds, for any finite sequence (βn)n∈Z∗ and T sufficiently large,
C(T, α)
n∈Z∗
|βn|2e−ωε|n|2α ≤ T
−T
n∈Z∗
βneλnt
2
dt, (1.19)
where ε ∈ (0,1), ω is an absolute positive constant and C a positive constant depending of T and α but independent of ε. From this point of view our article extends the results from [11,15,32], where Ingham-type inequalities are obtained under a more restrictive uniform sparsity condition of the sequence (λn)n∈Z∗. Indeed, one of the major difficulty in our study is related to the fact that the sequence of our eigenvalues (λn)n∈Z∗ is not included in a sector of the positive real axis and does not verify a uniform separation condition of the type
|λn−λm| ≥δ|nβ−mβ| (n, m∈Z∗),
for some β > 1 and δ > 0 independent of ε. The fact that C(T, α) in (1.19) does not depend of ε is of fundamental importance since it ensures the uniform boundedness of a family of controls (vε)ε∈(0,1) for (1.13) and the possibility to pass to the limit in order to obtain a controlvfor (1.1).
The rest of the paper is organized as follows. Section 2 gives the equivalent characterization of the control- lability property in terms of a moment problem. The core of the paper is Section 3 where two biorthogonal sequences to the family
eλnt
n∈Z∗ are constructed and evaluated. The proof of Theorem 1.1 is provided in Section 4. The article ends with an Appendix in which a technical lemma is proved.
2. The moment problem
In this section we show the equivalence between the controllability problem (1.13)−(1.15) and the moment problem (1.16). In order to do this we need first a result concerning the existence of solutions for equation (1.13).
More precisely we have the following property.
A SINGULAR CONTROLLABILITY PROBLEM WITH VANISHING VISCOSITY
Proposition 2.1. Given anyT >0,α∈[0,1],ε≥0,h∈L1(0, T;L2(0, π))and(u0, u1)∈H01(0, π)×L2(0, π), there exists a unique weak solution (u, ut)∈C([0, T], H01(0, π)×L2(0, π))of the problem
⎧⎨
⎩
utt+ (−∂xx2 )u+ε2(−∂xx2 )2αu+ 2ε(−∂xx2 )αut=h(t, x) (x, t)∈(0, π)×(0, T)
u(t,0) =u(t, π) = 0 t∈(0, T)
u(0, x) =u0(x) ut(0, x) =u1(x) x∈(0, π).
(2.1)
Proof. For anyα∈[0,1] andε≥0, let X =
⎧⎨
⎩
D((−∂xx2 )α)×L2(0, π) ifε >0 andα > 12
H01(0, π)×L2(0, π) ifε= 0 orε >0 andα≤12,
(for the definition of the operator (D((−∂xx2 )α),(−∂xx2 )α), see (1.11)). We suppose that the spaceX is endowed with the inner product
((u1, u2),(v1, v2))X = (∂xu1, ∂xv1)L2(0,π)+ε2((−∂xx2 )αu1,(−∂xx2 )αv1)L2(0,π)+ (u2, v2)L2(0,π).
Note that, sinceD((−∂xx2 )1/2) =H01(0, π), it results thatX ⊆H01(0, π)×L2(0, π). Now, we define the unbounded operator (D(A), A) inX as follows,
D(A) =
⎧⎨
⎩
D((−∂xx2 )2α)×D((−∂xx2 )α) ifε >0 andα > 12
H2(0, π)∩H01(0, π)×H01(0, π) ifε= 0 orε >0 andα≤ 12, A=
0 −I
−∂xx2 +ε2(−∂xx2 )2α2ε(−∂xx2 )α
.
The operator (D(A), A) is maximal and monotone inX. Indeed, for anyU = (u1, u2)∈D(A), we have that (AU, U)X = 2ε((−∂xx2 )α2u2,(−∂xx2 )α2u2)L2(0,π)≥0
from which we deduce that (D(A), A) is monotone in X.
To prove that (D(A), A) is maximal inX we only have to show that there existsλ >0 with the property that for anyF= (f1, f2)∈ X there exitsU = (u1, u2)∈D(A) such that
(A+λI)U =F, (2.2)
which is equivalent to
u2=λu1−f1
−∂xx2 u1+ε2(−∂xx2 )2αu1+λ2u1+ 2ελ(−∂xx2 )αu1=f2+ 2ε(−∂xx2 )αf1+λf1. (2.3) Nextly, we define the bilinear forma:X × X → Cand the linear formL:X → Cgiven by
a(ϕ, ψ) =
π
0 ϕx(x)ψx(x)dx+ε2
π
0 (−∂xx2 )αϕ(x)(−∂xx2 )αψ(x)dx + 2ελ
π
0 (−∂xx2 )α2ϕ(x)(−∂xx2 )α2ψ(x)dx+λ2
π
0 ϕ(x)ψ(x)dx, L(ψ) =
π 0
f2(x) + 2ε(−∂xx2 )αf1(x) +λf1(x)
ψ(x)dx,
I.F. BUGARIU AND S. MICU
where
X=
⎧⎨
⎩
D((−∂xx2 )α) ifε >0 andα > 12
H01(0, π) ifε= 0 orε >0 andα≤12. From Lax–Milgram Theorem it follows that the problem
a(ϕ, ψ) =L(ψ) (ψ∈X)
has a unique solutionϕ∈X. Consequently, (2.2) has a solutionU ∈D(A) and the maximality of the operator (D(A), A) follows. From Hille−Yosida’s Theorem we deduce that the operator (D(A), A) generates a semigroup of contraction (S(t))t≥0inX. At the same time, [4], Lemma 4.1.5 ensures the existence and uniqueness of a weak solution for (2.1) which belongs toC([0, T],X). SinceX ⊆H01(0, π)×L2(0, π)), the proof is completed.
Remark 2.2. A more general equation than (2.1) is studied in [5,6]. It is proved that, if 12 ≤ α ≤ 1, the generated semigroup is analytic on a triangular sector ofCcontaining the positive real axis. This is not true if 0≤α < 12·
Remark 2.3. A lumped control of the formvε(t)f(x) has been chosen to act on our system (1.13). Of course, other types of controls can be proposed (interior, boundary etc.). At least formally, these new controllability problems may be reduced to prove an inequality of type (1.19) and the same technique could be used to study all of them. However, in the boundary control case we need to work in a space ˜X not included in anyD((−∂x2)α), α >0, which requires much more care. We recall that the fractional Laplace operator may also be introduced by considering the Dirichlet to Neumann map for the two-dimensional cylinder (0, π)×R+(see [2,3] for details).
This method is commonly used in the recent literature since it allows to write nonlocal problems in a local way and this permits to use the variational techniques for these kind of problems. It is known that, for functions in the spaces D((−∂x2)α) from above, this definition is coherent with the spectral one (1.11) used by us (see [2], Cor. 3.6). Consequently, homogeneous Dirichlet boundary condition may be easily treated by using the spectral definition of the fractional Laplace operator. On the contrary, to study the nonhomogeneous boundary problems for the same operator (or for general nonlocal operator) the strategy should be different. These problems are known to be more difficult and even ill-posed [9].
Now we can give the characterization of the controllability property of (1.13)−(1.15) in terms of a moment problem. Based on the Fourier expansion of solutions, the moment problems have been widely used in linear control theory. We refer to [1,22,34,36] for a detailed discussion of the subject.
Theorem 2.4. Let T >0, ε∈(0,1), (u0, u1)∈ H0andf ∈L2(0, π). There exists a control vε∈L2(0, T)such that the solution (u, ut)of equation (1.13) verifies (1.15), if and only if, vε∈L2(0, T)satisfies
f|n|
T2
−T2
vε
t+T
2
eλntdt=−e−λnT2
u1|n|+λnu0|n|
(n∈Z∗), (2.4)
whereλn=in+ε|n|2α, for anyn∈Z∗. Proof. We consider the “adjoint” equation
⎧⎨
⎩
ϕtt+ (−∂xx2 )ϕ+ε2(−∂2xx)2αϕ−2ε(−∂xx2 )αϕt= 0 (x, t)∈(0, π)×(0, T)
ϕ(t,0) =ϕ(t, π) = 0 t∈(0, T)
ϕ(T, x) =ϕ0(x) ϕt(T, x) =ϕ1(x) x∈(0, π).
(2.5) If we multiply (1.13) by ϕ and we integrate by parts over (0, T)×(0, π), we deduce that vε ∈ L2(0, T) is a control for (1.13) if, and only if, it verifies
T 0 vε(t)
π
0 f(x)ϕ(t, x)dxdt=− π
0 u1(x)ϕ(0, x)dx+
π 0 u0(x)
ϕt(0, x)−2ε
−∂xx2
α
ϕ(0, x)
dx, (2.6)
A SINGULAR CONTROLLABILITY PROBLEM WITH VANISHING VISCOSITY
for every (ϕ, ϕt) solution of (2.5) with the initial data (ϕ0, ϕ1). Since (sin(nx))n≥1 is a basis for L2(0, π) we have to check (2.6) only for the initial data of the form (ϕ0, ϕ1) = (sin(nx),0) and (ϕ0, ϕ1) = (0,sin(nx)), for eachn≥1. In the first case the solution of (2.6) is given by
ϕ(t, x) = λn
λn−λne(t−T)λn+ λn
λn−λne(t−T)λn
sin(nx) (n∈N∗), (2.7)
whereas in the second case it becomes ϕ(t, x) =
1 λn−λn
e(t−T)λn+ 1 λn−λn
e(t−T)λn
sin(nx) (n∈N∗). (2.8)
By tacking in (2.6)ϕ of the form (2.7) and (2.8), we obtained that vε ∈L2(0, T) is a control of (1.13) if and
only if it verifies (2.4).
Remark 2.5. Note that (λn)n∈Z∗ introduced in the previous theorem are the eigenvalues of the differential operator corresponding to the “adjoint” equation (2.5).
Remark 2.6. The conditionfn = 0 for anyn∈N∗ is necessary in order to solve the moment problem (2.4) for any initial data in H0. Indeed, if there exists n0 ∈N∗ such thatfn0 = 0, then (2.4) has a solution only if the initial data (u0, u1) verify the additional conditionu1n0+λn0u0n0= 0.
We recall that (θm)m∈Z∗ ∈ L2(−T2,T2) is a biorthogonal sequence to the family of exponential func- tions
eλnt
n∈Z∗ ∈L2(−T2,T2) if and only if
T2
−T2 θm(t)eλntdt=δmn (m, n∈Z∗).
It is easy to see from (2.4) that, if (θm)m∈Z∗ is a biorthogonal sequence to the family of exponential functions eλnt
n∈Z∗ inL2(−T2,T2),then a control of (1.13) is given by vε(t) =−
m∈Z∗
e−λmT2 f|m|
u1|m|+λmu0|m|
θm
t−T
2
(t∈(0, T)), (2.9)
provided that the right hand side converges inL2(0, T).Now the main problem is to show that there exists a biorthogonal sequence (θm)m∈Z∗ to the family of exponential functions
eλnt
n∈Z∗inL2(−T2,T2) and to evaluate its norm, in order to prove the convergence of the right hand side of (2.9) for any (u0, u1)∈ H0.
3. Construction of a biorthogonal sequence
The aim of this section is to construct and evaluate an explicit biorthogonal sequence to the family etλn
n∈Z∗
in L2
−T2,T2
, whereλn=in+ε|n|2α are the eigenvalues introduced in Theorem2.4. In order to do that, we define a family (Ψm(z))m∈Z∗ of entire functions of exponential type independent of ε (see, for instance, [35]) such that Ψm(iλn) = δmn.The inverse Fourier transform of (Ψm)m∈Z∗ will give us the biorthogonal sequence (θm)m∈Z∗ that we are looking for. Each Ψm is obtained from a Weierstrass product Pm multiplied by an appropriate functionMmwith rapid decay on the real axis. Such a method was used for the first time by Paley and Wiener [29] and, in the context of control problems, by Fattorini and Russell [12,13]. The main difficulty in our analysis is to obtain good estimates for the behavior ofPmon the real axis and to construct an appropriate multiplier Mm in order to ensure the boundedness ofΨm on the real axis. As we shall see in Proposition3.6 below, the behavior of ln|Pm(x)| is always dominated by a subunitary power of|x|, ifα < 12 which facilitates
I.F. BUGARIU AND S. MICU
the entire construction and analysis. In the more difficult caseα > 12, ln|Pm(x)|behaves like|x| on an interval of length O1
ε
2α−11
. It is precisely this property which makes the construction of Mm more problematic and imposes the necessity of a careful analysis ofPm(x). Finally, the bounds obtained on the real axis forΨm
and the Plancherel’s Theorem, will provide the desired estimates forθmL2(0,T) and their dependence of the parametersm,εandα.
3.1. An entire function
In this subsection we construct the Weierstrass product Pm mentioned above and we study some of its properties. For everym∈Z∗, we define the function
Pm(z) =
n∈Z∗ n=m
1 + zi
λn
λn
λn−λm
· (3.1)
Firstly, let us state the following technical result concerning the second part of the productPm, whose proof will be given in the Appendix.
Lemma 3.1. There exists a constant C >0 such that, for all ε∈(0,1)andm∈Z∗, we have
n∈Z∗ n=m
λn
λn−λm
≤16 exp(Cεm2α). (3.2)
Now we pass to study the basic properties of the productPm. Proposition 3.2. Let α∈[0,1)\1
2
andε∈(0,1). For eachm∈Z∗,Pmis an entire function of exponential type at most L1, where
L1:=
⎧⎪
⎪⎨
⎪⎪
⎩ max
√ 2π 2 ,1−24εα
α∈
0,12 max
√ 2π 2 ,2α8−1
α∈1
2,1 , with the property that
Pm(iλn) =δmn (n∈Z∗). (3.3)
Remark 3.3. Note that Proposition3.2 does not consider the case α = 12· In fact, if α = 12, the family of exponential functions
eλnt
n∈Z∗ is complete inL2(0, a), for anya >0.Indeed, since
n∈Z∗
(λn)
1 +|λn|2 =∞, (3.4)
the completeness is a consequence of the Theorem Sz´asz–M¨untz [33]. Since this property remains true if we eliminate a finite number of elements, we deduce that
eλnt
n∈Z∗ is not minimal in L2(0, a) and there exists no biorthogonal sequence to it inL2(0, a). From the controllability point of view, it follows that (1.13) is not spectrally controllable ifα=12·
Proof of Proposition3.2. By taking into account the estimate (3.2) from Lemma3.1, we only have to study the function
Em(z) =
n∈Z∗ n=m
1 + zi
λn
·
A SINGULAR CONTROLLABILITY PROBLEM WITH VANISHING VISCOSITY
We have that
|Em(z)|= 1 + zi
λm
n∈N∗ n=|m|
1 + zi
λn
1 + zi λn
≤
1 + |z|
|λm| ∞
n=1
|λn|2+ 2|(λn)||z|+|z|2
|λn|2
=
1 + |z|
|λm|
exp [Nz]
n=1
ln
1 +2|(λn)||z|+|z|2
|λn|2
S1
+ ∞ n=[Nz]+1
ln
1 + 2|(λn)||z|+|z|2
|λn|2
S2
,
whereNz= |z|
2ε
2α1
.It follows that S1 ≤
[Nz]
n=1
ln
1 + 2|z|2
|λn|2
≤ Nz
0 ln
1 + 2|z|2 t2+ε2t4α
dt≤ Nz
0 ln
1 +2|z|2 t2
dt
≤√
2|z| ∞
0 ln
1 + 1 t2
dt=
√2π 2 |z|.
Thus, we have that
S1≤
√2π
2 |z|. (3.5)
Forα∈ 0,12
we have that
S2≤4|z| ∞
n=[Nz]+1
εn2α
n2+ε2n4α ≤4|z|∞
n=2
εn2α−2≤ 4ε 1−2α|z|.
Forα∈1
2,1
we defineγε= 1
ε 2α−11
and we deduce that S2 ≤ ∞
n=[Nz]+1
4|(λn)||z|
|λn|2 = 4|z| ∞
n=[Nz]+1
εn2α
n2+ε2n4α ≤4|z|
⎛
⎝[γε]
n=1
+ ∞ n=[γε]+1
⎞
⎠ εn2α n2+ε2n4α
≤4|z|
⎛
⎝[γε]
n=1
ε n2−2α+
∞ n=[γε]+1
1 εn2α
⎞
⎠≤ 8 2α−1|z|.
It follows that
S2≤
⎧⎨
⎩
4ε
1−2α|z|α∈ 0,12
2α8−1|z|α∈1
2,1 .
(3.6) From (3.5) and (3.6) we deduce thatEm(z) is an entire function of exponential type at mostL1 and the proof
ends.
3.2. Evaluation of P
mon the real axis
This subsection is devoted to study the behavior of the entire functionPm on the real axis. The main result will be presented in Proposition3.6below. Let us begin with the following two simple lemmas. We recall that, forα > 12, we have introduced the notationγε=
1 ε
2α−11 . Lemma 3.4. Let ε∈(0,1)be fixed and α∈[0,1)\1
2
. For any x≥0 there exists a unique xε≥0 such that x2=x2ε+ε2x4εα.Moreover, if α∈
0,12
orα∈1
2,1
andx≤γε, then xε≤x≤√
2xε, (3.7)
