HAL Id: hal-00906208
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Approximation of the controls for the beam equation with vanishing viscosity
Ioan Florin Bugariu, Sorin Micu, Ionel Roventa
To cite this version:
Ioan Florin Bugariu, Sorin Micu, Ionel Roventa. Approximation of the controls for the beam equation
with vanishing viscosity. 2014. �hal-00906208v2�
Approximation of the controls for the beam equation with vanishing viscosity
Ioan Florin Bugariu
∗Sorin Micu
†Ionel Rovent¸a
‡January 22, 2014
Abstract
We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.
Keywords: beam equation, control approximation, vanishing viscosity, moment problem, biorthogonals.
Mathematical subject codes: 93B05, 58J45, 65N06, 30E05.
1 Introduction
The controlled transversal vibrations of a 1–D beam with hinged boundary conditions are modelled by the following equation
(1.1)
u′′(t, x) +uxxxx(t, x) = 0 (t, x)∈(0, T)×(0,1) u(t,0) =u(t,1) =uxx(t,0) = 0 t∈(0, T)
uxx(t,1) =v(t) t∈(0, T)
u(0, x) =u0(x) x∈(0,1)
u′(0, x) =u1(x) x∈(0,1),
where by′we denote the derivative in time. In (1.1) the vector u
u′
represents the state andvis the control acting on the extremityx = 1 of the beam. GivenT > 0 we say that (1.1) is null–controllable in time T if, for every initial data
u0 u1
∈ H :=H01(0,1)×H−1(0,1), there exists a controlv ∈L2(0, T) such that the corresponding solution of (1.1) verifies
(1.2) u(T, x) =u′(T, x) = 0 (x∈(0,1)).
∗Department of Mathematics, University of Craiova, 200585, Romania. E-mail: florin bugariu 86@yahoo.com.
†Department of Mathematics, University of Craiova, 200585, Romania and Institute of Mathematical Statistics and Applied Mathematics, 70700, Bucharest, Romania. E-mail: sd micu@yahoo.com.
‡Department of Mathematics, University of Craiova, 200585, Romania. E-mail: ionelroventa@yahoo.com.
A classical duality argument reduces the null-controllability property of (1.1) to the observability inequality (1.3)
ϕ ϕ′
(0)
2
H
≤C Z T
0 |ϕx(t,1)|2dt, for any
ϕ0 ϕ1
∈ H, ϕ
ϕ′
being the solution of the following homogeneous adjoint equation
(1.4)
ϕ′′(t, x) +ϕxxxx(t, x) = 0 (t, x)∈(0, T)×(0,1) ϕ(t,0) =ϕ(t,1) =ϕxx(t,0) =ϕxx(t,1) = 0 t∈(0, T)
ϕ(T, x) =ϕ0(x) x∈(0,1)
ϕ′(T, x) =ϕ1(x) x∈(0,1).
Boundary observability inequality (1.3) can be proved by using Fourier series or multipliers techniques (see, for instance, [19, 21]). Consequently, equation (1.1) is null-controllable in any timeT.
The main objective of this work is to study the controllability properties of the semi–discrete space approx- imation of (1.1). For each N ∈N∗ we consider an equidistant partition of the interval (0,1), x0 = 0 < x1 = h < ... < xj = jh < ... < xN+1 = 1, where the mesh-size is h = N1+1. To discretize the boundary condi- tions of the problem using centered finite differences, we also introduce two external pointsx−1=x0−hand xN+2=xN+1+h. The semi–discretization by finite differences of (1.1) is then given by the following system
(1.5)
u′′j(t) +uj+2(t)−4uj+1(t)+6uhj4(t)−4uj−1(t)+uj−2(t) = 0 1≤j ≤N, t∈(0, T)
u0(t) = 0, uN+1(t) = 0 t∈(0, T)
u−1(t) =−u1(t), uN+2(t) =h2vh(t)−uN(t) t∈(0, T) uj(0) =u0j(x), u′j(0) =u1j(x) 1≤j≤N.
System (1.5) consists ofN linear equation withN unknownsu1, u2, . . . , uN. The quantitiesuj(t) approximate u(t, xj), the solution of (1.1), if
u0j u1j
1≤j≤N
is an approximation of the initial data of (1.1). For instance, in our numerical examples in which the initial data are sufficiently regular, we shall choose
(1.6) u0j=u0(jh), uj1=u1(jh) (1≤j≤N).
We consider the following controllability property for (1.5): givenT > 0 and u0j
u1j
1≤j≤N
∈C2N, we look for a controlvh∈L2(0, T)such that the corresponding solution
uj
u′j
1≤j≤N
of (1.5)verifies
(1.7) uj(T) =u′j(T) = 0 (1≤j≤N).
If this property is verified for every initial data u0j
u1j
1≤j≤N
∈C2N, we say that (1.5) isnull-controllable in timeT.
It is not difficult to show that the null–controllability property for (1.5) holds in any timeT >0. However, as it has been proved in [20], the corresponding observability inequality is not uniform inh. More precisely, if we denote byk · k1,−1the discrete version of the normk · kH (see Section 3, formula (3.7)), it can be shown that, for anyh >0, there exists a constantC=C(T, h) such that
(1.8)
ϕj
ϕ′j
1≤j≤N
(0)
2
1,−1
≤C Z T
0
ϕN(t)
h
2
dt,
for any ϕ0j
ϕ1j
1≤j≤N
∈C2N and ϕj
ϕ′j
1≤j≤N
solution of the backward equation
(1.9)
ϕ′′j(t) +ϕj+2(t)−4ϕj+1(t)+6ϕhj4(t)−4ϕj−1(t)+ϕj−2(t)= 0 1≤j≤N, t∈(0, T)
ϕ0(t) = 0, ϕN+1(t) = 0 t∈(0, T)
ϕ−1(t) =−ϕ1(t), ϕN+2(t) =−ϕN(t) t∈(0, T) ϕj(T) =ϕ0j(x), ϕ′j(T) =ϕ1j(x) 1≤j ≤N, but
(1.10) lim
h→0 sup
(ϕ, ϕ′) solution of (1.9)
ϕj
ϕ′j
(0)
2
1,−1
Z T
0
ϕN(t)
h
2
dt
=∞.
Note that (1.8) is the discrete version of the observability inequality (1.3) and (1.10) shows that the constant C(T, h) explodes as h tends to zero. In this situation we say that (1.9) is not uniformly (with respect to h) observableor, equivalently, system (1.5) isnot uniformly controllable.
As in the continuous case, inequality (1.8) ensures the null–controllability property of (1.5) in timeT for any fixedh >0. On the other hand, (1.10) implies that there exist initial data
u0 u1
∈ Hsuch that any sequence of discretes controls (vh)h>0of (1.5) diverges inL2(0, T) ashtends to zero, when the initial data are discretized as in (1.6). In order to obtain a uniform observability inequality, two possibilities have been proposed and analyzed in [20]:
(a) the class of solutions of (1.9) has been restricted to a space in which the high frequencies have been filtered out. Under this assumption, the corresponding observability inequality becomes uniform and, consequently, the projection of the solution of (1.5) over this filtered space is controlled to zero uniformly;
(b) the observed quantity in the right side of (1.8) has been reinforced by introducing an extra term. This shows that an additional boundary control, which vanishes in limit, makes the system uniformly control- lable.
The above analysis shows that the spurious high frequencies introduced by the discretization process are re- sponsible for the nonuniform controllability properties of (1.5) and their elimination or diminution can guarantee the convergence of the scheme. For more general uniform controllability results, by using filtered spaces, the interested reader is referred to [6, 7, 24].
Our paper studies a third possibility to restore the uniform controllability property which is more natural and avoids the necessity of working with projections or with an additional resources consuming control. To this aim, we introduce in the discrete equation (1.5) a numerical viscosity which vanishes in the limit. Since this term damps out the high frequencies which are responsible for (1.10), we can expect that it will also help us to restore the desired uniform observability inequality and to improve the convergence properties of the discrete controls. More precisely, in this paper we study the null–controllability of the following different discretization of (1.1) given by
(1.11)
uj′′(t) +uj+2(t)−4uj+1(t)+6uhj4(t)−4uj−1(t)+uj−2(t)−εu′j+1(t)−2uh′j2(t)+u′j−1(t) = 0 1≤j≤N, t∈(0, T)
u0(t) = 0, uN+1(t) = 0 t∈(0, T)
u−1(t) =−u1(t), uN+2(t) =h2vh(t)−uN(t) t∈(0, T)
uj(0) =u0j(x), u′j(0) =u1j(x) 1≤j≤N.
Note that, ifε= 0, the system (1.11) coincides with the system (1.5). Ifε >0, the ratioεu′j+1(t)−2uh′j2(t)+u′j−1(t)
represents a viscous term which damps out the spurious high frequencies. The parameterεdepends on the step sizehand verifies
(1.12) lim
h→0ε(h) = 0.
Hence, scheme (1.11) represents in fact (1.5) with an added numerical vanishing viscosity. Note that the parameter ε should be chosen small, in order to preserve the convergence and the accuracy of the numerical scheme, but also sufficiently large to improve the observability properties of the system. Our analysis allows to obtain the minimal range of the parameterεanswering to both desiderates. Indeed, we shall prove that
(1.13) ε≥ h2
2T ln 1 h
is a necessary condition for the uniform observability inequality. Moreover, our main result ensures, for each T >0, the existence of two positive constantsh0andc0such that, for anyh∈(0, h0) andε∈ c0h2ln h1
, h , system (1.11) is uniformly controllable in timeT(see Theorem 6.1 below). This result is obtained by constructing and evaluating some particular controls corresponding to a single eigenmode initial data.
The artificial viscosity is a common tool in many numerical schemes. Let us mention here only the works [5, 25, 28, 29, 30, 33] where the uniform stabilization of discrete hyperbolic equations is achieved and [15] where uniform Strichartz’s estimates for the discrete Schr¨odinger equation are obtained by using this method.
The vanishing viscosity method has been used in the context of control problems for the wave equation in [23]. As in the case of the semi–discrete beam equation (1.11), uniform observability inequality for the wave equation has been obtained by adding a viscous term of the same type withε=h. Although the main ideas are similar, many technical details and final results are different. Indeed, in the present article an important aim is to obtain a uniform controllability result in arbitrarily small time. This is not possible in the context of the wave equation but it is perfectly realistic for the beam equation. However, in order to achieve it, additional work (like the construction of two different biorthogonals in Section 5) is required. Also, we manage to prove the uniform controllability result with any viscosity parameterεgreater thanc0h2ln 1h
which, according to (1.13), belongs to the lowest allowed range of values. The corresponding result for the semi-discrete wave equation is still an open problem.
The numerical viscosity method can be easily implemented for more general domains and boundary conditions.
However, how to prove a uniform controllability result in these new settings in which the spectrum is not known explicitly and/or the control depends both on time and space variables is a challenging and still open problem.
The rest of the paper is organized as follows. In Section 2 we recall some well–known facts concerning the controllability properties of the continuous beam equation. In Section 3 we introduce the semi–discrete model, we discuss its corresponding controllability properties and we present a spectral analysis. Section 4 is devoted to some remarks concerning the discrete observability inequality and the transformation of our control problem into a moment problem. Section 5 gives the construction of a biorthogonal sequence and its main properties.
These results are used in Section 6 to prove the main result of uniform controllability of the scheme (1.11).
Section 7 is devoted to some numerical experiments based on this discrete scheme.
2 The continuous problem
In this section we recall some well–known properties concerning the boundary null–controllability problem for the continuous 1−D linear beam equation and we characterize the control by using the moment theory. More precisely, given any T > 0 and any initial data
u0 u1
∈ H := H01(0,1)×H−1(0,1) we look for a control
v∈L2(0, T) such that the solution of the beam equation
(2.1)
u′′(t, x) +uxxxx(t, x) = 0 (t, x)∈(0, T)×(0,1) u(t,0) =u(t,1) =uxx(t,0) = 0 t∈(0, T)
uxx(t,1) =v(t) t∈(0, T)
u(0, x) =u0(x) x∈(0,1)
u′(0, x) =u1(x) x∈(0,1),
verifies
(2.2) u(T, x) =u′(T, x) = 0 (x∈(0,1)).
If for any initial data u0
u1
∈ H there exists a control v ∈ L2(0, T) for (2.1)–(2.2), we say that problem (2.1) is null–controllable in timeT >0. We mention that, in this context, the null–controllability property is equivalent to the exact controllability property for which the target may be any state from the spaceH, not only the origin.
As we have mentioned above, we intend to write the control problem of the beam equation in terms of a moment problem. To do that we need the following variational result.
Lemma 2.1. Let T >0and the initial data u0
u1
∈ H. The functionv∈L2(0, T)is a control which drives to zero the solution of (2.1)in time T if, and only if, the following relation holds
(2.3)
Z T
0
v(t)ϕx(t,1)dt=−
u1, ϕ(0, ·)
D+
u0, ϕ′(0,·)
D
ϕ0 ϕ1
∈ H
,
where ϕ
ϕ′
∈ His the solution of the following adjoint backward problem
(2.4)
ϕ′′(t, x) +ϕxxxx(t, x) = 0 (t, x)∈(0, T)×(0,1) ϕ(t,0) =ϕ(t,1) =ϕxx(t,0) =ϕxx(t,1) = 0 t∈(0, T)
ϕ(T, x) =ϕ0(x) x∈(0,1)
ϕ′(T, x) =ϕ1(x) x∈(0,1),
andh, iD denotes the duality product between the spaces H01(0,1) andH−1(0,1).
Proof: If we multiply in (2.1) byϕand we integrate by parts over (0, T)×(0,1) we obtain thatv∈L2(0, T) is a null–control for (2.1) if and only if it verifies (2.3).
By denotingW = ϕ
ϕ′
, equation (2.4) is equivalent with
(2.5)
W′+AW = 0 W(0) =W0=
ϕ0 ϕ1
,
whereA:D(A)⊂ H → H is the unbounded, maximal and monotone operator inHgiven by D(A) =
ϕ0 ϕ1
∈H3(0,1)∩H01(0,1)×H01(0,1) : ϕ0xx(t,0) =ϕ0xx(t,1) = 0
,
A=
0 −I
∂x4 0
.
Moreover, the eigenvalues ofAare given by
(2.6) ηn=isgn(n)n2π2 (n∈Z∗),
and the corresponding eigenvectors are
(2.7) Φn= 1
sgn(n)nπ 1
−ηn
sin(nπx) (n∈Z∗).
We note that the sequence (Φn)1≤|n|≤N forms an orthonormal basis inH. The following lemma transforms the control problem into a moment problem by using the Fourier expansion of the solution of (2.1). We recall that the moment problems have been, from the very beginning, one of the most successful method used to study controllability properties (see [1, 4, 19, 31]).
Lemma 2.2. Problem (2.1)is null–controllable in timeT >0if, and only if, for any initial data u0
u1
∈ H with the Fourier expansions
(2.8)
u0 u1
= X
n∈Z∗
a0nΦn,
there exists a controlv∈L2(0, T)such that (2.9)
Z T 0
v(t)etηndt= (−1)ni a0n (n∈Z∗).
Proof: Since (Φn)n∈Z∗ given by (2.7) is a basis inH, from Lemma 2.1 it follows that (2.1) is null–controllable if, and only if, there exists v∈L2(0, T) such that (2.3) holds for any initial data which is an eigenfunction of the operatorA. If
ϕ0 ϕ1
= Φn, the corresponding solution of (2.5) is given by ϕ
ϕ′
(t) =eηn(t−T)Φn. In this case, for any initial data
u0 u1
of the form (2.8) we obtain that
−
u1, ϕ(0,·)
D+
u0, ϕ′(0,·)
D=isgn(n)a0ne−T ηn,
and Z T
0
v(t)ϕx(t,1)dt= (−1)n sgn(n)e−T ηn
Z T
0
v(t)etηndt.
The last two relations give (2.9) and the proof ends.
3 The discrete control problem
In this section we study a finite-difference space discretization of equation (2.1). In order to do this let us consider N ∈N∗, a step h= N1+1 and a equidistant mesh of the interval (0,1), 0 = x0 < x1 < . . . < xN < xN+1 = 1 withxj =jhfor anyj∈[0, N+ 1]. To discretize the boundary conditions of the problem using centered finite differences, we also introduce two external points x−1=x0−hand xN+2 =xN+1+h. As we said before in this paper we study the null–controllability of the following discretization of (2.1)
(3.1)
uj′′(t) +uj+2(t)−4uj+1(t)+6uhj4(t)−4uj−1(t)+uj−2(t)−εu′j+1(t)−2uh′j2(t)+u′j−1(t) = 0 1≤j≤N, t∈(0, T)
u0(t) = 0, uN+1(t) = 0 t∈(0, T)
u−1(t) =−u1(t), uN+2(t) =h2vh(t)−uN(t) t∈(0, T)
uj(0) =u0j(x), u′j(0) =u1j(x) 1≤j≤N,
whereεis a nonnegative parameter which depends on the mesh sizehand vanishes ashgoes to zero. Later on, we shall choose conveniently the parameterεin order to guarantee the uniform controllability, with respect to h, of scheme (3.1).
Now, let us write (3.1) as an abstract Cauchy form by using the matricesAh, Bh∈ MN×N(R) given by
Ah= h12
2 −1 0 0 . . . 0 0
−1 2 −1 0 . . . 0 0
0 −1 2 −1 . . . 0 0 ... ... . .. . .. . .. ... ... 0 0 . . . −1 2 −1 0 0 0 . . . 0 −1 2 −1
0 0 . . . 0 0 −1 2
and Bh=A2h.
If we denote by
Uh0=
u01 u02 ... u0N
, Uh1=
u11 u12 ... u1N
, Uh(t) =
u1(t) u2(t)
... uN(t)
and Fh(t) = h12
0 0 ... 0
−vh(t)
,
then system (3.1) may be written vectorially as follows:
(3.2)
( Uh′′(t) +BhUh(t) +εAhUh′(t) =Fh(t) t∈(0, T) Uh(0) =Uh0, Uh′(0) =Uh1.
We say that (3.2) is null–controllable in time T > 0 if for any initial data Uh0
Uh1
∈ C2N, we can find a controlvh∈L2(0, T) such that the corresponding solution
Uh
Uh′
of (3.2) verifies
(3.3) Uh(T) =Uh′(T) = 0.
Let us consider inCN the canonical inner product
(3.4) hf, gi=h
XN k=1
fkgk f = (fk)1≤k≤N ∈CN, g= (gk)1≤k≤N ∈CN .
Moreover, we introduce two additional discrete inner products
(3.5) (f, g)1=hAhf, gi f = (fk)1≤k≤N ∈CN, g= (gk)1≤k≤N ∈CN , with the corresponding normk · k1 and
(3.6) (f, g)−1=hAh−1f, gi f = (fk)1≤k≤N ∈CN, g= (gk)1≤k≤N ∈CN , with the corresponding normk · k−1.
Finally, we define the discrete inner product in C2N
(3.7)
(f1, f2),(g1, g2)
1,−1= (f1, g1)1+ (f2, g2)−1 f1, f2, g1, g2∈CN
with the corresponding normk · k1,−1.
The controllability properties of (3.2) are directly related to the properties of the corresponding homogeneous
“adjoint” backward problem:
(3.8)
( Wh′′(t) +BhWh(t)−εAhWh′(t) = 0 t∈(0, T) Wh(T) =Wh0, Wh′(T) =Wh1,
where the initial data Wh0
Wh1
∈C2N are given. Indeed, we have the following discrete version of Lemma 2.1.
Lemma 3.1. Given T > 0, system (3.2) is null–controllable in time T if, and only if, for any initial data Uh0
Uh1
∈C2N there existsvh∈L2(0, T)which verifies
(3.9)
Z T
0
vh(t) WhN(t) h dt=
Uh1, Wh(0)
−
Uh0, Wh′(0)−εAhWh(0) Wh0 Wh1
∈C2N
,
where Wh
Wh′
is the solution of (3.8).
Proof: If we multiply in (3.2) by Wh and we integrate in time we obtain immediately thatvh∈L2(0, T) is a control which drives to zero the solution of (3.2) if, and only if, (3.9) takes place for any
Wh0 Wh1
∈C2N.
Now, if we put Z=
Wh(t) Wh′(t)
, then system (3.8) has the following equivalent vectorial form
(3.10)
−Z′+AhZ = 0 Z(T) =ZT, where the operatorAh is given byAh=
0 −I A2h −εAh
.
We pass to study the eigenvalues and eigenfunctions of the matriceal operator corresponding to (3.10). We begin with the operatorAh, for which it is known that (see, for instance, [18]) the eigenvalues are given by
(3.11) µn= 4
h2sin2 nπh
2
(1≤n≤N), with the corresponding eigenvectors
(3.12) φnh= (sin(knhπ))1≤k≤N ∈RN (1≤n≤N).
We have the following result.
Lemma 3.2. The eigenvalues of the operatorAh are given by
(3.13) λn=µ|n|
ε+isgn(n)√ 4−ε2
2 (1≤ |n| ≤N),
and the corresponding eigenvectors are
(3.14) Φnh= 1
õ|n|
1
−λn
φ|hn| (1≤ |n| ≤N),
where φ|hn| are given by (3.12). Moreover, the vectors (Φhn)1≤|n|≤N form an orthonormal basis in C2N with respect to the discrete inner product (3.7).
Proof: The eigenvalues and the eigenvectors can be compute immediately. As for the last part, which is also an explicit computation, the reader is referred to [22, Proposition 3.1].
Remark 3.1. From (3.13)it follows that
(3.15) |λn|=µ|n| (1≤ |n| ≤N).
Hence, the eigenvalues λn are obtained from the purely imaginary values i µ|n| by a simple rotation of angle sgn(n) arctan√
4−ε ε
in the complex plane.
In what follows, we analyze more closely the eigenvalues of the operatorAh and, more precisely, the distance between them. Firstly, we introduce the sets:
F={n∈Z∗ : 1≤ |n| ≤M andN−M + 1≤ |n| ≤N}, G={n∈Z∗ : M+ 1≤ |n| ≤N−M}.
Letγ >0 be fixed and define the following quantity
(3.16) M =hγ
4 i+ 1.
In the sequel we shall suppose thatN is sufficiently large such thatM < N2 or, equivalently,
(3.17) h < h10= 2
6 +γ.
Since γ is a positive number independent of N and h, this can be done without any loos of generality. The following result shows that we have a uniform gapγ, at least for a part of the spectrum ofAh.
Lemma 3.3. ForM given by (3.16)we have that
(3.18) |λn+1| − |λn|> γ (M ≤ |n| ≤N−M + 1).
Proof: We remark that
(3.19) |λn+1| − |λn|= 4
h2sin πh
2
sin
(2n+ 1)πh 2
.
Taking into account that sin(2n+1)πh2 is an increasing function on [1, N/2] and a decreasing one on (N/2,1/h), it is sufficient to verify that we have a gapγonly in the extremities. Indeed, for n=M we have that
|λM+1| − |λM|= 4 h2sin
πh 2
sin
(2M + 1)πh 2
≥4(2M+ 1)≥γ, and forn=N−M+ 1 we deduce that
|λN−M+2| − |λN−M+1|= 4 h2sin
πh 2
sin
π−(2(N−M + 1) + 1)πh 2
≥4(2M−1)≥γ, which ends the proof of our lemma.
Remark 3.2. Relation (3.18) shows that the distance (the “gap”) between two consecutive eigenvalues of the family(λn)n∈G is greater than γ. Sinceγ can be taken arbitrarily large, we obtain that this distance may be as large as we want. Also, note that the eigenvalues from the complementary set, (λn)n∈F, form a finite family with a number of elements depending only ofγ.
4 The moment problem and the observability inequality
As mentioned at the beginning, our aim is to prove the uniform controllability property of (3.2) or, equivalently, to show that a uniform observability inequality for the corresponding adjoint system (3.8) holds. In order to fulfil our objectives, we transform first the control problem in an equivalent moment problem (see Theorem 4.1 below) and we provide a solution for it by using a biorthogonal sequence to the family of complex exponential functions eλnt
1≤|n|≤N. In the second part of this section we show how can we estimate the norm of this solution by using the observability inequality for (3.8) (see Proposition 4.1 below). This analysis allows us to provide a lower bound for the parameterεas a function of the step sizeh, which represents a necessary condition for the uniform controllability property of (3.2).
Let us begin with the following theorem which transforms the null–controllability problem into a moment problem.
Theorem 4.1. Let T > 0. System (3.2) is null–controllable in time T if, and only if, for any initial data Uh0
Uh1
∈C2N of the form (4.1)
Uh0 Uh1
= X
1≤|n|≤N
a0nhΦnh, there exists a controlvh∈L2(0, T)such that
(4.2) Z T
0
vh(t)eλntdt= (−1)n 1 2 cos nπh2
ip
4−ε2a0nh−εsgn(n)a0nh+εsgn(n)a0−nh
(1≤ |n| ≤N).
Proof: The proof of this theorem follows immediately from Lemma 3.1. Indeed, since according to Lemma 3.2 (Φnh)n∈Z∗ is a basis in C2N, it is sufficient to verify (3.9) for
Wh0 Wh1
= Φnh, 1≤ |n| ≤ N. In this case, the corresponding solution of (3.8) will be
Wh
Wh′
(t) = e(t−T)λnΦnh, which gives us that (4.3)
Z T 0
vh(t)
h WhN(t)dt= (−1)n+1sin(nπh) h√µ|n|
e−T λn Z T
0
vh(t)etλndt,
(4.4)
Uh1, Wh(0)
−
Uh0, Wh′(0)−εAhWh(0)
=−e−T λn 2
isgn(n)p
4−ε2a0nh−εa0nh+εa0−nh . From (4.3) and (4.4) it follows that (4.2) takes place and the proof finishes.
The notion of biorthogonality is very useful in the study of moment problems like (4.2). We recall that a sequence (θm)1≤|m|≤N ⊂ L2 −T2,T2
is biorthogonal to the family of exponential functions eλnt
1≤|n|≤N in L2 −T2,T2
if (4.5)
Z T2
−T2
θm(t)eλntdt=δmn (1≤ |m|,|n| ≤N).
It is easy to see from (4.2) that, if (θm)1≤|m|≤N is a biorthogonal sequence to the family of exponential functions eλnt
1≤|n|≤N inL2 −T2,T2
, then a solutionvh of (4.2) will be given by (4.6) vh(t) =
XN
|n|=1
(−1)ne−λnT2 2 cos nπh2
ip
4−ε2a0nh−εsgn(n)a0nh+εsgn(n)a0−nh θn
t−T
2
(t∈(0, T)).
Now, the main task is to show that there exists a biorthogonal sequence (θm)1≤|m|≤Nto the family eλnt
1≤|n|≤N
inL2 −T2,T2
and to evaluate itsL2−norm. This allows us to estimate the norm ofvh from (4.6) and to study the convergence properties of the family (vh)h>0.
Let us define
(4.7) C(T, h) = sup
Wh0 Wh1
∈C2N
Wh
Wh′
(0)
2
1,−1
Z T 0
WhN(t) h
2
dt ,
where Wh
Wh′
∈C2N denotes the solution of (3.8) with initial data Wh0
Wh1
. We have the following result.
Proposition 4.1. The constantC(T, h)defined by (4.7)is finite and, for each Uh0
Uh1
∈C2N, there exists a solutionvh∈L2 −T2,T2
of (4.2)with the property that
(4.8) kvhkL2≤(1 +ε)p
C(T, h)
Uh0 Uh1
1,−1
.
Proof: Let us first show thatC(T, h) is finite. By taking the initial data of the form Wh0
Wh1
= XN
|n|=1
b0nhΦnh, we obtain that
(4.9)
Wh
Wh′
(0)
2
1,−1
= XN
|n|=1
b0nh2e−2Tℜ(λn),
(4.10)
Z T 0
WhN(t) h
2
dt= Z T2
−T2
XN
|n|=1
b0nh(−1)n+1sin(nπh)e−T λn h√µ|n|
etλn
2
dt.
Both quantities from the right hand sides in (4.9) and (4.10) are norms inC2N. Hence, they are equivalent and the finiteness of the constantC(T, h) is proved.
Now, sinceC(T, h)<∞, we deduce from (4.9) and (4.10) that the following weighted Ingham type inequality takes place
(4.11)
XN
|n|=1
eb0nh
2 e−Tℜ(λn)
cos2 nπh2 ≤C(T, h) Z T2
−T2
XN
|n|=1
eb0nhetλn
2
dt,
for every finite sequence (˜b0nh)1≤|n|≤N.
On the other hand, by taking into account that we are dealing with a finite family of distinct exponential functions, it follows that there exists a unique biorthogonal sequence (θm)1≤|m|≤N to the family eλnt
1≤|n|≤N
which belongs to Spann eλnt
1≤|n|≤N
o. This biorthogonal sequence is constructed by using the Projection Theorem (see, for instance, [8, 13]). Moreover, from (4.11), we obtain that
(4.12)
XN
|m|=1
αmθm
2
L2(−T2,T2)
≤C(T, h) XN
|m|=1
|αm|2eT|ℜ(λm)|cos2 mπh
2
,
for any finite sequence (αm)1≤|m|≤N. It follows that vh defined by (4.6) is a solution of (4.2) which verifies (4.8).
Now, we pass to analyze the behavior of the observability constant C(T, h) from (4.7) with respect to h.
According to (4.8), if the constantC(T, h) is bounded inh(which is equivalent to say that (3.8) is uniformly observable), then the family (vh)h>0is bounded inL2(0, T). This represents a key condition for the convergence of our numerical scheme. In [20] it was proved that, forε= 0, the observability constantC(T, h) is not uniform inh. More precisely, in this particular case
(4.13) C(T, h)≥ 1
Tcos2 N πh2 ≥ 4 π2T h2.
Whenε >0 is sufficiently large, due to the viscous term, we can obtain a uniform estimate of the observability constant. This is a difficult problem which will be proved at the end of this paper (see Theorem 5.4 below).
However, it is easy to show that at least some particular initial data can be observed uniformly, ifε is large enough.
Proposition 4.2. Let T >0. We have that
(4.14) C(T, h) = sup
Wh0 Wh1
∈C2N
Wh
Wh′
(0)
2
1,−1
Z T
0
WhN(t) h
2
dt
≥ 16ε
π2h4
e4T εh2 −1.
Proof: Let us consider the initial data Wh0
Wh1
= Φnh. Since the corresponding solution of (3.8) is given by Wh
Wh′
(t) =e(t−T)λnΦnh, we deduce from (4.9) and (4.10) that
(4.15)
Wh
Wh′
(0)
2
1,−1
Z T 0
WhN(t) h
2
dt
= 1
cos2 nπh2 εµ|n|
eT εµ|n|−1.
The proof is completed by taking into account that the function eT xx−1 is decreasing in (0,∞) andµn< h42. Remark 4.1. From Proposition 4.2 it follows that a necessary condition for the uniformity of the observability constant C(T, h) is that the constant εh−4
e4T εh2 −1−1
is bounded as h tends to zero. From this we deduce that a necessary condition for the uniformity of the observability constantC(T, h)is
(4.16) ε > 1
2Th2ln 1
h
.
On the other hand the proof of Proposition 4.2 shows that the solutions of (3.8)with initial data Wh0
Wh1
= Φnh, 1≤ |n| ≤N, are uniformly observable for any
(4.17) ε > 1
Th2ln 1
h
.
Finally, note that (4.16) implies that the real parts of the large eigenvalues λn are not uniformly bounded in h. Consequently, in order to obtain a uniform controllability result for (3.2), the perturbation introduced by the viscosity term should be unbounded.
5 Construction of the biorthogonal sequence
This section is devoted to construct and evaluate a biorthogonal sequence to the family Λ = eλnt
1≤|n|≤N
in L2 −T2,T2
, where (λn)1≤|n|≤N are given by (3.13). Since eλnt
1≤|n|≤N is a finite family of exponential functions, it follows immediately that it has infinitely many biorthogonal sequences inL2 −T2,T2
. However we are interested not only on their existence, but also on their dependence of the parametershandε. Therefore, we shall construct an explicit biorthogonal and we shall evaluate itsL2−norm. In order to succeed in our purpose we need to treat separately the biorthogonal sequences for the families Λ1 = eλnt
n∈F and Λ2 = eλnt
n∈G. This fact is motivated by the different behavior of the eigenvalues λn from these two families. Indeed, the exponents of Λ2 have a gap at leastγ (see Lemma 3.3 and Remark 3.2) whereas those of Λ1 are not so well separated. On the other hand, the cardinality of these two families is different: Λ2 has a number of elements which goes to infinity whenhtends to zero whereas Λ1has a number of elements depending only onγ.
Firstly, we present some notations and results that will be used along this section. For any n∈Z∗ such that
|n| ∈(M, N−M] we introduce the following quantities
(5.1) kn=
|λn| γ
,
(5.2) ekn=
kn if |λn| −γkn≤ γ2 kn+ 1 if γ(kn+ 1)− |λn| ≤ γ2. Lemma 5.1. We have the following properties:
(5.3) ˜knγ−µ|n|
≤ γ
2 (|n| ∈(M, N −M]),
(5.4) ˜kM+1γ−µM ≥ γ
2,
(5.5) µN−M+1−k˜N−Mγ≥γ
2.
Proof: From (3.15), by taking into account (5.1) and (5.2) we obtain immediately (5.3). For (5.4) and (5.5) we obtain, by using (3.18), that
k˜M+1γ−µM =µM+1−µM + ˜kM+1γ−µM+1≥µM+1−µM−γ 2 ≥ γ
2, and, respectively,
µN−M+1−k˜N−Mγ=µN−M+1−µN−M −˜kN−Mγ+µN−M ≥µN−M+1−µN−M −γ 2 ≥γ
2, which ends the proof of this lemma.
In the sequel, C >0 denotes a generic constant which may changes from one row to another but it is always independent ofh,εandm.
5.1 A biorthogonal sequence to the family Λ
1In this subsection we construct a biorthogonal sequence to the family Λ1 = eλnt
n∈F. Let us consider the following product
(5.6) Pm1(z) =
YM
|j|=1 j6=m
z−iλj
iλm−iλj
YN
|j|=N−M+1 j6=m
z−iλj
iλm−iλj
(z∈C, m∈F)
Firstly, we remark that
(5.7) Pm1(iλn) =δmn (m, n∈F).
In the following proposition we obtain some estimates for the product Pm1. Proposition 5.1. Let γ >0. There existsh10>0 such that for anyh∈ 0, h10
andε∈(0, h)the function Pm1 defined by (5.6)has the following property
(5.8) |Pm1(z)| ≤
C(1 +|z−iλm|)2M−1
1 +|z−Niλ2m|
2M
(1≤ |m| ≤M, z∈C) C(1 +|z−iλm|)2M
1 + |z−Niλ2m|
2M−1
(N−M+ 1≤ |m| ≤N, z∈C), whereC andω are two positive constants independent of h,εandm.
Proof: Leth10 be the constant given by (3.17). In order to evaluatePm1 we remark that (5.9) |Pm1(z)|=
YM
|n|=1 n6=m
z−iλn
i(λm−λn)
YN
|n|=N−M+1 n6=m
z−iλn
i(λm−λn) ≤
≤ YM
|n|=1 n6=m
1 + |z−iλm|
|λm−λn|
YN
|n|=N−M+1 n6=m
1 + |z−iλm|
|λm−λn|
.
By taking into account that, for 1≤ |m| ≤M we have λm−λn
>
C 1≤ |n| ≤M, n6=m CN2 N−M+ 1≤ |n| ≤N, and, forN−M+ 1≤ |m| ≤N we have
λm−λn>
CN2 1≤ |n| ≤M
C N−M + 1≤ |n| ≤N, n6=m, then from (5.9) we obtain immediately (5.8).
We are now in position to construct a biorthogonal sequence to the family of exponential functions Λ1. This is accomplished in the following theorem.
Theorem 5.1. Let T, γ > 0. There exists h10,c˜0 > 0 such that for any h ∈ (0, h10) and ε ∈ (˜c0h2ln1h, h) there exists a biorthogonal sequence(ψm)m∈F to the family of exponential functionsΛ1 inL2 −T4,T4
with the following properties
(5.10) kψmkL2(−T4,T4)≤C1eT8|ℜ(λm)| (m∈F),
(5.11) |ψb1(x)(x−iλ1)| ≤C1 (x∈R), (5.12) |ψb1(iλk)(iλk−iλ1)| ≥C2 (k∈G), whereC1 andC2 are two positive constants independent ofh,ε andm.
