# Numerical approximation of the averaged controllability for the wave
equation with unknown velocity of propagation

Mouna Abdelli and Carlos Castro Laboratory of Mathematics Informatics and
Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria
(mouna.abdelli@univ-tebessa.dz)Departamento de Matemática e Informática, ETSI
Caminos, Canales y Puertos, Polytecnical University of Madrid, 28040 Madrid,
Spain<EMAIL_ADDRESS>
Abstract: We propose a numerical method to approximate the exact averaged
boundary control of a family of wave equations depending on an unknown
parameter $\sigma$. More precisely the control, independent of $\sigma$, that
drives an initial data to a family of final states at time $t=T$, whose
average in $\sigma$ is given. The idea is to project the control problem in
the finite dimensional space generated by the first $N$ eigenfunctions of the
Laplace operator. When applied to a single (nonparametric) wave equation, the
resulting discrete control problem turns out to be equivalent to the Galerkin
approximation proposed by F. Bourquin et al. in [2]. We give a convergence
result of the discrete controls to the continuous one. The method is
illustrated with several examples in 1-d and 2-d in a square domain and allows
us to give some conjectures on the averaged controllability for the continuous
problem.

Keywords: Exact control, Numerical approximation, Averaged control, Projection
method.

##  1 Introduction

Consider the wave equation with a missing parameter $\sigma$ and a control $f$
acting on one part of the boundary:

$\left\\{\begin{array}[]{l}u_{tt}-a(\sigma)\Delta u=0\\\
u=f\chi_{\Gamma_{0}}\\\ u(x,0)=u^{0},\quad
u_{t}(x,0)=u^{1}\end{array}\begin{array}[]{c}in\text{ }Q\\\ on\text{
}\Sigma\\\ in\text{ }\Omega\end{array}\right.$ (1)

where $\Omega$ is an open bounded domain in $\mathbb{R}^{d}$ with smooth
boundary $\Gamma$ $,\Gamma_{0}$ be an open nonempty subset of $\Gamma,$
$\chi_{\Gamma_{0}}$ the characteristic function of the set $\Gamma_{0}$,
$t\in\left[0,T\right],T>0,Q=\Omega\times(0,T),a(\sigma)\in L^{\infty}$ is the
square of the velocity of propagation and we assume

$0<a_{m}\leq a(\sigma)\leq
a_{M}<\infty,\quad\sigma\in\Upsilon\subset\mathbb{R},$

for some constants $a_{m},a_{M}$ and $\Upsilon$ an interval of $\mathbb{R}$.
The function $f=f(x,t)$ is a control, independent of the unknown parameter
$\sigma$ and which acts on $\Gamma_{0}$.

For each value of the parameter $\sigma\in\Upsilon$, problem (1) has a unique
solution $(u,u_{t})\in C(\left[0,T\right],L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right))$ for any $f\in
L^{2}\left((0,T)\times\Gamma_{0}\right)$ and $(u^{0},u^{1})\in
L^{2}\left(\Omega\right)\times H^{-1}\left(\Omega\right)$. Of course, these
solutions will depend on the parameter $\sigma\in\Upsilon$ and we will write
$u(x,t;\sigma)$, when we want to make this dependence explicit.

Moreover, there exists $C=C(T)>0$, independent of $\sigma\in\Upsilon$, such
that the solution of (1) satisfies the following inequality

$\left\|(u,u_{t})\right\|_{L^{\infty}(\left[0,T\right];L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right))}\leq
C\left(\left\|(u^{0},u^{1})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}+\left\|f\right\|_{L^{2}\left((0,T)\times\Gamma_{0}\right)}\right).$
(2)

We are interested in the numerical approximation of the following
controllability problem: Given $T>0$, the initial data $(u^{0},u^{1})\in
L^{2}(\Omega)\times H^{-1}(\Omega)$ and a final target
$(u^{0}_{T},u^{1}_{T})\in L^{2}(\Omega)\times H^{-1}(\Omega)$, find $f\in
L^{2}(0,T;\Gamma_{0})$ such that the solution of (1) satisfies

$\int_{\Upsilon}u(x,T;\sigma)d\sigma=u^{0}_{T},\quad\int_{\Upsilon}u_{t}(x,T;\sigma)d\sigma=u^{1}_{T},\quad
x\in\Omega.$ (3)

When such a control exists we say that the initial data $(u^{0},u^{1})\in
L^{2}\left(\Omega\right)\times H^{-1}\left(\Omega\right)$ is controllable in
average to the target $(u^{0}_{T},u^{1}_{T})$.

This notion of controllability in average (or averaged controllability) was
first introduced by E. Zuazua in [24], where sufficient conditions were given
for the controllability of finite dimensional systems. In this work the author
also highlight the difficulty to extend classical controllability results for
PDE to obtain averaged controls. Precisely, the fact that the dynamics of the
average is no longer solution of the same PDE. There have been a considerable
effort to overcome this difficulty and obtain results in this direction for
different distributed systems (see for example [18], [22]). For the particular
case of the wave equation, we can address to [14] where the authors consider
two different wave equation with the same control, or [16] where a more
general family of wave equations is considered. More precisely, they deal with
a parametric family of wave equations where the parameter is a measure
perturbation of a Dirac mass.

Control problems for parameter families of partial differential equations have
attracted a considerable interest in the last years due to their applicability
in engineering processes, where uncertainty in one or several parameters of
the model is common. In this case, it is natural to look for controls valid
for any value of the unknown parameter $\sigma\in\Upsilon$ and therefore,
independent of $\sigma$. In general, such controls are not likely to produce
solutions which attain the same target for all values of the parameters, as
for instance in our case
$(u(x,T;\sigma),u_{t}(x,T;\sigma))=(u^{0}_{T},u^{1}_{T})$ for all values of
$\sigma$. Instead, a weakened objective must be considered. Here we focus on
the notion of control in average for which the target is the average in
$\sigma$ of the solutions at time $t=T$. The controllability in average is
only one of the multiple ways to deal with an unknown parameter. There are
other notions in the literature according to specific applications as robust
control, where one looks for an approximate control valid for a range of
parameters, or the risk averse control where the control tries to avoid some
undesirable behaviors in the solutions. We refer to [19] and the references
therein where such problems are considered in the context of the optimal
control.

To illustrate the mathematical difficulties behind the particular situation we
consider in this work, i.e. the control in average for the wave equation, we
mention that for the one-dimensional model with the simplest choice
$a(\sigma)=\sigma,$ $\sigma\in[\sigma_{0},\sigma_{1}]$, the existence of
controls is still open. A partial result is given in [16] assuming, among
other properties, that $\sigma_{0}^{-1}\sigma_{1}\notin\mathbb{Q},$ which does
not seem a natural hypothesis in this context.

In this paper we do not address this difficult problem. We rather focus on the
numerical approximation of the averaged controls, assuming that they exist.
However, this is also far from being a simple exercise since the usual
approach based on the discretization of the controlled wave equation via
finite differences or finite elements does not inherit the properties of the
continuous model. Even for a single wave equation (i.e. without a parameter
dependence), the discrete controls obtained for the associated finite
dimensional systems may become unbounded as the discretization parameter goes
to zero. This was first observed in [13] where the authors considered a finite
dimensional version of the Hilbert Uniqueness Method introduced in [15]. Since
then, several cures have been proposed to recover convergence approximations
of the controls as bigrid algorithms, Tychonoff regularization, filtering,
mixed finite elements, etc (see for instance [12], [8], [10]). We refer to the
review [23] for a detailed description of such problems and references.

More recently, a direct approximation of the optimality system associated to
the continuous controllability problem with a conformal finite element method
has proven to be convergent in quite general situations as general geometries,
nonconstant coefficients and systems (see [6], [7]). In this approach, a
suitable Lagrangian formulation is introduced where the controlled and adjoint
equation are imposed as a constraint with a suitable Lagrange multiplier.
Following a similar Lagrangian approach, a convergence result for nonconformal
finite elements has been recently obtained in [5].

Here we follow a different approach based on a projection method of the
control problem on the finite dimensional subspace constituted by the first
eigenfunctions of the Laplacian. Then, we apply the Hilbert Uniqueness Method
to characterize the minimal $L^{2}$-norm controls. It turns out that the
variational formulation of the resulting finite dimensional control problem
can be interpreted as a Galerkin approximation for the continuous one. For
nonparametric PDE’s such Galerkin approximation was proposed by F. Bourquin
and co-workers in the nineties to approximate the boundary control (see [2],
[4], [3]). Here we adapt it to obtain averaged controls. As in the
nonparametric case, the existence and convergence of controls can be deduced
by the usual theory of Galerkin approximations for variational problems. The
finite dimensional control problem can be solved using the finite dimensional
theory introduced in [24] which provides an explicit expression for the
control in average.

The main advantage of this approach is that it provides a simpler and
efficient method to obtain numerical approximations of controls without any
regularization technique. Moreover, it can be easily adapted to parametric
systems as (1) that require to compute a large number of control problems to
approximate the averages involved. In fact, it can be used in any situation
where the control can be characterized variationally (simultaneous control,
parametric regional control [1], etc.). An important drawback is that it
requires to compute the eigenfunctions of the Laplace operator. This is simple
in one-dimensional problems or two dimensional ones in special domains
(rectangles or disks), but not for general domains or variable coefficients
equations. For general domains, there is an added difficulty in the
approximation of the normal derivatives in the variational characterization of
the controls (see formula (29) below). In [2] a second projection is proposed
for this approximation, this time on the subspace generated by the
eigenvectors of the associated capacitance operator on the boundary. We give
an alternative method that avoids to approximate this eigenvalue problem.

To illustrate the method we present some numerical experiments in one
dimension and two dimensions in a square domain. We also give numerical
evidences of several unknown properties for the dynamics of the average and
the control in average. In particular we find the following:

  1. 1.

The wave equation is controllable in average, with no restrictions on the
parameters interval, as long as the time $T$ is sufficiently large. In
particular this is true when the wave equation is controllable for all values
of $\sigma\in\Upsilon$, but it is also true even when this property holds only
for a subinterval in $\Upsilon$.

  2. 2.

The dynamics of the average decays in time, without control, to a constant
which depends on the initial data. When we apply a control, it takes advantage
of this decay and acts mainly at the end of the time interval.

  3. 3.

The averaged control converges to the control of the averaged parameter when
the parameter belongs to an interval of length $\varepsilon\to 0$.

The rest of the paper is divided as follows: in section 2 we adapt the usual
variational characterization of controls for the wave equation to the control
in average. In section 3 we introduce the observability inequality that allows
to prove the existence of controls and a particular class of controls that is
obtained by minimization of a suitable functional. In section 4 we introduce
the numerical method. In section 5 we prove the convergence of discrete
controls to the continuous one. In section 6 we deduce matrix formulation
equivalent to the discrete system and the controllability of the finite
dimensional system. Finally, in section 7 we present some numerical examples
in one dimension and in two dimensions on a square domain.

##  2 Variational characterization of the control in average

In this section we introduce a variational characterization of the controls in
average that we use later to find their numerical approximation. In particular
we prove that a class of controls in average can be obtained as minimizers of
a quadratic functional defined on a Hilbert space. The results of this section
are not new and can be found in [18] (appendix A.2). We include them here for
completeness.

For technical reasons we restrict ourselves to controls which are zero near
$t=0,T$. This affects to the quadratic functional that we define below. We
follow the approach in [20] for the wave equation, that we adapt to the
parametric system (1) for the control in average.

Let us consider the following backwards wave equation, for each
$\sigma\in\Upsilon$,

$\left\\{\begin{array}[]{l}\phi_{tt}-a(\sigma)\Delta\phi=0\\\ \phi=0\\\
\phi(x,T)=\phi^{0},\varphi_{t}(x,T)=\phi^{1}\end{array}\begin{array}[]{c}in\text{
}Q\\\ on\text{ }\Sigma\\\ in\text{ }\Omega\end{array}\right.$ (4)

where $(\phi^{0},\phi^{1})\in H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)$ are independent of $\sigma$.

We also define the duality product between $L^{2}(\Omega)\times
H^{-1}(\Omega)$ and $H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ by

$\left\langle(\phi^{0},\phi^{1}),(u^{0},u^{1})\right\rangle=\int_{\Omega}u^{0}\phi^{1}d\gamma-\left\langle
u^{1},\phi^{0}\right\rangle_{-1,1},$

where $\left\langle\cdot,\cdot\right\rangle_{-1,1}$ is the usual duality
product between $H_{0}^{1}$ and its dual space $H^{-1}$.

The following result provides a variational characterization of the control in
average.

###### Lemma 1

Assume that for $T>0$ and the data $(u^{0},u^{1}),(u^{0}_{T},u^{1}_{T})\in
L^{2}\times H^{-1}$ there exists a control function $f\in
L^{2}\left((0,T)\times\Gamma_{0}\right)$ for which the solutions of (1)
satisfy (3). Then, $f$ is solution of the variational identity

$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma
fd\gamma
dt-\left\langle\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle$
$\displaystyle+\left\langle(\phi^{0},\phi^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle=0,$
(5)

for all $(\phi^{0},\phi^{1})\in H_{0}^{1}\times L^{2}(\Omega)$, where
$(\phi,\phi_{t})$ is the solution of the backwards wave equation (4).
Reciprocally, if $f$ satisfies (5) then (3) holds.

Proof. Let us first suppose that
$(u^{0},u^{1}),(\phi^{0},\phi^{1})\in\mathcal{D}(\Omega)\times\mathcal{D}(\Omega),f\in$
$\mathcal{D}((0,T),\Gamma_{0})$ and let $u$ and $\phi$ be the smooth solutions
of (1) and (4) respectively.

Multiplying the equation of $u$ by $\phi$ and integrating by parts one obtains

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Omega}\int_{\Upsilon}\left(u_{tt}-a(\sigma)\Delta
u\right)\phi d\sigma dxdt=\left.\int_{\Omega}\int_{\Upsilon}\left(\phi
u_{t}-\phi_{t}u\right)d\sigma dx\right|_{0}^{T}$ (6)
$\displaystyle+\int_{0}^{T}\int_{\Gamma}\int_{\Upsilon}a(\sigma)\left(-\frac{\partial
u}{\partial\upsilon}\phi+\frac{\partial\phi}{\partial\upsilon}u\right)d\sigma
d\gamma dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma
fd\gamma
dt+\int_{\Omega}\left(\phi^{0}\int_{\Upsilon}u_{t}(T;\sigma)d\sigma-\phi^{1}\int_{\Upsilon}u(T;\sigma)d\sigma\right)dx$
$\displaystyle-\int_{\Omega}\left(u^{0}\int_{\Upsilon}\phi\left(0;\sigma\right)d\sigma-u^{1}\int_{\Upsilon}\phi_{t}(0;\sigma)d\sigma\right)dx$

By a density argument we deduce that an analogous formula holds for any
$(u^{0},u^{1})\in L^{2}\left(\Omega\right)\times H^{-1}\left(\Omega\right)$,
$(\phi^{0},\phi^{1})\in H_{0}^{1}\times L^{2}(\Omega)$ and $f\in
L^{2}(0,T;\Gamma_{0})$. We only have to replace the integrals by duality
products when corresponding. Now, if $f$ is a control such that (3) holds then
the weak form of (6) is equivalent to (5). Reciprocally, if (5) holds then the
weak form of (6) is equivalent to

$\left\langle(\phi^{0},\phi^{1}),\left(\int_{\Upsilon}u_{t}(T;\sigma)d\sigma,\int_{\Upsilon}u(T;\sigma)d\sigma\right)\right\rangle=\left\langle(\phi^{0},\phi^{1}),\left(u^{T,0},u^{T,1}\right)\right\rangle,$

for all $(\phi^{0},\phi^{1})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega),$ and
this is equivalent to (3).

One possibility to construct controls $f$ that satisfy the variational
condition (5) is as minimizers of a particular quadratic functional. We define
the following cost functional $J:H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)\rightarrow\mathbb{R}$ by:

$\displaystyle J(\phi^{0},\phi^{1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}\eta(t)\int_{\Gamma_{0}}\left|\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt-\left\langle\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle$
(7)
$\displaystyle+\left\langle(\phi^{0},\phi^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle,$

where $(\phi,\phi_{t})$ is the solution of (4) with the final data
$(\phi^{0},\phi^{1})\in H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)$. The function $\eta(t)$ is a prescribed smooth
function in $[0,T]$ introduced to guarantee that the controls vanish in a
neighborhood of $t=0,T$. Thus, we consider $\delta>0$ arbitrarily small and,

$\eta(t)=\left\\{\begin{array}[]{c}1\\\ 0\end{array}\begin{array}[]{l}in\text{
}\left[\delta,T-\delta\right]\\\ in\text{
}t=[0,\delta/2]\cup[T+\delta/2,T].\end{array}\right.$ (8)

The function $\eta$ will depend on $\delta>0$ but we do not make explicit this
dependence in the notation to simplify.

###### Theorem 1

Let $(u^{0},u^{1}),(u^{0}_{T},u^{1}_{T})\in L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)$ and suppose that
$(\hat{\phi}^{0},\hat{\phi}^{1})\in H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)$ is a minimizer of $J$ . If $\hat{\phi}$ is the
corresponding solution of (4) with final data
$(\hat{\phi}^{0},\hat{\phi}^{1})$ then

$f(x,t)=\eta(t)\int_{\Upsilon}a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}_{|\Gamma_{0}}d\sigma,$
(9)

is a control such that the solution of (1) satisfies (3).

Proof. The Gateaux derivative ot $J$ at $(\hat{\phi}^{0},\hat{\phi}^{1})$ in
the direction $({\phi}^{0},{\phi}^{1})$ is given by

$\displaystyle\lim_{h\rightarrow
0}\frac{1}{h}J((\hat{\phi}^{0},\hat{\phi}^{1})+h(\phi^{0},\phi^{1}))-J(\hat{\phi}^{0},\hat{\phi}^{1})$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\eta(t)\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}d\sigma\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma
d\gamma
dt-\left\langle\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle$
$\displaystyle+\left\langle(\phi^{0},\phi^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle.$

If $J$ achieves its minimum at $(\hat{\phi}^{0},\hat{\phi}^{1})$ we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\eta(t)\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}d\sigma\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma
d\gamma
dt-\left\langle\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle$
$\displaystyle+\left\langle(\phi^{0},\phi^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle,$

for all $(\phi^{0},\phi^{1})\in H^{1}_{0}(\Omega)\times L^{2}(\Omega)$.

From Lemma 1 it follows that
$f=\eta(t)\int_{\Upsilon}a(\sigma)\dfrac{\partial\hat{\phi}}{\partial\upsilon}_{|\Gamma_{0}}d\sigma$
is a control for which (3) holds.

##  3 The observability inequality in average

Let us now give a general condition which ensures the existence of a minimizer
for $J$ and therefore a control in average for system (1).

###### Definition 1

The system (4) is observable in average in time $T>0$ if for some
$\varepsilon>0$ there exists a constant $C_{1}>0$ such that

$C_{1}\left\|(\phi^{0},\phi^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}^{2}\leq\int_{\varepsilon}^{T-\varepsilon}\int_{\Gamma_{0}}\left|\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt$ (10)

for any $(\phi^{0},\phi^{1})\in H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)$ where $\phi$ is the solution of (4) with final data
$(\phi^{0},\phi^{1}).$

In particular, when system (4) is observable in average in time $T>0$ we can
choose $\eta(t)$ as in (8), with $\delta>0$ sufficiently small, in such a way
that

$C_{1}\left\|(\phi^{0},\phi^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}^{2}\leq\int_{0}^{T}\int_{\Gamma_{0}}\eta(t)\left|\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt.$ (11)

It is enough to consider $\delta=\varepsilon$ where $\varepsilon$ is the
constant in Definition 1, since in this way $\eta(t)=1$ for
$t\in[\varepsilon,T-\varepsilon]$ and formula (11) is a direct consequence of
(10).

From now on, when a system is observable in average we will assume that $\eta$
is chosen in such a way that (11) holds.

###### Remark 1

The parameter $\varepsilon>0$ in the previous definition is necessary to
deduce (11) from (10). If we consider $\varepsilon=0$, we obtain a more
natural version of the observability inequality but now it is not clear how to
deduce (11). This is in contrast with a single wave equation (i.e. without
parameter dependence) where (11) and (11) are equivalent, for sufficiently
large time $T$.

We prove below that the observability in average that we have defined above is
a sufficient condition for the controllability in average of system (1). But
we first state a technical lemma that will be used later.

###### Lemma 2

There exists a constant $C>0$, independent of $\sigma\in\Upsilon$, such that

$\left\|\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right)\right\|_{H_{0}^{1}(\Omega)\times
L^{2}(\Omega)}\leq C\|(\phi^{0},\phi^{1})\|_{H_{0}^{1}(\Omega)\times
L^{2}(\Omega)},$ (12)

for all solutions $\phi$ of the adjoint system (4) with final data
$(\phi^{0},\phi^{1})\in H_{0}^{1}(\Omega)\times L^{2}(\Omega)$.

Proof. Observe that,

$\displaystyle\left\|\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right)\right\|_{H_{0}^{1}\times
L^{2}}\leq\int_{\Upsilon}\left\|\left(\phi(\cdot,0;\sigma),\phi_{t}(\cdot,0;\sigma)\right)\right\|_{H_{0}^{1}\times
L^{2}}d\sigma$
$\displaystyle\quad\leq\int_{\Upsilon}\left(\frac{a(\sigma)}{a_{m}}\left\|\phi(\cdot,0;\sigma)\right\|_{H_{0}^{1}}^{2}+\left\|\phi_{t}(\cdot,0;\sigma)\right\|_{L^{2}}^{2}\right)^{\frac{1}{2}}d\sigma$
$\displaystyle\quad\leq\left(\frac{1}{a_{m}}+1\right)^{\frac{1}{2}}\int_{\Upsilon}\left(a(\sigma)\left\|\phi(\cdot,0;\sigma)\right\|_{H_{0}^{1}}^{2}+\left\|\phi_{t}(\cdot,0;\sigma)\right\|_{L^{2}}^{2}\right)^{\frac{1}{2}}d\sigma$
$\displaystyle\quad=\left(\frac{1}{a_{m}}+1\right)^{\frac{1}{2}}\int_{\Upsilon}\left(a(\sigma)\left\|\phi^{0}\right\|_{H_{0}^{1}}^{2}+\left\|\phi^{1}\right\|_{L^{2}}^{2}\right)^{\frac{1}{2}}d\sigma\leq\left(\frac{1}{a_{m}}+1\right)^{\frac{1}{2}}\left(a_{M}+1\right)^{\frac{1}{2}}\left(\left\|\phi^{0}\right\|_{H_{0}^{1}}^{2}+\left\|\phi^{1}\right\|_{L^{2}}^{2}\right)^{\frac{1}{2}}.$

Then, inequality (12) holds with
$C=\left(\dfrac{1}{a_{m}}+1\right)^{\frac{1}{2}}\left(a_{M}+1\right)^{\frac{1}{2}}$
and $a_{m},a_{M}$ are the minimum and maximum value of $a(\sigma)$,
respectively.

###### Theorem 2

If the system (4) is observable in average then the functional $J$ defined by
(7) has an unique minimizer $(\hat{\phi}^{0},\hat{\phi}^{1})\in
H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$

Proof. It is easy to see that $J$ is continuous and convex. The continuity of
the second term in (7) is a consequence of Lemma 2 while for the first term
one just observe that

$\displaystyle\left\|\eta(t)\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma\right\|_{L^{2}((0,T)\times\Gamma_{0})}\leq\int_{\Upsilon}\left\|\eta(t)a(\sigma)\frac{\partial\phi}{\partial\upsilon}\right\|_{L^{2}((0,T)\times\Gamma_{0})}d\sigma$
$\displaystyle\quad\leq\int_{\Upsilon}a(\sigma)C(\sigma,T)\left\|(\phi^{0},\phi^{1})\right\|_{H^{1}_{0}(\Omega)\times
L^{2}(\Omega)}d\sigma\leq
C^{\prime}\left\|(\phi^{0},\phi^{1})\right\|_{H^{1}_{0}(\Omega)\times
L^{2}(\Omega)},$ (13)

where $C(\sigma,T)$ is the constant in the classical direct (regularity)
inequality for the constant coefficients wave equation (1). The dependence on
$\sigma$ of this constant is easily obtained from its proof (see [15]). In
particular, $C(\sigma,T)=C(1+T\sigma)/\sigma$ that is integrable in
$\Upsilon$, and the continuity of this first term in $J$ holds.

The existence of a minimum is ensured if we prove that $J$ is also coercive
i.e.

$\lim_{\left\|(\phi^{0},\phi^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}\rightarrow\infty}J(\phi^{0},\phi^{1})=\infty$ (14)

The coercivity of the functional $J$ follows immediately from (12) and (11).
Indeed,

$\displaystyle J(\phi^{0},\phi^{1})$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{0}^{T}\eta(t)\int_{\Gamma_{0}}\left|\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt-\left\|(\phi^{0},\phi^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}\left\|(u^{0}_{T},u^{1}_{T})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}$
$\displaystyle-\left\|\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right)\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}\left\|(u^{0},u^{1})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}$ $\displaystyle\geq$ $\displaystyle
C_{1}\left\|(\phi^{0},\phi^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}^{2}-\left\|(\phi^{0},\phi^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}\left\|(u^{0}_{T},u^{1}_{T})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}$
$\displaystyle-C\|(\phi^{0},\phi^{1})\|_{H_{0}^{1}\times
L^{2}}\left\|(u^{0},u^{1})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}.$

Condition (14) follows inmediatly from this inequality.

We now give two properties of the control obtained by minimization of the
functional $J$.

###### Proposition 3

For some initial data $(u^{0},u^{1})\in L^{2}(\Omega)\times H^{-1}(\Omega)$,
let
$f(t)=\eta(t)\int_{\Upsilon}a(\sigma)\dfrac{\partial\hat{\phi}}{\partial\upsilon}_{|\Gamma_{0}}d\sigma$
be the averaged control given by the minimizer
$(\hat{\phi}^{0},\hat{\phi}^{1})$ of the functional $J$. If $g$ $\in
L^{2}\left((0,T)\times\Gamma_{0}\right)$ is any other control driving to zero
the average of the state of system (1) then

$\int_{0}^{T}\int_{\Gamma_{0}}|f|^{2}ds\;\frac{dt}{\eta(t)}\leq\int_{0}^{T}\int_{\Gamma_{0}}|g|^{2}ds\;\frac{dt}{\eta(t)}$
(15)

###### Remark 2

This result establishes that the control given from the minimizer of $J$ is
the one that minimizes the $L^{2}-$weighted norm associated to the measure
$\frac{1}{\eta(t)}dt$.

Proof. Let $(\hat{\phi}^{0},\hat{\phi}^{1})$ be the minimizer for the
functional $J$ . Consider now relation (5) for the control
$\eta(t)\int_{\Upsilon}a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}_{|\Gamma_{0}}d\sigma$
. By taking $(\hat{\phi}^{0},\hat{\phi}^{1})$ as test function we obtain that

$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}|f|^{2}ds\;\frac{dt}{\eta(t)}$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}\eta(t)\left|\int_{\Upsilon}a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt$ $\displaystyle=$
$\displaystyle\left\langle\left(\int_{\Upsilon}\hat{\phi}(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\hat{\phi}_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle-\left\langle(\hat{\phi}^{0},\hat{\phi}^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle$

On the other hand, relation (5) for the control $g$ and test function
$(\hat{\phi}^{0},\hat{\phi}^{1})$ gives

$\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}d\sigma
g(x,t)\;d\gamma\;dt=\left\langle\left(\int_{\Upsilon}\hat{\phi}(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\hat{\phi}_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle-\left\langle(\hat{\phi}^{0},\hat{\phi}^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle$

Combining the last two identities we obtain that

$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}|f|^{2}ds\;\frac{dt}{\eta(t)}$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}\eta(t)a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}d\sigma
g\;ds\;\frac{dt}{\eta(t)}$ $\displaystyle\leq$
$\displaystyle\left(\int_{0}^{T}\int_{\Gamma_{0}}\left|\int_{\Upsilon}\eta(t)a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}d\sigma\right|^{2}ds\;\frac{dt}{\eta(t)}\right)^{1/2}\left(\int_{0}^{T}\int_{\Gamma_{0}}|g|^{2}ds\;\frac{dt}{\eta(t)}\right)^{1/2}.$

Dividing this inequality by
$\left(\int_{0}^{T}\int_{\Gamma_{0}}|f|^{2}ds\;\frac{dt}{\eta(t)}\right)^{1/2}$
we easily obtain (15).

The next result provides a bound of the control from the initial and target
data.

###### Proposition 4

Let
$f(t)=\eta(t)\int_{\Upsilon}a(\sigma)\dfrac{\partial\hat{\phi}}{\partial\upsilon}_{|\Gamma_{0}}d\sigma$
be the averaged control, associated to the initial data $(u^{0},u^{1})$ and
target $(u^{0}_{T},u^{1}_{T})$, obtained by minimization of the functional
$J$. Then there exists a constant, independent of $\sigma\in\Upsilon$, such
that

$\int_{0}^{T}\frac{1}{\eta(t)}\left|f\left(t\right)\right|^{2}dt\leq
C\left(\left\|(u^{0},u^{1})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}^{2}+\left\|(u^{0}_{T},u^{1}_{T})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}^{2}\right)$ (16)

Proof. As $\hat{\phi}$ is the solution of the adjoint system (4) associated to
the the minimizer $(\hat{\phi}^{0},\hat{\phi}^{1})$ of $J$ in
$H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$, we have in
particular

$J(\hat{\phi}^{0},\hat{\phi}^{1})\leq J(0,0).$ (17)

Then, taking into account Lemma 2, we have

$\displaystyle\frac{1}{2}\int_{0}^{T}\int_{\Gamma_{0}}\eta(t)\left|\int_{\Upsilon}a(\sigma)\frac{\partial\hat{\phi}}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt$ $\displaystyle\leq$
$\displaystyle\left\langle(\hat{\phi}^{0},\hat{\phi}^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle-\left\langle\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle$
$\displaystyle\leq$
$\displaystyle\left\|(\hat{\phi}^{0},\hat{\phi}^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}\left\|(u^{0}_{T},u^{1}_{T})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}$
$\displaystyle+\left\|\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right)\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}\left\|(u^{0},u^{1})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}$ $\displaystyle\leq$ $\displaystyle
C\left\|(\hat{\phi}^{0},\hat{\phi}^{1})\right\|_{H_{0}^{1}\left(\Omega\right)\times
L^{2}\left(\Omega\right)}\left(\left\|(u^{0},u^{1})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}+\left\|(u^{0}_{T},u^{1}_{T})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}\right)$ $\displaystyle\leq$
$\displaystyle\frac{C}{\sqrt{C_{1}}}\left(\int_{0}^{T}\int_{\Gamma_{0}}\eta(t)\left|\int_{\Upsilon}a(\sigma)\frac{\partial\phi}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt\right)^{\frac{1}{2}}\left(\left\|(u^{0},u^{1})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}+\left\|(u^{0}_{T},u^{1}_{T})\right\|_{L^{2}\left(\Omega\right)\times
H^{-1}\left(\Omega\right)}\right)$

Dividing by
$\left(\int_{0}^{T}\int_{\Gamma_{0}}\eta(t)\left|\int_{\Upsilon}a(\sigma)\dfrac{\partial\phi}{\partial\upsilon}d\sigma\right|^{2}d\gamma
dt\right)^{\frac{1}{2}}$ we easily obtain (16).

##  4 Numerical approximation of the control problem

In this section we introduce a suitable discretization of the control problem
(1) that we use later to find numerical approximations of the controls. Unlike
previous works (see [2], [4]), where the projection method is applied directly
to the variational formulation of the control problem (5), we follow a more
natural strategy which consists in applying the projection operator to the
original control system (1). In this way, we obtain a discrete control problem
whose controls approximate the continuous one. In the next section we prove
that both strategies are equivalent.

Let $\\{w_{k}\\}_{k\in\mathbb{N}}$ be an orthonormal basis of $L^{2}(\Omega)$
constituted by eigenfunctions of the Laplace operator with Dirichlet boundary
conditions, $-\Delta_{D}$, and $\\{\lambda_{k}^{2}\\}_{k\in\mathbb{N}}$ the
corresponding eigenvalues. We assume that they are ordered increasingly, i.e.
$0<\lambda_{1}^{2}<\lambda_{2}^{2}\leq\lambda_{3}^{2}\leq,...$ We define the
scaled spaces $\mathcal{H}^{\alpha}(\Omega)$ as

$\mathcal{H}^{\alpha}=\left\\{\sum_{k=1}^{\infty}c_{k}w_{k}(x),\quad\mbox{
with }\sum_{k=1}^{\infty}\lambda_{k}^{2\alpha}|c_{k}|^{2}<\infty\right\\}.$

Note that $\mathcal{H}^{0}=L^{2}(\Omega)$, $\mathcal{H}^{1}=H^{1}_{0}(\Omega)$
and $\mathcal{H}^{-1}=H^{-1}(\Omega)$.

We also introduce the finite dimensional space $X^{N}$ generated by the first
$N$ eigenfunctions, i.e.

$X^{N}=\left\\{\sum_{k=1}^{N}c_{k}w_{k}(x),\quad\mbox{ with
}c_{k}\in\mathbb{R}\right\\}\subset\mathcal{H}^{\alpha},\quad\alpha=-1,0,1.$

and the projection operator

$P^{N}:L^{2}(\Omega)\to X^{N},$

defined by

$P^{N}(u)=\sum_{k=1}^{N}u_{k}w_{k}(x),\quad\mbox{ for
}u=\sum_{k=1}^{\infty}u_{k}w_{k}(x).$

Clearly, this operator can be extended to the analogous projection in
$\mathcal{H}^{\alpha}$ for $\alpha=-1,1$, that we still denote by $P^{N}$.

The idea is to project system (1) into the finite dimensional space $X^{N}$,
but we first transform this system to introduce a more suitable representation
of the boundary control.

For each $t\in[0,T]$ we introduce the following elliptic problem with the
control at the boundary,

$\left\\{\begin{array}[]{ll}\Delta h=0,&x\in\Omega,\\\
h=f\chi_{\Gamma_{0}},&x\in\Gamma_{0},\\\
h=0,&x\in\Gamma\backslash\Gamma_{0}.\end{array}\right.$ (18)

Then, we have at least formally that $v=\Delta_{D}^{-1}u$ satisfies the system

$\left\\{\begin{array}[]{l}v_{tt}-a(\sigma)\Delta v=-a(\sigma)h(x,t)\\\ v=0\\\
v(x,0)=v^{0},\qquad v_{t}(x,0)=v^{1},\end{array}\begin{array}[]{c}in\text{
}Q\\\ on\text{ }\Sigma\\\ in\text{ }\Omega\end{array}\right.$ (19)

where $(v^{0},v^{1})=(\Delta_{D}^{-1}u^{0},\Delta_{D}^{-1}u^{1})$. Now, we
apply the projection operator to system (19). Taking into account that the
Laplace operator conmmutes with $P^{N}$ and leave invariant the subspace
$X^{N}$ we easily obtain the finite dimensional system

$\left\\{\begin{array}[]{l}v_{tt}^{N}-a(\sigma)D^{N}v^{N}=-a(\sigma)P^{N}h(x,t)\\\
v^{N}\in X^{N}\\\
v^{N}(x,0)=v^{0,N},v_{t}^{N}(x,0)=v^{1,N}\end{array}\begin{array}[]{c}in\text{
}Q\\\ for\text{ }t\in[0,t]\\\ in\text{ }\Omega\end{array}\right.$ (20)

where
$(v^{0,N},v^{1,N})=(P^{N}\Delta_{D}^{-1}u^{0},P^{N}\Delta_{D}^{-1}u^{1})$ and
$D^{N}=P^{N}\Delta:X^{N}\rightarrow X^{N}$. Finally, we apply the operator
$D^{N}$, and we obtain for the new variable $u^{N}=D^{N}v^{N}$ the system

$\left\\{\begin{array}[]{l}u_{tt}^{N}-a(\sigma)D^{N}u^{N}=-a(\sigma)D^{N}P^{N}h(x,t),\\\
u^{N}\in X^{N},\\\
u^{N}(x,0)=u^{0,N},u_{t}^{N}(x,0)=u^{1,N},\end{array}\begin{array}[]{c}in\text{
}Q,\\\ for\text{ }t\in[0,T],\\\ in\text{ }\Omega,\end{array}\right.$ (21)
$\left\\{\begin{array}[]{ll}\Delta h=0,&x\in\Omega,\\\
h=f\chi_{\Gamma_{0}},&x\in\Gamma_{0},\\\
h=0,&x\in\Gamma\backslash\Gamma_{0}.\end{array}\right.$ (22)

where $(u^{0,N},u^{1,N})=(P^{N}u^{0},P^{N}u^{1})$.

We adopt (21)-(22) as the discrete approximation of the control system (1).
More precisely, the discrete control problem reads: Given $T>0$,
$(u^{0,N},u^{1,N})\in X^{N}\times X^{N}$ and a target
$(u^{0,N}_{T},u^{1,N}_{T})\in X^{N}\times X^{N}$, find a control $f^{N}\in
L^{2}(0,T;\Gamma_{0})$ such that the solution of (21)-(22) with $f=f^{N}$
satisfies

$\int_{\Upsilon}u^{N}(T;\sigma)d\sigma=u^{0,N}_{T},\quad\int_{\Upsilon}u_{t}^{N}(T;\sigma)d\sigma=u^{1,N}_{T}.$
(23)

Note that the elliptic problem in (22) is not discretized in the above
formulation. For one dimensional problems this is not necessary since $h$ can
be computed explicitly from the control $f$. However, in more general problems
a further discretization of the controls and this elliptic problem would be
also required. In [2] the authors propose a second projection of the controls
in the subspace of $L^{2}(\Gamma_{0})$ constituted by the eigenfunctions of
the associated capacitance operator in the boundary. However, this requires to
compute or approximate numerically this new eigenvalue problem. A simpler
alternative consists in parametrizing the boundary $\Gamma_{0}$ (assuming some
smoothness on $\Omega$) and then use a projection on the trigonometric basis
associated to the parameter interval. More precisely, if we are in dimension
$d=2$ and assume (to simplify) that $\Gamma_{0}$ can be parametrized by a
single invertible $C^{1}$-function ${\bf r}:(0,1)\to\Gamma_{0}$, with $\|{\bf
r}^{\prime}(s)\|>\alpha>0$ for all $s\in(0,1)$, then any function $g\in
L^{2}(\Gamma_{0})$ can be associated with another function $\sqrt{\|{\bf
r}^{\prime}(s)\|}g({\bf r}(s))\in L^{2}(0,1)$. Now we can project this
function on the finite dimensional space generated by the first functions of
the trigonometric basis $\\{\sin(j\pi s)\\}_{j=1}^{M}$, $M>0$ given, and we
obtain the following discretization of $g\in L^{2}(\Gamma_{0})$,

$g^{M}(x)=\sum_{j=1}^{M}g_{j}\beta_{j}(x),\quad
g_{j}=2\int_{0}^{1}\sqrt{\|{\bf r}^{\prime}(s)\|}g({\bf r}(s))\sin(j\pi
s)\;ds,\quad\beta_{j}(x)=\frac{\sin(j\pi{\bf r}^{-1}(x))}{\sqrt{\|{\bf
r}^{\prime}({\bf r}^{-1}(x))\|}}.$ (24)

The control $f^{N}$ is then approximated by

$f^{N,M}(x,t)=\sum_{j=1}^{M}f^{N,M}_{j}(t)\beta_{j}(x),\quad x\in\Gamma_{0}.$
(25)

Now, for each $j=1,...,M$ we define $h_{j}(x)$ the solution of (22) with
boundary data $\beta_{j}(x)$ in $\Gamma_{0}$, and replace the discrete control
problem (21)-(22) by

$\left\\{\begin{array}[]{l}u_{tt}^{N}-a(\sigma)D^{N}u^{N}=-a(\sigma)\sum_{j=1}^{M}f_{j}^{N,M}(t)D^{N}P^{N}h_{j}(x),\\\
u^{N}\in X^{N},\\\
u^{N}(x,0)=u^{0,N},u_{t}^{N}(x,0)=u^{1,N},\end{array}\begin{array}[]{c}in\text{
}Q,\\\ for\text{ }t\in[0,T],\\\ in\text{ }\Omega.\end{array}\right.$ (26)

Here the controls are the $M$ functions $f_{j}^{N,M}(t)$, $j=1,...,M$. Note
that in general the number of controls $M$ is much smaller than the dimension
of the control problem $N$. Observe also that one can divide $\Gamma_{0}$ in
several curves and use a different parametrization for each one. The discrete
problem has a similar structure in this case but the number of controls will
be larger.

##  5 Convergence of discrete averaged controls

We first introduce a variational characterization of the discrete controls
following the same approach as in the continuous system. Let us consider the
following backwards wave equation:

$\left\\{\begin{array}[]{l}\phi_{tt}^{N}-a(\sigma)D^{N}\phi^{N}=0\\\
\phi^{N}\in X^{N}\\\
\phi^{N}(x,T)=\phi^{0,N},\phi_{t}^{N}(x,T)=\phi^{1,N}\end{array}\begin{array}[]{c}in\text{
}Q\\\ for\text{ }t\in[0,t]\\\ in\text{ }\Omega\end{array}\right.$ (27)

where $(\varphi^{0,N},\varphi^{1,N})\in X^{N}\times X^{N}$.

Note that, due to our discretization scheme, the solution $\phi^{N}$ of system
(27) coincides with the solution of the continuous system (4) with the initial
data $(\varphi^{0,N},\varphi^{1,N})\in X^{N}\times X^{N}$.

We also introduce the following scalar product in $X^{N}\times X^{N}$

$\left\langle(\phi^{0,N},\phi^{1,N}),(u^{0,N},u^{1,N})\right\rangle_{N}=\left\langle(\phi^{0,N},\phi^{1,N}),(u^{0,N},u^{1,N})\right\rangle,$
(28)

that coincides with the duality product between $L^{2}(\Omega)\times
H^{-1}(\Omega)$ and $H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ for functions in
$X^{N}\times X^{N}$.

The following result is the analogous to Lemma 1 for the discrete control
problem.

###### Lemma 3

Assume that for $T>0$ and the data
$\left(u^{0,N},u^{1,N}\right),\left(u^{0,N}_{T},u^{1,N}_{T}\right)\in
X^{N}\times X^{N}$ the control $f^{N}\in L^{2}(0,T;\Gamma_{0})$ makes the
solution of the discrete system (21)-(22) to satisfy (23). Then,

$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\phi^{N}}{\partial\upsilon}d\sigma
f^{N}(x,t)d\gamma
dt-\left\langle\left(\int_{\Upsilon}\phi^{N}(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi^{N}_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0,N},u^{1,N})\right\rangle_{N}$
$\displaystyle+\left\langle(\phi^{0,N},\phi^{1,N}),(u^{0,N}_{T},u^{1,N}_{T})\right\rangle_{N}=0$
(29)

for all $(\phi^{N,0},\phi^{N,1})\in X^{N}\times X^{N}$, where
$(\phi^{N},\phi^{N}_{t})$ is the solution of the discrete backwards system
(27).

Proof. Following the proof of Lemma 1 we multiply the equation of $u^{N}$ in
(21) by the solution of the adjoint problem $\phi^{N}$ and integrate by parts.
We easily obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\Omega}\int_{\Upsilon}a(\sigma)\phi^{N}d\sigma
D^{N}P^{N}h^{N}(x,t)dxdt-\left\langle\left(\int_{\Upsilon}\phi^{N}(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi^{N}_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle_{N}$
$\displaystyle+\left\langle(\phi^{N,0},\phi^{N,1}),(u^{0}_{T},u^{1}_{T})\right\rangle_{N}$

where $h^{N}$ is the solution of (22) with $f=f^{N}$ the averaged control.

Therefore, it is enough to prove

$\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\phi^{N}}{\partial\upsilon}d\sigma
f^{N}d\gamma
dt=-\int_{0}^{T}\int_{\Omega}\int_{\Upsilon}a(\sigma)\phi^{N}D^{N}P^{N}d\sigma
h^{N}(x,t)\;dxdt.$

Note that

$\phi^{N}\left(x,t,\sigma\right)=\sum_{j=1}^{N}\phi_{j}^{N}(t;\sigma)w_{j}(x)$

where $w_{j}(x),\lambda_{j}$ are the eigenfunctions and eigenvalues of the
operator $-\Delta.$

Using Green formula we easily obtain,

$\displaystyle\int_{\Gamma_{0}}\frac{\partial
w_{j}}{\partial\upsilon}f^{N}d\gamma$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{0}}\frac{\partial
w_{j}}{\partial\upsilon}h^{N}d\gamma$ $\displaystyle=$
$\displaystyle\int_{\Omega}\Delta w_{j}h^{N}dx-\int_{\Omega}w_{j}\Delta
h^{N}dx+\int_{\Gamma_{0}}w_{j}\frac{\partial h^{N}}{\partial\upsilon}dx$
$\displaystyle=$ $\displaystyle-\int_{\Omega}\lambda_{j}w_{j}h^{N}dx$

Then,

$\displaystyle\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial\phi^{N}}{\partial\upsilon}d\sigma
f^{N}dxdt$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\int_{0}^{T}\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\phi_{j}^{N}\frac{\partial
w_{j}}{\partial\upsilon}d\sigma f^{N}dxdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\sum_{j=1}^{N}\int_{\Upsilon}a(\sigma)\phi_{j}^{N}(t;\sigma)\int_{\Omega}\lambda_{j}w_{j}h^{N}d\sigma
dxdt$ $\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\Omega}\int_{\Upsilon}a(\sigma)D^{N}\phi^{N}d\sigma
h^{N}dxdt.$

This concludes the proof.

###### Remark 3

As the solutions of systems (4) and (27) coincide, the variational
characterization of the discrete controls in (29) is analogous to the one
associated to the continuous control problem, but for functions in the subset
$X^{N}\times X^{N}\subset H^{1}_{0}(\Omega)\times L^{2}(\Omega)$. This means
in particular, that any control of the continuous system is also a control for
the discrete one.

Of course, the reciprocal is not true. In the rest of the paper we construct a
sequence of discrete controls that converges to the control that minimizes
$J$, assuming that the continuous system is observable in average.

We define the following quadratic cost functional $J^{N}:X^{N}\times
X^{N}\rightarrow\mathbb{R}$ by:

$\displaystyle J^{N}(\phi^{N,0},\phi^{N,1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}\int_{\Gamma_{0}}\eta(t)\left|\int_{\Upsilon}a(\sigma)\frac{\partial\varphi^{N}}{\partial\upsilon}d\sigma\right|^{2}dxdt$
(30)
$\displaystyle-\left\langle\left(\int_{\Upsilon}\phi^{N}(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi^{N}_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0,N},u^{1,N})\right\rangle_{N}$
$\displaystyle+\left\langle(\phi^{0,N},\phi^{1,N}),(u^{0,N}_{T},u^{1,N}_{T})\right\rangle_{N}$

where $\phi^{N}$ is the solution of (27). Once again, the fact that the
solutions of systems (4) and (27) coincide for initial data in $X^{N}\times
X^{N}$, allows us to interpret $J^{N}$ as the restriction of $J$ to
$X^{N}\times X^{N}$. In particular, existence of minimizers is guaranteed as
soon as the continuous system is controllable in average. The characterization
of these minimizers as controls for the finite dimensional system is
straightforward, following the proof of Theorem 1. In particular we have the
following result:

###### Theorem 5

Let $\left(u^{0,N},u^{1,N}\right),\;\left(u^{0,N}_{T},u^{1,N}_{T}\right)\in
X^{N}\times X^{N}$ be some discrete data and suppose that
$(\hat{\phi}^{0,N},\hat{\phi}^{1,N})\in X^{N}\times X^{N}$ is a minimizer of
$J^{N}$. If $\hat{\phi}^{N}$ is the corresponding solution of (27) with final
data $\left(\hat{\phi}^{0,N},\hat{\phi}^{1,N}\right)$ then
$f^{N}(t)=\eta(t)\int_{\Upsilon}\left.\frac{\partial\hat{\phi}^{N}}{\partial\upsilon}\right|_{\Gamma_{0}}d\sigma$
is a control such that the solution of (22) satisfies (29). In particular,

$\int_{\Upsilon}u^{N}\left(x,T;\sigma\right)d\sigma=u^{0,N}_{T},\qquad\int_{\Upsilon}u_{t}^{N}\left(x,T;\sigma\right)d\sigma=u^{1,N}_{T}.$

We now state the convergence result for the discrete controls.

###### Theorem 6

Let $T>0$ be large enough in order to have averaged observability of the
continuos wave equation. For a given initial data and target
$(u^{0},u^{1}),(u^{0}_{T},u^{1}_{T})\in L^{2}(\Omega)\times H^{-1}(\Omega)$,
let $f$ be the averaged control of the continuous system provided by the
minimizer of $J$, and $f^{N}(x,t)$ be the sequence of averaged controls
obtained by minimizing the discrete functional $J^{N}$ for the initial data
and target $(P^{N}u^{0},P^{N}u^{1}),(P^{N}u^{0}_{T},P^{N}u^{1}_{T})\in
X^{N}\times X^{N}$ respectively. Then

$f^{N}\to f,\text{ in }L^{2}(0,T;\Gamma_{0}).$ (31)

Proof. We show that the numerical approximation can be stated as a Ritz-
Galerkin approximation of the variational characterization of the control. The
result will follow from the classical convergence result for such
approximations of variational problems (see for example [21]).

We consider the Hilbert space $V=H^{1}_{0}\times L^{2}$ and the following
bilinear form on $V$,

$A((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))=\int_{0}^{T}\eta(t)\int_{\Gamma_{0}}\int_{\Upsilon}a(\sigma)\frac{\partial{\phi}}{\partial\upsilon}d\sigma\int_{\Upsilon}a(\sigma)\frac{\partial\psi}{\partial\upsilon}d\sigma
dxdt,$

where $\phi,\;\psi$ are the solutions of the adjoint system (4) with final
data $(\phi^{0},\phi^{1}),\;(\psi^{0},\psi^{1})$ respectively. We also
consider the linear form on $V$,

$L((\phi^{0},\phi^{1}))=\left\langle\left(\int_{\Upsilon}\phi(\cdot,0;\sigma)d\sigma,\int_{\Upsilon}\phi_{t}(\cdot,0;\sigma)d\sigma\right),(u^{0},u^{1})\right\rangle-\left\langle(\phi^{0},\phi^{1}),(u^{0}_{T},u^{1}_{T})\right\rangle.$

The minimizer of $J$, $(\hat{\phi}^{0},\hat{\phi}^{1})\in V$ solves the
variational equation,

$A((\hat{\phi}^{0},\hat{\phi}^{1}),(\phi^{0},\phi^{1}))=L(\phi^{0},\phi^{1}),\quad\mbox{
for all }(\phi^{0},\phi^{1})\in V.$

Both, the bilinear form $A$ and the linear one $L$ are continuous on $V$. For
the continuity of $A$ we refer to the proof of Theorem 2 while the continuity
of $L$ is a direct consequence of Lemma 2. The bilinear form is also coercive,
as a consequence of the averaged observability.

Now we consider the finite dimensional subspace of $V^{N}=X^{N}\times
X^{N}\subset V$. Note that, by our choice of $X^{N}$, the space $V^{N}$
becomes dense in $V$ as $N\to\infty$ in the sense that

$\forall(\phi,\psi)\in V,\quad\lim_{N\to\infty}\inf_{(\phi^{N},\psi^{N})\in
V^{N}}\|(\phi,\psi)-(\phi^{N},\psi^{N})\|_{V}=0.$

We only have to prove that the minimizer of the discrete functional $J^{N}$,
$(\hat{\phi}^{0,N},\hat{\phi}^{1,N})\in V^{N}$ is solution of

$A((\hat{\phi}^{0,N},\hat{\phi}^{1,N}),(\phi^{0,N},\phi^{1,N}))=L(\phi^{0,N},\phi^{1,N}),\quad\mbox{
for all }(\phi^{0,N},\phi^{1,N})\in V^{N}.$ (32)

But this is straightforward from the variational characterization of the
minimizers for $J^{N}$, Lemma 3, formula (28) and the fact that the solutions
of the adjoint systems (27) and (4) coincide for final data
$(\phi^{0,N},\phi^{1,N})\in V^{N}$.

According to the Ritz-Galerkin convergence result we have the following
estimate for the minimizers of $J$ and $J^{N}$

$\|(\hat{\phi}^{0},\hat{\phi}^{1})-(\hat{\phi}^{0,N},\hat{\phi}^{1,N})\|_{V}\leq
C\inf_{(\psi^{0},\psi^{1})\in
V^{N}}\|(\hat{\phi}^{0},\hat{\phi}^{1})-(\psi^{0},\psi^{1})\|_{V},$ (33)

for some constant $C$ independent of $N$. The classical approximation result
for spectral projections gives the convergence of the right hand side as
$N\to\infty$, and therefore the convergence of minimizers.

Finally, the $L^{2}-$convergence of controls is a direct consequence of the
continuity of the bilinear form $A$.

###### Remark 4

Convergence rates in (31) can be easily obtained for smooth initial data. In
fact, as pointed out in [9, 11] for a single wave equation one can produce
smoother controls for smoother initial data. In particular, with initial data
$(u^{0},u^{1})\in\mathcal{H}^{\alpha}\times H^{\alpha-1}$ one can modify the
functional $J$ to recover minimizers that satisfy
$(\hat{\phi}^{0},\hat{\phi}^{1})\in\mathcal{H}^{\alpha+1}\times\mathcal{H}^{\alpha}$
(see [11]). We only have to add to the functional $J$ a suitable compact
support cutoff function $\tilde{\eta}(x)\in C^{\infty}(\Gamma_{0})$ to
guarantee that the control is compact support both in time and space.
Following [11], it is straightforward to check that the same property holds
for averaged controls. In this case the right hand side in (33) can be
estimated by

$\inf_{(\psi^{0},\psi^{1})\in
V^{N}}\|(\hat{\phi}^{0},\hat{\phi}^{1})-(\psi^{0},\psi^{1})\|_{V}\leq\lambda^{-\alpha}_{N}\inf_{(\psi^{0},\psi^{1})\in
V^{N}}\|(\hat{\phi}^{0},\hat{\phi}^{1})-(\psi^{0},\psi^{1})\|_{\mathcal{H}^{\alpha+1}\times\mathcal{H}^{\alpha}},$

which is of the order $N^{-\alpha/d}$, according to the Weyl asymptotic
formula for the Dirichlet Laplacian eigenvalues. Therefore, we obtain

$\|f^{N}-f\|_{L^{2}(0,T;\Gamma_{0})}\leq CN^{-\alpha/d}.$

This remark was already observed in [4] for $\alpha=1$.

An analogous convergence result can be obtained for the discrete control
problem (26) following the ideas in [2]. To simplify, we restrict ourselves to
the case where $\Gamma_{0}$ is parametrized by a unique function ${\bf
r}:(0,1)\to\Gamma_{0}$. A straightforward calculus shows that in this case we
can define a class of controls $(f^{N,M}_{j})_{j=1}^{M}$ in the form

$f^{N,M}_{j}(t)=\int_{0}^{1}\int_{\Upsilon}a(\sigma)\frac{\partial{\hat{\phi}^{N,M}}}{\partial\upsilon}({\bf
r}(s),t)\;|{\bf r}^{\prime}(s)|^{1/2}\sin(j\pi s)\;d\sigma\;ds,$ (34)

where $\hat{\phi}^{N,M}$ is the solution of the adjoint equation (4) with
final data $(\hat{\phi}^{0,N,M},\hat{\phi}^{1,N,M})$. This final data is the
solution of a new variational equation, analogous to (32), but replacing the
bilinear form $A$ by,

$\displaystyle A^{M}((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\eta(t)\sum_{j=1}^{M}\left(\int_{0}^{1}\int_{\Upsilon}a(\sigma)\frac{\partial{\phi}}{\partial\upsilon}({\bf
r}(s),t)|{\bf r}^{\prime}(s)|^{1/2}\sin(j\pi s)d\sigma\;dx\right)$
$\displaystyle\left(\int_{0}^{1}\int_{\Upsilon}a(\sigma)\frac{\partial\psi}{\partial\upsilon}({\bf
r}(s),t)|{\bf r}^{\prime}({\bf r^{-1}})|^{1/2}\sin(j\pi s)d\sigma
dx\right)dt.$

We prove below the convergence of solutions of both variational equations and
we obtain the following result:

###### Theorem 7

Let $(f^{N,M}_{j})_{j=1}^{M}$ be the sequence of controls defined in (34),
$\beta_{j}$ the functions defined in (24) and $f^{N}$ the control obtained by
minimizing the functional $J^{N}$. Then,

$\left\|\sum_{j=1}^{M}f^{N,M}_{j}(t)\beta_{j}(x)-f^{N}(x,t)\right\|_{L^{2}(0,T;\Gamma_{0})}\to
0,\mbox{ as $M\to\infty$.}$

Proof. Note that it is sufficient to prove that $|{\bf
r}^{\prime}(s)|^{1/2}\int_{\Upsilon}\frac{\partial{\hat{\phi}^{N,M}}}{\partial\upsilon}({\bf
r}(s),t)d\sigma\to|{\bf
r}^{\prime}(s)|^{1/2}\int_{\Upsilon}\frac{\partial{\hat{\phi}^{N}}}{\partial\upsilon}({\bf
r}(s),t)d\sigma$ in $L^{2}((0,1)\times(0,T))$, which is easily deduced from
the convergence of the final data
$(\hat{\phi}^{0,N,M},\hat{\phi}^{1,N,M})\to(\hat{\phi}^{0,N},\hat{\phi}^{1,N})$
in $H^{1}_{0}\times L^{2}$ as $M\to\infty$ and the direct inequality (13).
Thus, we only have to prove that the solutions of the variational equation
(32) converge to those of the same equation when replacing the bilinear form
$A$ by $A^{M}$. This is a consequence of the following convergence result for
the associated bilinear forms,

$\sup_{(\phi^{0},\phi^{1}),(\psi^{0},\psi^{1})\in H^{1}_{0}\times
L^{2}}\frac{|A^{M}((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))-A((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))|}{\|(\phi^{0},\phi^{1})\|_{H^{1}_{0}\times
L^{2}}\|(\psi^{0},\psi^{1})\|_{H^{1}_{0}\times L^{2}}}\to 0,\mbox{ as
$N\to\infty$.}$

To prove this result we consider $(\phi^{0},\phi^{1}),\;(\psi^{0},\psi^{1})$
of norm 1 and observe that, with the parametrization of $\Gamma_{0}$,

$A((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))=\int_{0}^{T}\eta(t)\int_{0}^{1}\int_{\Upsilon}({\bf
r}(s),t)a(\sigma)\frac{\partial{\phi}}{\partial\upsilon}({\bf
r}(s),t)d\sigma\int_{\Upsilon}a(\sigma)\frac{\partial\psi}{\partial\upsilon}({\bf
r}(s),t)d\sigma|{\bf r}^{\prime}(s)|\;ds\;dt.$ (35)

Now, we write for each $(\phi^{0},\phi^{1})\in H^{1}_{0}\times L^{2}$,

$\displaystyle|{\bf
r}^{\prime}(s)|^{1/2}\int_{\Upsilon}a(\sigma)\frac{\partial{\phi}}{\partial\upsilon}({\bf
r}(s),t)d\sigma=\sum_{j=1}^{\infty}\alpha_{j}(\phi)\sin(j\pi s),$
$\displaystyle\alpha_{j}(\phi)=\int_{0}^{1}|{\bf
r}^{\prime}(s)|^{1/2}\int_{\Upsilon}a(\sigma)\frac{\partial{\phi}}{\partial\upsilon}({\bf
r}(s),t)d\sigma\sin(j\pi s)\;ds,$

and substitute in (35) to obtain,

$A((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))=\int_{0}^{T}\eta(t)\sum_{j=1}^{\infty}\alpha_{j}(\phi)\alpha_{j}(\psi)\;dt.$

Therefore,

$\displaystyle|A^{M}((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))-A((\phi^{0},\phi^{1}),(\psi^{0},\psi^{1}))|=\left|\int_{0}^{T}\eta(t)\sum_{j=M+1}^{\infty}\alpha_{j}(\phi)\alpha_{j}(\psi)\;dt\right|$
$\displaystyle\leq\left(\int_{0}^{T}\eta(t)\sum_{j=M+1}^{\infty}|\alpha_{j}(\phi)|^{2}\right)^{1/2}\left(\int_{0}^{T}\eta(t)\sum_{j=M+1}^{\infty}|\alpha_{j}(\psi)|^{2}\right)^{1/2}.$

Here both terms converges to zero as $M\to\infty$ due to the direct inequality
(13) and the dense approximation in $L^{2}(0,1)$ of the trigonometric basis
$\\{\sin{j\pi x}\\}_{j=1}^{M}$, as $M\to\infty$.

##  6 Matrix formulation and finite dimensional control

In this section we consider some implementation issues related with the finite
dimensional approximation of the averaged control given by (21)-(22). There
are several ways to do that. For instance one can follow the original method
in [13] where the authors compute the minimizer of the discrete functional
$J^{N}$ with a conjugate gradient algorithm. This method should be adapted to
the present setting with the averaged control. A more direct approach is to
write a matrix formulation of the discrete problem and compute the control
using the explicit expression given for the finite dimensional control theory
(see [24]). From the practical point of view this second approach requires
more computing time but it has a simpler implementation. In this paper we
follow this second approach. We divide this section in two subsections where
we consider separately the 1-d and 2-d wave equations respectively.

###  6.1 The 1-d wave equation

We assume $\Omega=(0,1)$ and that the control acts at the extreme $x=1$. The
eigenfunctions of the Laplace operator in this domain are given by
$w_{i}(x)=\sqrt{2}\sin(i\pi x)$ with $i\in\mathbb{N}$ and the corresponding
eigenvalues are $\lambda_{i}^{2}=i^{2}\pi^{2}$. Note that $w_{i}$ are
normalized in the $L^{2}(\Omega)$ norm. In particular, in this case $X^{N}$ is
the subspace generated by those eigenfunctions $w_{i}$ with $i\leq N$ and the
dimension of $X^{N}$ is therefore $N$.

The function $h^{N}(x)$, solution of (21), can be computed explicitly,

$h^{N}(x)=xf^{N}(t)$ (36)

and

$P^{N}h^{N}(x)=f^{N}(t)P^{N}x=f^{N}(t)\sum_{i=1}^{N}\frac{(-1)^{i+1}}{i\pi}w_{i}(x),$

Let us denote by $\\{u_{i}^{N}(t)\\}_{i=1}^{N}$ the components of $u^{N}$ in
the basis of eigenfunctions of the Laplace operator. Then, system (22) is
equivalent to the following system of equations for these components,

$\left\\{\begin{array}[]{l}(u_{i}^{N})^{\prime\prime}+a(\sigma)\lambda_{i}^{2}u_{i}^{N}=-\lambda_{i}^{2}f^{N}(t)\frac{(-1)^{i+1}}{i\pi},\qquad
i=1,...,N\\\
u_{i}^{N}(0)=u_{i}^{N,0},\quad(u_{i}^{N})^{\prime}(0)=u_{i}^{N,1},\qquad
i=1,...,N,\end{array}\right.$

with $u_{i}^{N,k}=\int_{0}^{1}u^{k}w_{i}\;dx$, that can be written in matrix
form as,

$\left\\{\begin{array}[]{l}U^{\prime\prime}+a(\sigma)DU=DGf^{N}(t),\\\
U(0)=U^{0},\quad U^{\prime}\left(0\right)=U^{1}.\end{array}\right.$ (37)

where,

$U=(u_{1}^{N},u_{2}^{N},...,u_{N}^{N})^{T},\quad
D=diag(\lambda_{1}^{N},\lambda_{2}^{N},...,\lambda_{N}^{N}),\quad
G=\left(-\frac{1}{\pi},\frac{1}{2},...,\frac{(-1)^{N}}{\pi N}\right)^{T},$
(38)

and the initial data,

$U^{k}=(u_{1}^{N,k},u_{2}^{N,k},...,u_{N}^{N,k})^{T},\quad k=0,1.$ (39)

Note that we have removed the dependence on $N$ in the notation to simplify,
but all the vectors and matrixes in system (37) depend on $N$. The components
of the target are also written in vector form $(U_{T}^{0},U_{T}^{1})$ where,

$U_{T}^{k}=(u_{T,1}^{N,k},u_{T,2}^{N,k},...,u_{T,N}^{N,k})^{T},\quad k=0,1.$
(40)

To apply the general theory of controllability for finite dimensional systems
we write (37) as a first order one

$\left\\{\begin{array}[]{l}Z^{\prime}+A(\sigma)Z=B(\sigma)f^{N}(t),\\\
Z(0)=Z^{0}\end{array}\right.$ (41)

with $Z=\left(\begin{array}[]{c}U\\\ U^{\prime}\end{array}\right),$
$A(\sigma)=\left(\begin{array}[]{cc}0&I\\\
a(\sigma)D&0\end{array}\right),B(\sigma)=\dbinom{0}{a(\sigma)DG},$
$Z^{0}=\left(\begin{array}[]{c}U^{0}\\\ U^{1}\end{array}\right),$ and the
target $Z_{T}=\left(\begin{array}[]{c}U^{0}_{T}\\\
U^{1}_{T}\end{array}\right).$

According to the control theory for finite dimensional systems (see [24]), a
control that drives the initial data $Z^{0}$ to the target $Z_{T}$ is given by

$f^{N}(t)=-\eta(t)\int_{\Upsilon}B^{\top}(\sigma)e^{-tA^{\top}\left(\sigma\right)}d\sigma\;Q_{T}^{-1}\;\left(\int_{\Upsilon}e^{tA(\sigma)}Z^{0}\;d\sigma-
Z_{T}\right),$ (42)

where $Q_{T}$ is the average controllability gramiam (see [24]) given by

$Q_{T}=\int_{0}^{T}\eta(t)\int_{\Upsilon}e^{(T-t)A(\sigma)}B(\sigma)\;d\sigma\int_{\Upsilon}B^{\top}(\sigma)e^{(T-t)A^{\top}(\sigma)}d\sigma
dt.$ (43)

The fact that $f^{N}(t)$ in (42) is a control for (41) can be easily checked.
We only have to substitute it in the solution of the system (41), given by

$Z(t,\sigma)=e^{A(\sigma)t}Z_{0}+\int_{0}^{t}e^{A(\sigma)(t-s)}B(\sigma)f^{N}(s)\;ds.$
(44)

Moreover, this control is the one obtained from the minimizer of $J^{N}$ in
(30).

The whole process can be implemented with the following algorithm:

Algorithm

  1. 1.

Choose $N$ (number of Fourier nodes),

  2. 2.

Define a spacial mesh $x_{i}=i\Delta x$, $i=0,...,M_{x}$, $\Delta x=1/M_{x}$
and a temporal mesh $t_{j}=j\Delta t$, $j=0,...,M_{t}$, $\Delta t=T/M_{t}$.
Choose $M_{x},M_{t}\geq 2N$ to avoid the aliasing phenomenon in the
approximation of Fourier modes.

  3. 3.

Define a parameter mesh $\sigma_{l}=\sigma_{0}+l\Delta\sigma$,
$l=0,...,M_{\sigma}$, $\Delta\sigma=(\sigma_{2}-\sigma_{1})/M_{\sigma}$.

  4. 4.

From the initial data $(u^{0},u^{1})$ at the nodes $x_{i}$ approximate the
vector of Fourier coefficients $Z^{0}$. If the initial data are piecewise
continuous we can simply use the trapezoidal rule to approximate
$u^{k}_{i}=\int_{0}^{1}u^{k}(x)w_{i}(x)\;dx$, for $k=0,1$.

  5. 5.

Compute the Grammian $Q_{T}$ from (43). For each $\sigma_{l}$,
$l=0,...,M_{l}$, compute the matrixes $A(\sigma_{l})$ and $B(\sigma_{l})$. Use
the matrix exponential (expm in MATLAB) and a trapezoidal rule to approximate
the integrals in time and $\sigma$.

  6. 6.

Use formula (42) to compute the control: For each time $t_{j}$,
$j=1,...,M_{t}$,

    1. (a)

Compute $G_{1}=\int_{\Upsilon}e^{t_{j}A(\sigma)}Z^{0}\;d\sigma-Z_{T}$

    2. (b)

Solve the linear system $Q_{T}G_{2}=G_{1}$ to compute $G_{2}$.

    3. (c)

Compute
$f(t_{j})=-\eta(t_{j})\int_{\Upsilon}B^{\top}(\sigma)e^{-t_{j}A^{\top}\left(\sigma\right)}d\sigma\;G_{2}$

###  6.2 The 2-d wave equation

In this section we consider the numerical approximation of the average control
for the 2-d wave equation in a square domain. We assume
$\Omega=(0,1)\times(0,1)$ and that the control acts in
$\Gamma_{0}=\\{1\\}\times(0,1)\cup(0,1)\times\\{1\\}$. For this problem the
averaged control of the wave equation is not known, for any time $T>0$. Our
experiments provide a numerical evidence that such controllability holds.

The eigenfunctions of the Laplace operator in this domain are given by
$w_{i,j}(x_{1},x_{2})=4\sin(i\pi x_{1})\sin(j\pi x_{2})$ with
$(i,j)\in\mathbb{N}\times\mathbb{N}$ and the corresponding eigenvalues are
$\lambda_{ij}^{2}=(i^{2}+j^{2})\pi^{2}$. Note that $w_{ij}$ are normalized in
the $L^{2}(\Omega)$ norm and that we have used a double index to represent the
eigenpair $(\lambda_{ij},w_{ij})$. Of course this does not affect the results
of the previous section. In particular, in this case $X^{N}$ is the subspace
generated by those eigenfunctions $w_{ij}$ with $i,j\leq N$ and the dimension
of $X^{N}$ is therefore $N^{2}$.

Any function $u\in C([0,T];L^{2}(\Omega))$ can be written as
$u(x_{1},x_{2},t)=\sum_{i,j=1}^{\infty}u_{ij}(t)w_{ij}(x_{1},x_{2}),$ where
$u_{ij}(t)$ are the Fourier coefficients, and the projection operator
$P^{N}:L^{2}(\Omega)\to X^{N}$ is defined as

$P^{N}u(x_{1},x_{2},t)=\sum_{i,j=1}^{N}u_{ij}(t)w_{ij}(x_{1},x_{2}).$

To discretize the control $f$ we divide $\Gamma_{0}$ in two parts, the right
and upper boundaries that we denote $\Gamma_{0}^{1}$ and $\Gamma_{0}^{2}$
respectively, and parametrized by,

$\displaystyle{\bf r}_{1}:(0,1)\to\Gamma_{0}^{1},\quad{\bf
r}_{1}(s)=(1,s)\in\Gamma_{0}^{1},$ $\displaystyle{\bf
r}_{2}:(0,1)\to\Gamma_{0}^{2},\quad{\bf r}_{2}(s)=(s,1)\in\Gamma_{0}^{2}.$

For each basis function $\sin(j\pi s)$ in $L^{2}(0,1)$ we define

$\beta^{k}_{j}(x)=\left\\{\begin{array}[]{ll}\sin(j\pi{\bf
r}_{k}^{-1}(x)),&x\in\Gamma_{0}^{k},\\\
0,&x\in\Gamma_{0}\backslash\Gamma_{0}^{k},\end{array}\right.\quad k=1,2.$

According to (25), we look for a discrete control in the form,

$f^{N}(x,t)=\sum_{j=1}^{M}(f_{j}^{1,N}(t)\beta_{j}^{1}(x)+f_{j}^{2,N}(t)\beta_{j}^{2}(x)),\quad
x\in\Gamma_{0},\quad t\in(0,T),$

for some functions $f^{k,N}_{j}$. We next solve the family of elliptic
problems

$\left\\{\begin{array}[]{ll}\Delta h=0,&\mbox{ $x$ in $\Omega$,}\\\
h=\beta_{j}^{k}(x),&\mbox{ $x$ on $\Gamma_{0}$,}\\\ h=0,&\mbox{ $x$ on
$\Gamma\backslash\Gamma_{0}$,}\end{array}\right.\quad k=1,2,\quad j=1,...,M.$

This give us a family of functions $h_{j}^{k}(x)$ in $\Omega$. We can compute
explicitly these functions,

$h_{j}^{1}(x_{1},x_{2})=\frac{\sinh(j\pi x_{1})}{\sinh(j\pi)}\sin(j\pi
x_{2}),\quad h_{j}^{2}(x_{1},x_{2})=\frac{\sinh(j\pi
x_{2})}{\sinh(j\pi)}\sin(j\pi x_{1}),\quad j=1,...,M.$

Now we compute $P^{N}h_{j}^{k}$,

$P^{N}h_{j}^{1}=\sum_{i=1}^{N}c_{ij}w_{ij},\quad
P^{N}h_{i}^{2}=\sum_{j=1}^{N}d_{ij}w_{ij},$

where

$c_{ij}=2\int_{0}^{1}\frac{\sinh(i\pi x_{1})}{\sinh(i\pi)}\sin(j\pi
x_{1})\;dx_{1}=\frac{2j(-1)^{j+1}}{(i^{2}+j^{2})\pi},\quad d_{ij}=c_{ij}.$

System (26) for this particular example reads,

$\left\\{\begin{array}[]{l}(u_{ij}^{N})^{\prime\prime}+a(\sigma)\lambda_{ij}^{2}u_{ij}^{N}=-a(\sigma)\lambda_{ij}^{2}(c_{ij}f_{i}^{1,N}+d_{ji}f_{j}^{2,N}),\\\
u_{ij}^{N}(0)=u_{ij}^{N,0},\quad(u_{ij}^{N})^{\prime}(0)=u_{ij}^{N,1}.\end{array}\right.$
(45)

Here we have taken $M=N$ since for a lower value of $M$ some equations would
have zero second hand term, i.e. without control.

Now we write the components $u_{ij}$ in a column matrix as follows,

$U=(u_{11},u_{21},...,u_{N1},u_{1,2},u_{22},....,u_{N,2},...,u_{1,N},...,u_{NN})^{T}.$

A straightforward computations allows us to write system (45) in the matrix
form (37) for suitable matrixes $D,G$ and a column vector with the Fourier
coefficients of the controls

$f^{N}(t)=(f_{1}^{1,N},f_{2}^{1,N},...,f_{N}^{1,N},f_{1}^{2,N},f_{2}^{2,N},...,f_{N}^{2,N})^{T}.$

The rest is similar to the 1-d case.

##  7 Numerical experiments

In this section we compute the control in average in different situations, for
the 1-d and 2-d wave equation in the square. In both cases we use formula (42)
for the finite dimensional systems obtained by the projection method described
above.

###  7.1 The 1-d wave equation

We consider in this section four experiments. In the first one we assume that
the unknown parameter is in an interval and we obtain the control in average,
that we compare with the control at a single value of the parameter. We also
check that the Grammiam associated to the discrete problem is uniformly
bounded by below as $N$ grows, which provides an evidence of the uniform
observability. In the second experiment we analyze the behavior of the control
in average for large time. In the third experiment we show the behavior of
this control when the length of the interval tends to zero. In particular we
observe that the control in average converges to the control of the average
parameter. The fourth experiment considers a parameter in the union of two
small intervals. We see that the control in average makes the solution
$u(x,T)$ to approximate either of two different symmetric profiles.

Experiment 1. We first consider the one-dimensional wave equation with initial
position and velocity given by $u^{0}(x)=1-2|x-1/2|$, $u^{1}(x)=0$, and the
equilibrium target $u^{0}_{T}=u_{1}^{T}=0$. We take final time $T=2.5$,
$a(\sigma)=\sigma$, $\sigma\in[1,2]$, the number of Fourier coefficients is
$N=30$ and time step $dt=10^{-2}$. The average in $\sigma$ is computed with
the trapezoidal rule and step $d\sigma=10^{-2}$.

In Figure 1 we show the control in average and the control of a single
realization for comparison. We observe that both seem to have the same
regularity as the initial position, i.e. continuous but not $C^{1}$. As the
initial data is in $H^{1}_{0}\times L^{2}$, the solution must be in the same
space and we consider this norm. We show the time behavior of the
$H^{1}_{0}\times L^{2}$-norm of the solutions to illustrate that, as expected,
the averages of controls or the control of the averaged parameter do not
produce a control in average. Finally, we have plotted the solution at time
$t=T$ for a few realizations of the parameter $\sigma$, with the control in
average to illustrate the effect of the control. Note that, even if the
average of solutions is zero at any $x\in[0,1]$, the solutions for different
values of the parameter may be far from zero. In this example the velocity
$u_{t}(x,t)$ becomes a discontinuous function in $x$, for $t>0$, even if we do
not apply any control. In fact, we can see the oscillatory behavior of a
trigonometric approximation for discontinuous functions in the lower right
experiment in Figure 1, which draws the controlled solutions
$u_{t}(x,T;\sigma)$, for different values of $\sigma$.

In table 1 we show the behavior of the norm of the control and the average of
$u(x,T;\sigma)$ for $1,000$ random values of $\sigma$, when the number of
Fourier coefficients $N$ grows. This gives a numerical evidence that the norm
of the controls remains bounded and that the action of these controls is
closer to the control in average as $N$ grows.

|
---|---
Control in average and control for | Time evolution of the $H^{1}_{0}\times L^{2}$-norm
the average parameter $\sigma_{m}=1.5$ | of the solutions with different controls.
|
Position of the controlled solutions $u(x,T)$ | Velocity of the controlled solutions at $t=T$,
Figure 1: Numerical results of experiment 1: the value of the unknown parameter $\sigma$ is in the interval $[1,2]$. The upper right figure compares the time evolution of the norm in the following cases: without control (solid blue), with the control in average (dash red), the average of the controls (dot yellow) and the average parameter $\sigma_{m}=1.5$ (dash-dot cyan). The two lower simulations contain several controlled solutions at time $t=T$ for different values of the parameter to illustrate that, even if the average is zero in $x\in[0,1]$ when the control is active, the solutions at time $t=T$ may be far from zero. We take $\sigma=1$ (light blue), $1.2$ (dark blue), $1.4$ (green), $1.6$ (red), $1.8$ (cyan) and $2$ (yelow). N | $\|\left(\int_{\Upsilon}u(x,T)d\sigma,\int_{\Upsilon}u_{t}(x,T)d\sigma\right)\|_{L^{2}\times H^{-1}}$ | $\|f\|_{L^{2}}$
---|---|---
$10$ | $2.3421\times 10^{-4}$ | $2.2243\times 10^{-1}$
$50$ | $3.5533\times 10^{-5}$ | $2.2245\times 10^{-1}$
$100$ | $8.8591\times 10^{-6}$ | $2.2245\times 10^{-1}$
Table 1: Experiment 1: Norm of the control in average and target as $N$ grows

We finish this experiment giving a numerical evidence of the uniform bound for
the observability inequality for the control in average. We compute the lowest
eigenvalue of the Grammian $Q_{T}$ in (43), $\alpha_{1}$, which corresponds to
the discrete version of the observability inequality (11). In Figure 2 we show
that this lowest eigenvalue remains uniformly bounded as $N$ grows.

Figure 2: Experiment 1: lowest eigenvalue of the discrete Grammiam $Q_{T}$ in
terms of the number of Fourier (log10 scale).

Experiment 2. In this experiment we compare the behavior of the averaged
control with the control corresponding to a single parameter (the average
value $\sigma_{m}$) as $T$ grows. We consider the same initial data and
parameters as in experiment 1. In Table 2 we compare the $L^{2}-$norm of both
controls. We observe that, as $T$ grows, both the norm of the averaged control
and the one corresponding to a single parameter control slightly decrease.

T | $\|f_{ave}\|_{L^{2}(0,T)}$ | $\|f_{\sigma_{m}}\|_{L^{2}}$
---|---|---
$2.5$ | $2.24\times 10^{-1}$ | $3.09\times 10^{-1}$
$20$ | $1.37\times 10^{-2}$ | $1.06\times 10^{-1}$
$40$ | $1.07\times 10^{-2}$ | $7.46\times 10^{-2}$
Table 2: Experiment 2: Norm of the control in average and the associated to
$\sigma_{m}=1.5$ as $T$ grows.

In figure 3 we compare the control in average with the control for a single
value of the parameter for large time $T=40$. We observe that the control in
average acts at the end of the time interval. This is due to the dissipative
character of the average evolution. As it is show in the lower simulations,
the norm of the average decays, even without any control, to a constant value
that depends on the intial data. Therefore, the control in average only have
to act at the end of the time interval.

|
---|---
Control in average vs control for $\sigma_{m}=1.5$ | Zoom near $t=0$.
|
Norm of the solutions | Zoom near $t=T$.
Figure 3: Numerical results of experiment 2: Control in average and the
control for a single value of the parameter $\sigma_{m}=1.5$ (upper left), and
a zoom near $t=0$ to illustrate the presence of oscillations in the control in
average (upper right). The two low simulations show the time evolution of the
$L^{2}\times H^{-1}$-norm of the average of some solutions (in log scale) when
we apply the control in average and without control.

Experiment 3. Here we illustrate the convergence of the control in average to
the control of the average parameter when the set of parameters is an interval
with decreasing length. In particular we consider $a(\sigma)=\sigma$,
$\sigma\in\Upsilon=[1-\varepsilon,1+\varepsilon]$ for different values of
$\epsilon\to 0$ and we compare the control in average with the control for
$\sigma_{0}=1$. The average in $\sigma$ is computed with the trapezoidal rule
and step $d\sigma=\varepsilon 10^{-2}$. The rest of the data are as in the
experiment 1. In table 3 we illustrate the convergence of the control in
average to the control of the average parameter.

$\varepsilon$ | $\|f_{ave}-f_{\sigma_{0}}\|_{L^{2}}$
---|---
$1/2$ | $2.0006\times 10^{-1}$
$10^{-1}$ | $2.1701\times 10^{-2}$
$10^{-2}$ | $3.8824\times 10^{-4}$
$10^{-3}$ | $3.3179\times 10^{-6}$
Table 3: Experiment 3: Convergence of the averaged control to the control for
the averaged parameter $\sigma_{0}$ when the lenght of the interval
$\varepsilon\to 0$.

Experiment 4. Now we illustrate the behavior of the control in average when
the set of parameters is the union of two intervals. In particular we consider
$a(\sigma)=\sigma$,
$\sigma\in\Upsilon=[1-\varepsilon,\;1+\varepsilon]\cup[3/2-\varepsilon,\;3/2+\varepsilon]$
with $\varepsilon=5\times 10^{-3}$. The average in $\sigma$ is computed with
the trapezoidal rule and step $d\sigma=\varepsilon 10^{-2}$. The rest of the
data are as in the Experiment 1. In Figure 4 we show the control in average
and the controls for the two parameters $\sigma=1,3/2$. We see that the
averaged control exhibits some small oscillations which are not present in the
other two controls. We also draw the controlled solutions at $t=T$ for some
values of the parameter. We observe how the solution is controlled either to
one of two symmetric final states.

We have also considered the case where one of the intervals contains values of
the parameter for which the wave equation is not controllable. In particular,
when
$\sigma\in\Upsilon=[1/2-\varepsilon,\;1/2+\varepsilon]\cup[3/2-\varepsilon,\;3/2+\varepsilon]$
the exact controllability result is not true for
$\sigma\in[1/2-\varepsilon,\;1/2+\varepsilon]$ if we maintain the time
$T=2.5$. However, the averaged controllability for $\sigma\in\Upsilon$ seems
to hold in this case.

|
---|---
Control in average | Controls for the parameters $\sigma=1,3/2$
|
Position for some controlled solutions | Velocity for some controlled solutions
Figure 4: Experiment 4. Here the parameter is equally distributed in two small disjoint intervals around $\sigma=1$ and $\sigma=3/2$. The control chooses to drive the solution to either of two different symmetric profiles, whose average is zero (the chosen target). |
---|---
Control in average | Control for the parameter $\sigma=3/2$
|
Position for some controlled solutions | Velocity for some controlled solutions
Figure 5: Experiment 4. Here the parameter is equally distributed in two small
disjoint intervals around $\sigma=1/2$ and $\sigma=3/2$. The control time is
$T=2.5$ and therefore the wave equation is not controllable when $\sigma=1/2$.
However, the averaged control exists.

###  7.2 The 2-d wave equation

In this section we consider a single experiment that illustrate that the
method can be easily adapted to multidimensional situations, as long as we
know the eigenfunctions of the associated Laplace operator in the domain.

Experiment 5. We consider a square domain $[0,1]\times[0,1]$. The control acts
at the two sides $\\{1\\}\times[0,1]\cup[0,1]\times\\{1\\}$ in the time
interval $t\in[0,2.5]$. We assume $a(\sigma)=\sigma$, $\sigma\in[3/2,2]$ and
the average is computed with the trapezoidal rule with step $d\sigma=10^{-2}$.
We consider the first $N=100$ Fourier coefficents. The initial data are
$u^{0}=\max\\{x_{1},(1-x_{1})\\}\max\\{x_{2},(1-x_{2})\\}$ and $u^{1}=0$. The
target is the equilibrium $u^{0}_{T}=u^{1}_{T}=0$. In Figure 6 we have drawn
the $L^{2}$-norm of the control acting in the two sides of the square and the
evolution of the norm
$\|(\int_{\Upsilon}u\;d\sigma,\int_{\Upsilon}u_{t}\;d\sigma)\|_{H^{1}\times
L^{2}}$.

|
---|---
Time evolution of the norm of $(\int_{\Upsilon}u\;d\sigma,\int_{\Upsilon}u_{t}\;d\sigma)$ |  $L^{2}$-norm of the control in average
Figure 6: Experiment 5 corresponding to the boundary control in a square
domain.

##  8 Conclusions

We have applied a spectral projection method to approximate the averaged
control for the wave equation. We prove the convergence of the method and how
it can be reduced to a classical finite dimensional control problem for which
there is an explicit formula for the control. This provides an efficient
method to compute averaged controls. We illustrate it with several numerical
experiments. The method is quite general and can be easily adapted to similar
situations where a variational characterization of controls exists.

Acknowledgements. The first author was supported by the Directorate-General
for Scientific Research and Technological Development (DGRSDT) (Algeria). The
second author is supported by project MTM2017-85934-C3-3-P from the MICINN
(Spain).

The second author is grateful to Jon Barcena for useful comments on the last
version of this paper that contributed to improve it. The authors are also
grateful to the anonymous referees for their detailed reading and comments.

## References

  * [1] M. Abdelli, A. Hafdallah, F. Merghadi and M. Louafi, Regional averaged controllability for hyperbolic parameter dependent systems, Control Theory Tech. 18(3), (2020) 307–314.
  * [2] C. Bardos, F. Bourquin and G. Lebeau, Calcul de dérivées normales et méthode de Galerkin appliqué au problème de contrôllabilité exacte, C.R. Acad. Sci. Paris, 313 I, (1991) 757–760.
  * [3] F. Bourquin, A numerical approach to the exact controllability of Euler-Navier- Bernoulli beams, Proceedings of the First World Conference on Structural Control, Pasadena (California), 1994, 120-129.
  * [4] F. Bourquin. Approximation theory for the problem of exact controllability of the wave equation with boundary control. In: Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993), pp. 103–112. SIAM, Philadelphia (1993).
  * [5] E. Burman, A. Feizmohammadi and L. Oksanen, A fully discrete numerical control method for the wave equation. SIAM Journal on Control and Optimization, 58(3), 1519-1546, (2020).
  * [6] N. Cindea and A. Münch, A mixed formulation for the direct approximation of the control of minimal $L^{2}$-norm for linear type wave equations, Calcolo, 52(3), 245-288, (2015).
  * [7] N. Cindea, E. Fernandez-Cara and A. Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, ESAIM: Control, Optimisation and Calculus of Variations, 19(4), 1076-1108, (2013).
  * [8] C. Castro, S. Micu, A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square, IMA J. Numer. Anal., 28 (2008), 186-214.
  * [9] B. Dehman and G. Lebeau, Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time, SIAM J. Control Optim., 48 (2009), pp. 521–550.
  * [10] S. Ervedoza, E. Zuazua, The wave equation: control and numerics, in Control of Partial Differential Equations, ed. by P.M. Cannarsa, J.M. Coron. Lecture Notes in Mathematics, CIME Subseries (Springer, New York, 2012), 245–340.
  * [11] S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 1375–1401.
  * [12] R. Glowinski, J.-L. Lions, J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Encyclopedia of Mathematics and its Applications, vol. 117 (Cambridge University Press, Cambridge, 2008)
  * [13] R. Glowinski, C. H. Li and J.-L. Lions,  A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods, Jap. J. Appl. Math. 7 (1990), 1–76.
  * [14] M. Lazar and E. Zuazua,  Averaged control and observation of parameter depending wave equations. C. R. Math. Acad. Sci. Paris, 352(6), (2014) 497–502.
  * [15] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris, 1988. With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.
  * [16] J. Lohéac and E. Zuazua, Averaged controllability of parameter dependent conservative semigroups. Journal of Differential Equations, 262 (3), (2017), 1540–1574
  * [17] J. Lohéac and E. Zuazua, From average to simultaneous controllability. Ann. Fac. Sci. Toulouse, Math. Ser. 6, 25(4), (2016) 785–828.
  * [18] Q. Lü and E. Zuazua, Averaged controllability for random evolution Partial Differential Equations. J. Math. Pures Appl., 105(3), (2016) 367–414.
  * [19] J. M-Frutos and F. P. Esparza, Optimal Control of PDEs under Uncertainty. Springer Briefs in Mathematics ,(2018), doi:10.1007/978-3-319-98210-6.
  * [20] S. Micu, E. Zuazua An introduction to the controllability of partial differential equations T. Sari (Ed.), Quelques questions de théorie du contrôle, Collection Travaux en Cours, Hermann (2005), pp. 67-150
  * [21] P.A. Raviart and J.-M. Thomas, Introduction à l’analyse numérique des équations aux derivées partielles, Dunond, Paris 1998.
  * [22] E. Zuazua, Stable observation of additive superpositions of Partial Differential Equations. Systems & Control Letters, 93 (2016), 21–29.
  * [23] E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Waves, _SIAM Review_ 47, (2005), 197–243.
  * [24] E. Zuazua, Averaged Control. Automatica 50(12), (2014) 3077–3087.