MATHEMATICAL ANALYSIS OF AN OBSERVER FOR SOLVING
INVERSE SOURCE WAVE PROBLEM

TIPHAINE DELAUNAY<sup>1,2</sup>, SÉBASTIEN IMPERIALE<sup>*1,2</sup> AND PHILIPPE MOIREAU<sup>1,2</sup>

<sup>1</sup>Inria-Saclay Ile de France, France

<sup>2</sup>École Polytechnique - CNRS - Institut Polytechnique de Paris6, France

(Communicated by Handling Editor)

ABSTRACT. The objective of this work is to propose a method using observers to estimate a source term of a wave equation from internal measurements in a subdomain. The first part of the work consists in proving an identifiabil- ity result from classical observability conditions for wave equations. We show that the source reconstruction is an ill-posed inverse problem of degree 1 or 2 depending on the measurements type. This inverse problem is solved us- ing observers – a sequential strategy – that is proven to be equivalent to a minimization of a cost functional with Tikhonov regularization.

1. **Introduction.** Inverse problems applied to wave-like propagation phenomena have received a lot of attention with numerous applications in geophysics [15, 60], tomography [14, 28, 26], medical imaging [1, 31] with the development of elastography [6, 32], non-destructive testing [2]. In fact, several subcategories of inverse problems can be distinguished: the reconstruction of initial conditions, the reconstruction of sources, and finally the identification of constitutive parameters. First, the reconstruction of initial conditions has been widely studied [50, 32] and is based on the proof of a so-called observability inequality. This inequality, which has been studied extensively by the control community [40, 61, 21] quantifies the initial condition to measurement operator invertibility which in this case is linear, allowing for the analysis of reconstruction strategies [20, 17]. At the other end of the spectrum, parameter reconstruction is a nonlinear inversion problem [33, 51], where identifiability and stability results are hard to obtain in the general cases and often rely on Carleman inequalities, see for example [8, 9, 10] and references therein. Finally, source reconstruction is the intermediate step between initial conditions reconstruction and parameter reconstruction. In a first step, we consider a source term with a separate variable, *i.e.*, the source is the product of a time-dependent function multiplying a space-dependent function to be estimated. This source term can take various forms, even being point sources [34, 35]. One of the advantages of estimating such source terms is that one can show identifiability by relying on initial condition observability inequalities [3, 57], while more general source terms may require the use of dedicated Carleman estimates [45, 58]. From a practical point

2020 Mathematics Subject Classification. Primary: 93C20, 35L05; Secondary: 93B07, 49J20.
Key words and phrases. Wave equation, Inverse source problem, Tikhonov regularization, Riccati dynamics, Reduced Kalman Estimator.

This work was supported by ANR Projet-ANR-19-CE48-0004 ODISSE.

*Corresponding author: Sébastien Imperiale.
of view, there are two main categories of methods for solving these inverse problems: Least squares minimization approaches [14, 16, 17] – also known as variational approaches or 4D-var in the data assimilation community [39, 54, 49] – or sequential approaches, where a new evolution problem integrating the measurements as a feedback pursues the target solution. The least squares approach is certainly the oldest strategy [52, 53] and is typically solved using adjoint equations [38]. The goal is to eventually find the optimal trajectory that minimizes the discrepancy between the solution and the observations. The sequential approach, also called the observer approach, is less studied. The principle is to introduce feedback into the dynamics based on measurement to create a new system, called the observer, which converges asymptotically to the observed system over time. Starting from the reconstruction of the initial conditions of wave problems, algorithms were developed and analyzed in [50] by combining Luenberger observer concepts and time-reversal strategies in a comparable spirit to the back-and-forth nudging introduced in data assimilation [4]. This strategy has been extended to unbounded domains by [32]. The proposed feedbacks are often easy to compute, and their efficiency is based on stabilization principles inspired by the so-called Luenberger observer introduced for finite-dimensional control problems [43, 42]. In the same community, it is also known that the principles of dynamic programming [11] can allow us to formulate observers that asymptotically solve the least squares minimization problem. Such a strategy is often referred to as optimal filtering, since the feedback is also called a filter, and is derived directly from an optimal criterion. It has been extended to the estimation of initial condition for infinite dimensional systems [12, 23] with recent works on numerical methods and their analysis, see e.g. [5] and references therein. Observers for reconstructing sources or identifying parameters have also been introduced in finite dimensional cases [59] with applications to discretized wave-like problems [44, 31].

Our goal is to extend the definition of such an optimal observer to source reconstruction in the infinite dimensional context of source inversion for wave problems. Our contribution includes the formulation of the observer, namely its existence, but also its asymptotic property, here called stability, and its robustness to measurement noise, as in all inverse problems. Notably, we base our analysis on general field measurements on a subdomain whereas most observability results for wave equations are classically based on velocity measurements.

This work is organized as follows. In the first section, we present the problem underlying the nature of the present observations. In the second section, we present how observability inequality allows us to quantify how poor the reconstruction of source terms is from the available field measurements. The next section is devoted to the formulation and analysis of the observer in the infinite dimensional context, developing an infinite dimensional Kalman filtering reduced to the source space. We then propose a discretization strategy based on the same principles at the discrete level. Finally, we illustrate the strategy with numerical experiments.
2. **Problem setting.** Let $\Omega$ be a bounded, connected open domain of class $C^2$ in $\mathbb{R}^d$. We consider the following wave equation

$$ egin{cases} 
\partial_{tt}u(x,t) - \Delta u(x,t) = \sigma(t)\theta(x), & (x,t) \in \Omega \times (0,T), \\
u(x,t) = 0, & (x,t) \in \partial\Omega \times (0,T), \\
u(x,0) = u_0(x), & x \in \Omega, \\
\partial_t u(x,0) = v_0(x), & x \in \Omega,
\end{cases} 
\quad (1) $$

where $(u_0, v_0)$ is the initial condition, $\sigma \in L^2(0, T)$ is known and $\theta \in \mathcal{P} \subseteq L^2(\Omega)$ – where $\mathcal{P}$ is a Hilbert space – is a parameter to be recovered from available measurements.

2.1. **Semigroup approach.** We rewrite (1) as a first order system in the state-space $\mathcal{Z} := H_0^1(\Omega) \times L^2(\Omega)$ that is the Hilbert space equipped with scalar product $(\cdot, \cdot)_\mathcal{Z}$ corresponding to the $H^1(\Omega)$ semi-norm for the first component and the $L^2(\Omega)$ norm for the second component,

$$ egin{cases} 
\dot{z}(t) = Az(t) + B(t)\theta, & t \in [0, T], \\
z(0) = z_0,
\end{cases} 
\quad (2) $$

where

$$ z_0 = \begin{pmatrix} u_0 \\ v_0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ \sigma(t)\mathrm{Id} \end{pmatrix}, \quad (3) $$

and the operator $A$ is an unbounded skew-adjoint operator from $\mathcal{D}(A) \subset \mathcal{Z}$ into $\mathcal{Z}$ defined by

$$ A = \begin{pmatrix} 0 & \mathrm{Id} \\ \Delta_0 & 0 \end{pmatrix}, \quad \mathcal{D}(A) = \mathcal{D}(-\Delta_0) \times H_0^1(\Omega), $$

where $\Delta_0$ denotes the Laplacian operator with homogeneous Dirichlet boundary conditions [13, Part II, Section 1.2.10] and $\mathcal{D}(-\Delta_0) = H^2(\Omega) \cap H_0^1(\Omega)$. The operator $A$ is maximal dissipative, hence it is a generator of a $C^0$-semigroup $(e^{tA})_{t \ge 0}$ on $\mathcal{Z}$. Note that $(B(t))_{t \ge 0}$ is a one parameter family of bounded operators from $\mathcal{P}$ to $\mathcal{Z}$. We recall the various concepts of solution and associated regularity.

**Theorem 2.1.** For all $T > 0$, the evolution equation (2) satisfies the properties:

1. For all $z_0 \in \mathcal{D}(A)$, $\sigma \in H^1(0, T)$ and $\theta \in \mathcal{P}$, there exists a unique strong solution $z \in C^0([0, T]; \mathcal{D}(A)) \cap C^1([0, T]; \mathcal{Z})$ of (2).

2. For all $z_0 \in \mathcal{Z}$, $\sigma \in L^1(0, T)$ and $\theta \in \mathcal{P}$, there exists a unique mild-solution $z \in C^0([0, T]; \mathcal{Z})$ of (2) given by the Duhamel's formula

$$ z(t) = e^{tA}z_0 + \int_0^t e^{(t-s)A} \begin{pmatrix} 0 \\ \sigma(s) \end{pmatrix} \theta \,ds, \quad t \in [0, T]. \quad (4) $$

Moreover, the solution is a weak solution, namely $z \in L^1((0, T); \mathcal{Z})$, for all $v \in D(A^*)$, $t \mapsto (v, z(t))_\mathcal{Z} \in W^{1,1}(0, T)$ and

$$ egin{cases} 
\frac{d}{dt}(v, z(t))_\mathcal{Z} = (A^*v, z(t))_\mathcal{Z} + (B(t)\sigma, v)_\mathcal{Z}, & t \in (0, T) \text{ a.e.,} \\
z(0) = z_0.
\end{cases} $$

*Proof.* See [13, Part II, Chapter 1, Section 3, Propositions 3.1 and 3.2]. □

<figure><img src="image_1.png" /></figure>

**Remark 2.2.** From Theorem 2.1, it follows naturally:
1. For all $(u_0, v_0) \in \mathcal{D}(-\Delta_0) \times H_0^1(\Omega)$, $\sigma \in H^1(0, T)$ and $\theta \in \mathcal{P}$, there exists a unique strong solution $u$ of (1) belonging to

$$C^0([0, T]; \mathcal{D}(-\Delta_0)) \cap C^1([0, T]; H_0^1(\Omega)) \cap C^2([0, T]; L^2(\Omega)).$$

2. For all $(u_0, v_0) \in H_0^1(\Omega) \times L^2(\Omega)$, $\sigma \in L^1(0, T)$ and $\theta \in \mathcal{P}$, there exists a unique mild-solution $u$ of (1) belonging to

$$C^0([0, T]; H_0^1(\Omega)) \cap C^1([0, T]; L^2(\Omega)).$$

We introduce the so-called sensitivity operators, namely the one parameter family $(L(t))_{t \ge 0}$ of bounded operator, such that

$$t \mapsto L(t)\theta := z(t) \in C^0([0, T]; \mathcal{Z}) \text{ with } z(t) = \int_0^t e^{(t-s)A} \begin{pmatrix} 0 \\ \sigma(s) \end{pmatrix} \theta \,ds, \quad (5)$$

then, the Duhamel's formula (4) also reads $z(t) = e^{tA}z_0 + L(t)\theta$, $t \in [0, T]$. For further use, we define $\bar{L}$ which corresponds to $L$ in the case of $\sigma \equiv 1$ namely the one parameter family $(\bar{L}(t))_{t \ge 0}$ of bounded operators, such that

$$t \mapsto \bar{L}(t)\theta := z(t) \in C^1([0, T]; \mathcal{Z}) \quad \text{with} \quad \begin{cases} \dot{z}(t) = Az(t) + \begin{pmatrix} 0 \\ \theta \end{pmatrix}, & t \in [0, T], \\ z(0) = 0. \end{cases} \qquad (6)$$

2.2. **Observation operator.** We must now set out the assumptions concerning the available observations. In this work, we suppose that we have at our disposal, at every time $t$ during the period $(0, T)$, measurements of the restriction of $u$ in $\omega \subset \Omega$ an open and nonempty subset of $\Omega$ with Lipschitz boundary $\partial\omega$ and $\partial(\Omega\setminus\omega)$ also being Lipschitz. To specify $\omega$, we first assume being given $\omega_{gcc}$ such that $(\omega_{gcc}, T_{gcc})$ satisfies the following condition [18] called Geometric Control Condition (GCC)

$$\exists T, \forall \xi \in \Omega, \forall m \in \mathbb{S}^d, \exists s \in [0, T], \exists \eta > 0; \omega_{gcc} \supset B(x_{\xi,m}(s); \eta), \quad (\text{GCC})$$

where $B(x_{\xi,m}(s); \eta)$ is a ball centered in $x_{\xi,m}(s)$ with radius $\eta$ and where a ray, starting from $\xi$ with unit direction vector $m$ and following the Descartes law of reflection in its most general senses [7, 18], is parametrized by $\{x_{\xi,m}(s), s \in [0, T]\}$. Note that is well known that such subdomain should contain a part of the boundary that we denote $\Gamma_{gcc} = \partial\Omega \cap \partial\omega$. Otherwise, some *whispering gallery* rays may prevent the GCC to be fulfilled.

Then, we assume that $\omega$ satisfies: there exists $\eta > 0$ such that

$$\bigcup_{x \in \omega_{gcc}} B(x; \eta) \cap \Omega \subset \omega. \qquad (7)$$

Such assumption is technical and allows later in the work to introduce a smooth cut-off function in $\Omega$, with support in $\omega$ and with value 1 in $\omega_{gcc}$.

Let us now present the definition of the observation operator $C$ usually introduced in control and observation theory – see for instance [55] and references therein. We consider a restriction operator in $\mathcal{L}(H_0^1(\Omega), \mathcal{Y})$, where $\mathcal{Y}$ is the observation space to be defined. Then formally, and in the absence of noise, at any time $t \in [0, T]$, the measurement procedure could produce a measurement $y(t)$. If we assume that it corresponds to the solution of the wave propagation problem $u(t) \in H_0^1(\Omega)$ for all $t$, it is then natural to consider that $\mathcal{Y} = H_\Gamma^1(\omega)$ – the subspace of functions in $H^1(\omega)$ vanishing on $\Gamma = \partial\Omega \cap \partial\omega$. However, in practice, the measurements are always
polluted by noise, hence in our deterministic context, the resulting measurement should belong to a less regular space, typically $\mathcal{Y} = L^2(\omega)$.

Summarizing, when considering regular measurements, the observation space is $\mathcal{Y} = H_\Gamma^1(\omega)$, and we introduce

$$C = (I_\omega^1 \quad 0) \in \mathcal{L}(\mathcal{Z}, \mathcal{Y}) \text{ where } I_\omega^1 \in \mathcal{L}(H_0^1(\Omega), H_\Gamma^1(\omega)), \quad (8)$$

is the restriction operator.

As mentioned when only noisy measurements are available one should introduce an observation operator where the image space is larger than in the previous case, namely $\mathcal{Y} = L^2(\omega)$. The observation operator becomes

$$C = (I_\omega^0 \quad 0) \in \mathcal{L}(\mathcal{Z}, \mathcal{Y}) \text{ where } I_\omega^0 \in \mathcal{L}(H_0^1(\Omega), L^2(\omega)), \quad (9)$$

is the restriction operator.

## 2.3. Inverse problem setting.
We can now formulate the inverse problems. We consider an actual trajectory $\tilde{u}$ modeled as the first component of a mild solution $\tilde{z}$ of (2) for a given and known $(u_0, v_0)$ and $\sigma(t)$ but an unknown $\tilde{\theta}$. Since the initial data are known, we can assume that they are null thanks to the linearity of the problem. Therefore, $\tilde{u}$ is a particular solution of the system

$$\left\{
\begin{array}{ll}
\partial_{tt}u(x,t) - \Delta u(x,t) = \sigma(t)\theta(x), & (x,t) \in \Omega \times (0,T), \\
u(x,t) = 0, & (x,t) \in \partial\Omega \times (0,T), \\
u(x,0) = 0, & x \in \Omega, \\
\partial_t u(x,0) = 0, & x \in \Omega,
\end{array}
\right.
\quad (10)$$

and its first-order formulation

$$\left\{
\begin{array}{ll}
\dot{z}(t) = Az(t) + B(t)\theta, & t \in (0, T), \\
z(0) = 0.
\end{array}
\right.
\quad (11)$$

This actual trajectory is subject to measurements modeled by the bounded observation operator $C \in \mathcal{L}(\mathcal{Z}, \mathcal{Y})$. We denote the available measurements (or observations) $y^\delta$ which, as they are perturbed by some measurements' error – also known as the noise in our deterministic framework – are assumed to belong to $L^2(0, T; \mathcal{Y})$, hence $y^\delta(t)$ is defined for almost all $t \in (0, T)$. In our deterministic framework, the measurement noise intensity is represented by saying that there exists $\delta \in \mathbb{R}^+$ such that

$$\|y^\delta - C\tilde{z}\|_{L^2(0,T;\mathcal{Y})} \le \delta^2 T.$$

Therefore, estimating $\tilde{\theta} \in \mathcal{P} \subset L^2(\Omega)$ from $y^\delta$ consists in inverting the linear input-output operator

$$\Psi_T: \quad \left|
\begin{array}{l}
L^2(\Omega) \to L^2((0, T); \mathcal{Y}), \\
\theta \mapsto (t \mapsto C L(t) \theta).
\end{array}
\right.
\quad (12)$$

We immediately find that $\Psi_T$ is bounded as $C$ is bounded from $\mathcal{Z}$ to $\mathcal{Y}$ and $L(t)$ is bounded from $\mathcal{P}$ to $\mathcal{Z}$ for all $t \in [0, T]$. Our objective is to study the condition for which $\Psi_T$ is injective. In fact, we are going to show that $\Psi_T$ has not a closed range in $L^2(0, T; \mathcal{Y})$ and our estimates suggest that its inversion defines a *mildly ill-posed* problem of degree 1 or 2 (see [25]) depending on the measurements space.

The objective of this work is to propose a sequential method – based on observer strategy – to solve the proposed source identification problem. For the analysis, we prove that the proposed sequential method is equivalent to the minimization of
a cost functional with Tikhonov regularization and show identifiability results by extending known classical observability conditions.

### 3. From observability to identifiability.
In this first part, we prove that we can extract enough information from the measurements to reconstruct $\theta$ and quantify the degree of ill-posedness of the inverse problem. A fundamental tool for measuring the degree of ill-posedness are identifiability inequalities. These inequalities are used to control parameters from observations. The most important elements are the norms that appear in this type of inequality which quantifies the ill-posedness [46, 25].

Our identifiability results are proven using an observability inequality that control initial data of the solution of a homogeneous wave propagation problem from its value on a subdomain during time. It relies on the geometric control condition. We adopt the observability condition proven in [19], here revisited for observation domains satisfying (7).

**Proposition 3.1.** Let $(T_{gcc}, \omega_{gcc})$ satisfies the GCC condition and $\omega$ satisfies (7). Then, for $T_0 > T_{gcc}$, there exists $c_o > 0$ such that for all $T \ge T_0$, there exists $c_{obs}(T) \ge c_o T$ such that for all $(u_0, v_0) \in H_0^1(\Omega) \times L^2(\Omega)$,

$$ \int_0^T \|u(\cdot, t)\|_{H^1(\omega)}^2 dt \ge c_{\text{obs}}(T) \left( \|u_0\|_{H_0^1(\Omega)}^2 + \|v_0\|_{L^2(\Omega)}^2 \right), \quad (13) $$

where $u$ is the mild solution of (1) with $\theta = 0$ and initial data $(u_0, v_0)$.

For the sake of completeness we provide a proof in Appendix A.1. This proposition provides a rough estimate of the behavior with respect to time $T$ of the observation constant $c_{obs}(T)$. A more precise estimate can be obtained using micro-local techniques, as in [30].

### 3.1. Measurements in $H_\Gamma^1(\omega)$.
In this section the observation operator $C$ is defined from $\mathcal{Z}$ to $\mathcal{Y} = H_\Gamma^1(\omega)$.

#### 3.1.1. Identifiability from observability.
From the observability condition adapted to initial condition reconstruction, we are now going to deduce the parameter identifiability of our source reconstruction problem. As a first step, we start by studying (11) when $\sigma \equiv 1$.

**Theorem 3.2.** Let $\omega$ satisfies (7). Then, there exists $c_i > 0$ such that for all $T$ sufficiently large, there exists $\bar{c}_{idf}(T) \ge c_i T$ such that

$$ \forall \theta \in \mathcal{P}, \quad \int_0^T \|C\bar{L}(t)\theta\|_{H^1(\omega)}^2 dt \ge \bar{c}_{idf}(T) \|\theta\|_{H^{-1}(\Omega)}^2, \quad (14) $$

where the observation operator is $C = (I_\omega^1, 0) \in \mathcal{L}(\mathcal{Z}, H_\Gamma^1(\omega))$ and $\bar{L}$ is given by (6).

*Proof.* The idea is to introduce the Riesz representation of $\theta$ seen as an element of $H^{-1}(\Omega)$ that we denote $g \in H_0^1(\Omega) \cap \mathcal{D}(-\Delta_0)$ and is given by $g = -\Delta_0^{-1}\theta$. We introduce the following dynamics which in fact is equivalent to (10) with $\sigma(t) \equiv 1$.

$$ \left\{ \begin{array}{ll} \partial_{tt} u_g(x, t) - \Delta u_g(x, t) = -\Delta_0 g(x), & (x, t) \in \Omega \times (0, T), \\ u_g(x, t) = 0, & (x, t) \in \partial \Omega \times (0, T), \\ u_g(x, 0) = 0, & x \in \Omega, \\ \partial_t u_g(x, 0) = 0, & x \in \Omega. \end{array} \right. \quad (15) $$
Note that for all $t \in [0, T]$, $u_g(t) \in D(\Delta_0)$ and $\partial_t u_g(t)$ belongs to $H^1(\Omega)$ (see Remark 2.2, and recall that $-\Delta_0 g = \theta \in \mathcal{P} \subset L^2(\Omega)$). We now proceed by standard energy estimate using the following energy functional

$$ \mathcal{E}_g(t) := \frac{1}{2} \|\nabla u_g(t)\|_{L^2(\Omega)}^2 + \frac{1}{2} \|\partial_t u_g(t)\|_{L^2(\Omega)}^2. $$

Integrating in space and time the first equation of (15) multiplied by $\partial_t u_g$, we get

$$ \int_0^T \int_{\Omega} \partial_{tt} u_g \partial_t u_g \, dx \, dt - \int_0^T \int_{\Omega} \Delta u_g \partial_t u_g \, dx \, dt = - \int_0^T \int_{\Omega} \Delta_0 g \partial_t u_g \, dx \, dt. $$

Integrating by parts while using that $g$ is time-independent, we obtain

$$ \mathcal{E}_g(T) = - \int_0^T \int_{\Omega} \Delta_0 g \partial_t u_g \, dx \, dt = - \int_{\Omega} \Delta_0 g u_g(T) \, dx = \int_{\Omega} \nabla g \nabla u_g(T) \, dx. $$

By Cauchy–Schwarz inequality, we get

$$ \mathcal{E}_g(T) \le \|\nabla g\|_{L^2(\Omega)} \|\nabla u_g(T)\|_{L^2(\Omega)} \le \sqrt{2} \|\nabla g\|_{L^2(\Omega)} \mathcal{E}_g^{\frac{1}{2}}(T). $$

This calculation finally leads to

$$ \|\partial_t u_g(T)\|_{L^2(\Omega)} \le 2 \|\nabla g\|_{L^2(\Omega)}. \quad (16) $$

We now introduce the system satisfied by $\tilde{u}_g = u_g - g$ in order to obtain an initial value problem for which we can use the observability condition of Proposition 3.1.

$$ \left\{ \begin{array}{ll} \partial_{tt} \tilde{u}_g(x, t) - \Delta \tilde{u}_g(x, t) = 0, & (x, t) \in \Omega \times (0, T), \\ \tilde{u}_g(x, t) = 0, & (x, t) \in \partial \Omega \times (0, T), \\ \tilde{u}_g(x, 0) = -g(x), & x \in \Omega, \\ \partial_t \tilde{u}_g(x, 0) = 0, & x \in \Omega. \end{array} \right. \quad (17) $$

Using the observability condition of Proposition 3.1, we get

$$ \int_0^T \|u_g - g\|_{H^1(\omega)}^2 \, dt = \int_0^T \|\tilde{u}_g\|_{H^1(\omega)}^2 \, dt \geq c_{\text{obs}}(T) \|g\|_{H_0^1(\Omega)}^2. \quad (18) $$

In order to get rid of the variable $g$ in the left-hand side, we write that

$$ \int_0^T \|u_g - g\|_{H^1(\omega)}^2 \, dt = \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt + \int_0^T \|g\|_{H^1(\omega)}^2 \, dt - 2 \int_0^T (u_g, g)_{H^1(\omega)} \, dt. \quad (19) $$

The last term can be controlled: multiplying the first equation of (15) by $\Delta_0^{-1}$ and integrating it in time and then, taking the scalar product with $g$, we have

$$ \int_0^T (u_g, g)_{H^1(\omega)} \, dt = T \|g\|_{H^1(\omega)}^2 + (\Delta_0^{-1} \partial_t u_g(T), g)_{H^1(\omega)}. $$

Injecting this result in (19), we get

$$ \int_0^T \|u_g - g\|_{H^1(\omega)}^2 \, dt = \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt - T \|g\|_{H^1(\omega)}^2 - 2(\Delta_0^{-1} \partial_t u_g(T), g)_{H^1(\omega)}. $$

From Cauchy–Schwarz inequality, we obtain

$$ \int_0^T \|u_g - g\|_{H^1(\omega)}^2 \, dt \le \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt - T \|g\|_{H^1(\omega)}^2 \\ + 2 \left( \| \Delta_0^{-1} \partial_t u_g(T) \|_{L^2(\Omega)}^2 + \| \nabla \Delta_0^{-1} \partial_t u_g(T) \|_{L^2(\Omega)}^2 \right)^{\frac{1}{2}} \|g\|_{H^1(\omega)}^2. $$

Using Poincaré inequality, we get
$$ \int_0^T \|u_g - g\|_{H^1(\omega)}^2 \, dt \le \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt - T \|g\|_{H^1(\omega)}^2 \\ + 2(c_p^2 + 1)^{\frac{1}{2}} \|\nabla \Delta_0^{-1} \partial_t u_g(T)\|_{L^2(\Omega)} \|g\|_{H^1(\omega)}. \quad (20) $$

Since $\nabla \Delta_0^{-1}$ is a bounded operator from $L^2(\Omega)$ to $L^2(\Omega)$, using Cauchy-Schwarz and Poincaré inequality, one can show that

$$ \|\nabla \Delta_0^{-1} \partial_t u_g(T)\|_{L^2(\Omega)} \le c_p \|\partial_t u_g(T)\|_{L^2(\Omega)}. \quad (21) $$

Injecting now (21) in (20), we have

$$ \int_0^T \|u_g - g\|_{H^1(\omega)}^2 \, dt \le \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt - T \|g\|_{H^1(\omega)}^2 \\ + 2 c_p (c_p^2 + 1)^{\frac{1}{2}} \|\partial_t u_g(T)\|_{L^2(\Omega)} \|g\|_{H^1(\omega)}. $$

Thanks to (16) and using Young inequality, we obtain, for any $\epsilon > 0$,

$$ \int_0^T \|u_g - g\|_{H^1(\omega)}^2 \, dt \le \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt - T \|g\|_{H^1(\omega)}^2 \\ + 2\epsilon c_p (c_p^2 + 1)^{\frac{1}{2}} \|g\|_{H_0^1(\Omega)}^2 + \frac{c_p}{\epsilon} (c_p^2 + 1)^{\frac{1}{2}} \|g\|_{H^1(\omega)}^2. $$

From the observability condition (18), we finally obtain

$$ \left(c_{\text{obs}}(T) - 2\epsilon c_p (c_p^2 + 1)^{\frac{1}{2}}\right) \|g\|_{H_0^1(\Omega)}^2 \le \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt - \left(T - \frac{c_p}{\epsilon} (c_p^2 + 1)^{\frac{1}{2}}\right) \|g\|_{H^1(\omega)}^2. $$

Choosing $\epsilon = \frac{1}{2} \frac{c_{\text{obs}}(T)}{2c_p(c_p^2+1)^{\frac{1}{2}}}$ gives

$$ \frac{c_{\text{obs}}(T)}{2} \|g\|_{H_0^1(\Omega)}^2 \le \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt - \left( T - \frac{4c_p^2(c_p^2+1)}{c_{\text{obs}}(T)} \right) \|g\|_{H^1(\omega)}^2. $$

For $T$ large enough we have

$$ \frac{c_{\text{obs}}(T)}{2} \|g\|_{H_0^1(\Omega)}^2 \le \int_0^T \|u_g\|_{H^1(\omega)}^2 \, dt. $$

We conclude by noting that, by definition, $\|g\|_{H_0^1(\Omega)} = \|\theta\|_{H^{-1}(\Omega)}$. $\square$

**Remark 3.3.** Inspecting the proof, one can see that $T$ should satisfy $T \ge T_0 > T_{gcc}$ and

$$ T \ge 4c_p^2(c_p^2+1)c_{\text{obs}}^{-1}(T), $$

where $c_p$ is the Poincaré constant and $c_{\text{obs}}(T)$ is the observability constant of Proposition 3.1. Using the estimate $c_{\text{obs}}(T) \ge c_o T$ one obtains that it is sufficient to have

$$ T \ge \max(T_0, 2c_p(c_p^2+1)^{\frac{1}{2}}c_o^{-\frac{1}{2}}), $$

and in that case $\bar{c}_{\text{idf}}(T) \ge c_{\text{obs}}(T)/2$.

From the identifiability property with $\sigma \equiv 1$, we deduce an identifiability property of the source term in (11) for a class of $\sigma \in H^1(0, T)$.
**Theorem 3.4.** Let $\omega$ satisfies (7), then for all $T$ sufficiently large and for all $\sigma(t) \in H^1(0, T)$ with $\sigma(0) \neq 0$, there exists $c_{idf}(T) > 0$ such that

$$ \forall \theta \in \mathcal{P}, \quad \int_{0}^{T} \|CL\theta\|_{H^1(\omega)}^2 dt \geq c_{idf}(T)\|\theta\|_{H^{-1}(\Omega)}^2, \qquad (22) $$

where the observation operator is $C = (I_{\omega}^1 \quad 0) \in \mathcal{L}(\mathcal{Z}, H_{\Gamma}^1(\omega))$ and $L$ is given by (5).

*Proof.* This proof is inspired by [3]. Given the scalar nature of $\sigma$, we use the first-order formulation of the system, so that observations of the solution read, for all $t \in [0, T]$

$$ \begin{aligned} CL(t)\theta &= \int_{0}^{t} Ce^{(t-s)A}\sigma(s) \begin{pmatrix} 0 \\ \theta \end{pmatrix} ds = \int_{0}^{t} Ce^{sA}\sigma(t-s) \begin{pmatrix} 0 \\ \theta \end{pmatrix} ds \\ &= \int_{0}^{t} \sigma'(t-s) \begin{pmatrix} \int_{0}^{s} Ce^{\tau A} \begin{pmatrix} 0 \\ \theta \end{pmatrix} d\tau \end{pmatrix} ds + \sigma(0) \begin{pmatrix} \int_{0}^{t} Ce^{\tau A} \begin{pmatrix} 0 \\ \theta \end{pmatrix} d\tau \end{pmatrix} \\ &= \int_{0}^{t} \sigma'(t-s)C\bar{L}(s)\theta ds + \sigma(0)C\bar{L}(t)\theta. \end{aligned} $$

We define a Volterra operator $S: L^2(0, T; \mathcal{Y}) \to L^2(0, T; \mathcal{Y})$ ([27, Part 1, Chapter 2]),

$$ Sy(t) = y(t) + \int_{0}^{t} k_{T}(t-s)y(s) ds, \quad \text{where } k_{T}(t) = \begin{cases} \frac{\sigma'(t)}{\sigma(0)} & t \in [0, T], \\ 0 & t > T. \end{cases} \qquad (23) $$

Using this Volterra operator, we have

$$ CL(t)\theta = \sigma(0) S \circ C\bar{L}(t)\theta, \quad t \in [0, T]. \qquad (24) $$

Since $k_T \in L^1(0, T)$, using [27, Part 1, Chapter 2, Theorem 3.1], it is well-known that $S$ is a linear bijective continuous operator from $L^2(0, T; \mathcal{Y})$ onto $L^2(0, T; \mathcal{Y})$. In particular, there exists a unique $r_T \in L^1(0, T)$ such that

$$ y(t) = Sy(t) - \int_{0}^{t} r_{T}(t-s)Sy(s) ds. \qquad (25) $$

Consequently, we obtain

$$ \|y\|_{L^2(0,T;\mathcal{Y})} \leq (1 + \|r_T\|_{L^1(0,T)}) \|Sy\|_{L^2(0,T;\mathcal{Y})}. \qquad (26) $$

Combining (24) and (26) with $y = C\bar{L}\theta$, we have

$$ \|C\bar{L}\theta\|_{L^2(0,T;\mathcal{Y})} \leq (1 + \|r_T\|_{L^1(0,T)}) \frac{1}{|\sigma(0)|} \|CL\theta\|_{L^2(0,T;\mathcal{Y})}. \qquad (27) $$

Combining the above equation with the observability condition of Proposition 3.2, we conclude the proof. □

We deduce from the identifiability inequality (22) that the observations should belong to $H^1(\omega)$ for a stable reconstruction in a $H^{-1}(\Omega)$ norm. As a consequence, we face an ill-posed inverse problem of degree 1 since we expect a reconstruction in $\mathcal{P} \subset L^2(\Omega)$.
**Remark 3.5.** In some cases, it is possible to prove that the identifiability constant of Theorem 3.4 has the same dependence in $T$ as that of the Theorem 3.2. For instance, when $\sigma$ generates a persistent excitation as follows

$$ \sigma(t) = \frac{1}{1 + \beta}(e^{-\alpha t} + \beta), \text{ where } \beta, \alpha \text{ are two positive constants.} $$

In fact, this corresponds to a perturbation of the case where $\sigma(t) \equiv 1$.

Following the proof of Theorem 3.4, one has to show that $\|r_T\|_{L^1(0,T)}$ is bounded uniformly with respect to $T$ where $r_T$ is defined implicitly by $\sigma$ (and $k_T$) in (25). First, we have

$$ \begin{aligned} \left\| \frac{\sigma'(t)}{\sigma(0)} \right\|_{L^1(0,T)} &= \|k_T\|_{L^1(0,T)} = \left\| -\frac{\alpha}{1+\beta} e^{-\alpha t} \right\|_{L^1(0,T)} &\le \frac{\alpha}{1+\beta} \int_0^T e^{-\alpha t} \, dt, \\ &\le \frac{1}{1+\beta} (1 - e^{-\alpha T}) \le \frac{1}{1+\beta}. \end{aligned} $$

From [27, Part 1, Chapter 2], we have an explicit expression for $r_T$ in terms of $k_T$,

$$ r_T = \sum_{j=1}^{\infty} (-1)^{j-1} (k_T)^{*j}, $$

where $(k_T)^{*j}$ is the $(j-1)$-fold convolution of $k_T$ with itself. Hence,

$$ \|r_T\|_{L^1(0,T)} \le \sum_{j=1}^{\infty} \|k_T\|_{L^1(0,T)}^j. $$

The common ratio $\|k_T\|_{L^1(0, +\infty)}$ is smaller than one, thus,

$$ \|r_T\|_{L^1(0,T)} \le \left(1 - \|k_T\|_{L^1(0,T)}\right)^{-1} \le \frac{1}{1 - \frac{1}{1+\beta}} = \frac{1+\beta}{\beta}, $$

which does not depend on time $T$.

As suggested in the previous remark, the dependency of the identifiability constant with respect to time is not always explicit. When considering the simpler case $\sigma(t) \equiv 1$, the identifiability constant behaves at worst linearly with respect to $T$. Following Remark 3.5 we also have $c_{idf}(T) \ge cT$ for some $c > 0$ as soon as

$$ \left\| \frac{d\sigma}{dt} \right\|_{L^1(\mathbb{R}^+)} < |\sigma(0)|. \qquad (28) $$

One can observe that any function of $H^1(\mathbb{R}^+)$ vanishing after a given time will not satisfy (28).

3.1.2. *From the identifiability result to a generalized Tikhonov regularization strategy.* Owing to Remark 3.5, we use in the theorems below the assumption that the dependence of the observability constant is in $O(T)$.

**Assumption 3.6.** For $T$ sufficiently large, the identifiability constant of Theorem 3.4 satisfies

$$ c_{idf}(T) \ge cT, \text{ for some } c > 0 \text{ independent of } T. $$
We recall that we assume that a target trajectory $\tilde{u}$ produced by a target source term parameter $\tilde{\theta}$ has resulted in a set of measurements $y^\delta$ where

$$ \int_{0}^{T} \|y^{\delta}(\cdot, t) - \tilde{u}(\cdot, t)\|_{H^{1}(\Omega)}^{2} dt \leq \delta^{2}T, \quad (29) $$

for a parameter $\delta > 0$. We introduce a cost functional to minimize $\mathcal{J}_T: H_0^1(\Omega) \to \mathbb{R}$ defined by

$$ \begin{aligned} \mathcal{J}_T(\theta) &= \frac{\epsilon}{2} \|\theta\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \int_0^T \|y^\delta(t) - u_{|_\theta}\|_{H^1(\omega)}^2 dt \\ &= \frac{\epsilon}{2} \|\theta\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \int_0^T \|y^\delta(t) - C z_{|_\theta}\|_{H^1(\omega)}^2 dt, \quad (30) \end{aligned} $$

where $u_{|_\theta}$ is the solution of (10) associated with the parameter $\theta$, $C \in \mathcal{L}(\mathcal{Z}, H_1^1(\omega))$ is defined by (8) and $\epsilon > 0$. Since the functional is quadratic and the application $\theta \mapsto u_{|_\omega}$ is continuous and $\theta \mapsto \|\theta\|_{H_0^1(\Omega)}^2$ is coercive, there exists a unique minimum of $\mathcal{J}_T$. We denote the unique minimizer in $H_0^1(\Omega)$ by $\bar{\theta}_T := \underset{\theta \in H_0^1(\Omega)}{\operatorname{argmin}} \mathcal{J}_T$.

**Theorem 3.7.** Let $\omega$ satisfies (7), then for all $T$ sufficiently large and for all $\sigma(t) \in H^1(0, T)$ with $\sigma(0) \neq 0$, there exists $c_{\text{stab}}(T) > 0$ such that for all $\tilde{\theta} \in H_0^1(\Omega)$, the minimizer $\bar{\theta}_T$ of the cost functional (30) with noise-free measurements, defined by (29) with $\delta = 0$, satisfies

$$ \|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)} \leq c_{\text{stab}}(T) \epsilon^{\frac{1}{4}} \|\tilde{\theta}\|_{H_0^1(\Omega)}. \quad (31) $$

Moreover, under Assumption 3.6, $c_{\text{stab}}(T) \leq c T^{-\frac{1}{4}}$, for some $c > 0$ independent of $T$.

*Proof.* In this proof, we write $a \lesssim b$ if there exists $c > 0$ independent of $T$, $\epsilon$ and $M$ such that $a \leq cb$.

Let $\tilde{u}$ be the solution of (10) associated with a target source term parameter $\tilde{\theta}$ and $\bar{u}$ the solution of (10) where the source term parameter corresponds to the minimizer of the cost functional (30). Let $\tilde{u} = \bar{u} - \tilde{u}$ be the solution of the following dynamics

$$ \left\{ \begin{array}{ll} \partial_{tt}\tilde{u}(x,t) - \Delta\tilde{u}(x,t) = \sigma(t)(\bar{\theta}_T(x) - \tilde{\theta}(x)), & (x,t) \in \Omega \times (0,T), \\ \tilde{u}(x,t) = 0, & (x,t) \in \partial\Omega \times (0,T), \\ \tilde{u}(x,0) = 0, & x \in \Omega, \\ \partial_t \tilde{u}(x,0) = 0, & x \in \Omega. \end{array} \right. \quad (32) $$

Identifying $L^2$ to its dual, we have

$$ \|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)} \leq \langle \bar{\theta}_T - \tilde{\theta}, \bar{\theta}_T - \tilde{\theta} \rangle_{H_0^1(\Omega), H^{-1}(\Omega)} \leq \|\bar{\theta}_T - \tilde{\theta}\|_{H_0^1(\Omega)} \|\bar{\theta}_T - \tilde{\theta}\|_{H^{-1}(\Omega)}. $$

Using the identifiability property (22) of Theorem 3.4, we get

$$ \|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)}^2 \leq (c_{\text{idf}}(T))^{-\frac{1}{2}} \|\bar{\theta}_T - \tilde{\theta}\|_{H_0^1(\Omega)}^2 \left( \int_0^T \|\bar{u} - \tilde{u}\|_{H^1(\omega)}^2 dt \right)^{\frac{1}{2}}. \quad (33) $$

By triangle inequality and since we consider free-noise observations, we have

$$ \|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)}^2 \leq (c_{\text{idf}}(T))^{-\frac{1}{2}} \left( \|\bar{\theta}_T\|_{H_0^1(\Omega)} + \|\tilde{\theta}\|_{H_0^1(\Omega)} \right) \left( \int_0^T \|\bar{u} - y^\delta\|_{H^1(\omega)}^2 dt \right)^{\frac{1}{2}}. $$
Using the definition of the cost functional (30), we get

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\mathrm{idf}}(T))^{-\frac{1}{2}} \left( (2\epsilon^{-1} \mathcal{J}_T(\bar{\theta}_T))^{\frac{1}{2}} + \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right) (2\mathcal{J}_T(\bar{\theta}_T))^{\frac{1}{2}}. $$

Remembering that $y^\delta(t) = \tilde{u}|_\omega(t)$, we can bound the cost functional as follows

$$ \mathcal{J}_T(\bar{\theta}_T) \le \mathcal{J}_T(\check{\theta}) \le \frac{\epsilon}{2} \|\check{\theta}\|_{H_0^1(\Omega)}^2. $$

Eventually, we obtain

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\mathrm{idf}}(T))^{-\frac{1}{2}} \left( (\|\check{\theta}\|_{H_0^1(\Omega)}^2)^{\frac{1}{2}} + \|\check{\theta}\|_{H_0^1(\Omega)} \right) (\epsilon \|\check{\theta}\|_{H_0^1(\Omega)}^2)^{\frac{1}{2}}, $$

which corresponds to the desired result (31). In addition, under Assumption 3.6, we have that there exists $c > 0$ independent of $T$ such that $c_{\mathrm{idf}}(T) \ge cT$. $\square$

**Theorem 3.8.** Let $\omega$ satisfies (7), then for all $T$ sufficiently large, for all $\sigma(t) \in H^1(0, T)$ with $\sigma(0) \ne 0$, for $\epsilon > 0$ small enough and for all $\check{\theta} \in H_0^1(\Omega)$, there exists $c_{\mathrm{stab}}(T) > 0$ such that the minimizer $\theta_T$ of the cost functional (30) satisfies

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)} \le c_{\mathrm{stab}}(T) \left( \epsilon^{\frac{1}{4}} \|\check{\theta}\|_{H_0^1(\Omega)} + \epsilon^{-\frac{1}{4}} \delta T^{\frac{1}{2}} \right). \quad (34) $$

Moreover, under Assumption 3.6, $c_{\mathrm{stab}}(T) \le cT^{-\frac{1}{4}}$, for some $c > 0$ independent of $T$.

*Proof.* In this proof, we write $a \lesssim b$ if there exists $c > 0$ independent of $T$, $\epsilon$, $\delta$ and $M$ such that $a \le cb$.

Proceeding exactly as in the proof of Theorem 3.7, we can obtain (33). Inserting $y^\delta - y^\delta = 0$ in (33) and, by triangular inequality, we find

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\mathrm{idf}}(T))^{-\frac{1}{2}} \left( \|\check{\theta}\|_{H_0^1(\Omega)} + \|\bar{\theta}_T\|_{H_0^1(\Omega)} \right) \times \left[ \left( \int_0^T \|u|_{\bar{\theta}_T} - y^\delta\|_{H^1(\omega)}^2 \mathrm{d}t \right)^{\frac{1}{2}} + \left( \int_0^T \|\tilde{u} - y^\delta\|_{H^1(\omega)}^2 \mathrm{d}t \right)^{\frac{1}{2}} \right]. $$

Using the estimate on the noise amplitude (29), we find

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\mathrm{idf}}(T))^{-\frac{1}{2}} \left( \|\check{\theta}\|_{H_0^1(\Omega)} + \sqrt{2\epsilon^{-1} \mathcal{J}_T(\bar{\theta}_T)} \right) \left( \sqrt{2\mathcal{J}_T(\bar{\theta}_T)} + \delta\sqrt{T} \right). $$

Using again (29), we can bound the cost functional as follows:

$$ \mathcal{J}_T(\bar{\theta}_T) \le \mathcal{J}_T(\check{\theta}) \le \frac{\epsilon}{2} \|\check{\theta}\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \delta^2 T. $$

Hence, we finally get

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\mathrm{idf}}(T))^{-\frac{1}{2}} \left( \|\check{\theta}\|_{H_0^1(\Omega)} + \sqrt{\|\check{\theta}\|_{H_0^1(\Omega)}^2 + \epsilon^{-1} \delta^2 T} \right) \times \left( \sqrt{\epsilon \|\check{\theta}\|_{H_0^1(\Omega)}^2} + \delta^2 T + \delta \sqrt{T} \right), \quad (35) $$

which implies

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (\epsilon c_{\mathrm{idf}}(T))^{-\frac{1}{2}} \left( \epsilon^{\frac{1}{2}} \|\check{\theta}\|_{H_0^1(\Omega)} + \delta T^{\frac{1}{2}} \right)^2, $$

leading to (34). Furthermore, under Assumption 3.6, we have that there exists $c > 0$ independent of $T$ such that $c_{\mathrm{idf}}(T) \ge cT$. $\square$
Finally, with the prior information that

$$ \|\tilde{\theta}\|_{H_0^1(\Omega)} \leq M, \qquad (36) $$

for some $M > 0$, by choosing $\epsilon = \delta^2 M^{-2}$ in (35), we obtain the following corollary.

**Corollary 3.9.** Under the hypothesis of Theorem 3.8 and (36), there exists $c_{opt}(T) > 0$ such that the minimizer $\bar{\theta}_T$ of the cost functional (30) with $\epsilon = \delta^2 M^{-2}$ satisfies

$$ \|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)} \leq c_{\text{opt}}(T) \sqrt{M} \sqrt{\delta}. \qquad (37) $$

3.2. **Measurements in $L^2(\omega)$**. In this section the observation operator $C$ is defined from $\mathcal{Z}$ to $\mathcal{Y} = L^2(\omega)$. Indeed, we assume – in order to model a more realistic practical case – that the available measurements are less regular as they are corrupted by noise. Namely, we suppose that the target trajectory $\tilde{u}$ generated by the target source term parameter $\tilde{\theta}$ has given rise to a collection of measurements $y^\delta$ such that

$$ \int_0^T \|y^\delta(\cdot, t) - \tilde{u}(\cdot, t)\|_{L^2(\omega)}^2 dt \leq \delta^2 T, \qquad (38) $$

for a parameter $\delta > 0$. We present a method to deal with this additional difficulty of regularity loss. This approach consists in modifying the definition of the observation operator with the second observation operator $C$ introduced in (9).

3.2.1. *Identifiability from observability.* The proof of the desired identifiability result for the identification estimation problem is based on the classical following initial condition observability inequality for a homogeneous wave equation in weakened norms (see for instance [41, Theorem 2.3]).

**Proposition 3.10.** Let $(T_{gcc}, \omega_{gcc})$ satisfies the GCC condition and $\omega$ satisfies (7). Then, for $T_0 > T_{gcc}$, there exists $c_o > 0$ such that for all $T \geq T_0$, there exists $c_{obs}(T) \geq c_o T$ such that for all $(u_0, v_0) \in L^2(\Omega) \times H^{-1}(\Omega)$,

$$ \int_0^T \|u(\cdot, t)\|_{L^2(\omega)}^2 dt \geq c_{\text{obs}}(T) \left( \|u_0\|_{L^2(\Omega)}^2 + \|v_0\|_{H^{-1}(\Omega)}^2 \right), \qquad (39) $$

where $u$ is the mild solution of (1) with $\theta = 0$ and initial data $(u_0, v_0)$.

From this standard observability result one deduce the following identifiability property.

**Theorem 3.11.** Let $\omega$ satisfies (7), then for all $T$ sufficiently large, and for all $\sigma(t) \in H^1(0, T)$ with $\sigma(0) \neq 0$, there exists $c_{idf}(T) > 0$ such that

$$ \forall \theta \in \mathcal{P}, \quad \int_0^T \|CL\theta\|_{L^2(\omega)}^2 dt \geq c_{\text{idf}}(T) \|\Delta_0^{-1}\theta\|_{L^2(\Omega)}^2, $$

where the observation operator is $C = (I_\omega^0 \ 0) \in \mathcal{L}(\mathcal{Z}, L^2(\omega))$ and $L$ is given by (5).

*Proof.* The proof of this Theorem is very similar to that of Theorem 3.2, particularly in terms of the calculation tools used. The main difference is based on the use of the observability inequality (39) instead of (13). The proof can be found in Appendix A.3. $\square$

**Remark 3.12.** Inspecting the proof one can see that $T$ should satisfy $T \geq T_0 > T_{gcc}$ and

$$ T \geq 2c_p^2 c_{\text{obs}}^{-1}(T), $$
where $c_p$ is the Poincaré constant and $c_{\text{obs}}(T)$ is the observability constant of Proposition 3.10. Using the estimate $c_{\text{obs}}(T) \ge c_o T$ one obtains that it is sufficient to have

$$T \ge \max (T_0, \sqrt{2} c_p c_o^{-\frac{1}{2}}),$$

and in that case $c_{\text{idf}}(T) \ge c_{\text{obs}}(T)/2$.

### 3.2.2. Updated convergence rate for the Tikhonov regularization.
As in the previous section, we assume that the dependence of the observability constant is in $O(T)$.

**Assumption 3.13.** For $T$ sufficiently large, the identifiability constant of Theorem 3.11 satisfies

$$c_{\text{idf}}(T) \ge cT, \text{ with } c > 0 \text{ independent of } T.$$

We infer from Theorem 3.11 that we now face an ill-posed problem of degree 2. This will modify the convergence rate of our Tikhonov regularization without further assumption on the prior regularity. Let us first recall the following classical interpolation lemma.

**Lemma 3.14.** Let $\theta \in H_0^1(\Omega)$, we have the interpolation property

$$\|\theta\|_{L^2(\Omega)}^2 \le \left( \|\Delta_0^{-1}\theta\|_{L^2(\Omega)}^2 \right)^{\frac{1}{3}} \left( \|\theta\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}}.$$

The proof can be found in Appendix A.2.

Keeping the same regularization in our criterion, we introduce the cost functional to minimize

$$\mathcal{J}_T(\theta) = \frac{\epsilon}{2} \|\theta\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \int_0^T \|y^\delta - C z_{|\theta}|_{L^2(\omega)}^2 \, dt, \quad (40)$$

where $C = (I_\omega^0 0) \in \mathcal{L}(\mathcal{Z}, L^2(\omega))$. We can then estimate the reconstruction error between $\bar{\theta}_T$ the functional minimizer and the target parameter $\tilde{\theta}$.

**Theorem 3.15.** Let $\omega$ satisfies (7), then for all $T$ sufficiently large and for all $\sigma(t) \in H^1(0, T)$ with $\sigma(0) \ne 0$, there exists $c_{\text{stab}}(T) > 0$ such that for all $\tilde{\theta} \in H_0^1(\Omega)$, the minimizer $\bar{\theta}_T$ of the cost functional (40) with noise-free measurements, defined by (38) with $\delta = 0$, satisfies

$$\|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)} \le c_{\text{stab}}(T) \epsilon^{\frac{1}{6}} \|\tilde{\theta}\|_{H_0^1(\Omega)}.$$

Moreover, under Assumption 3.13, $c_{\text{stab}}(T) \le cT^{-\frac{1}{6}}$, for some $c > 0$ independent of $T$.

*Proof.* We use the interpolation inequality of Lemma 3.14 to obtain

$$\|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)}^2 \le \left( \|\bar{\theta}_T - \tilde{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \left( \|\Delta_0^{-1}(\bar{\theta}_T - \tilde{\theta})\|_{L^2(\Omega)}^2 \right)^{\frac{1}{3}}.$$

Using the identifiability result given by Theorem 3.11, we have

$$\|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)}^2 \le (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( \|\bar{\theta}_T\|_{H_0^1(\Omega)}^2 + \|\tilde{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \times \left( \int_0^T \|u_{|\bar{\theta}_T} - \tilde{u}\|_{L^2(\omega)}^2 \, dt \right)^{\frac{1}{3}}. \quad (41)$$

The end of this proof is similar to that of Theorem 3.7. In particular, it is based on the definition of the cost functional. The complete proof is given in Appendix A.4.

□
**Theorem 3.16.** Let $\omega$ satisfies (7), then for all $T$ sufficiently large, for all $\sigma(t) \in H^1(0,T)$ with $\sigma(0) \neq 0$, for $\epsilon > 0$ small enough and for all $\tilde{\theta} \in H_0^1(\Omega)$, there exists $c_{\text{stab}}(T) > 0$ such that the minimizer $\tilde{\theta}_T$ of the cost functional (40) satisfies

$$ \|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)} \le c_{\text{stab}}(T) \left( \epsilon^{\frac{1}{6}} \|\tilde{\theta}\|_{H_0^1(\Omega)} + \epsilon^{-\frac{1}{3}} \delta T^{\frac{1}{2}} \right). \quad (42) $$

Moreover, under Assumption 3.13, $c_{\text{stab}}(T) \le cT^{-\frac{1}{6}}$, for some $c > 0$ independent of $T$.

*Proof.* The proof follows the same lines as in Theorem 3.8 and is based in particular on elements of proof from Theorem 3.15. □

Again, with the prior information that

$$ \|\tilde{\theta}\|_{H_0^1(\Omega)} \le M, \quad (43) $$

for some $M > 0$ then, by choosing $\epsilon = \delta^2 M^{-2}$ in (35), we obtain the following corollary

**Corollary 3.17.** Under the hypothesis of Theorem 3.16 and (43), there exists $c_{\text{opt}}(T) > 0$ such that the minimizer $\bar{\theta}_T$ of the cost functional (40), with $\epsilon = \delta^2 M^{-2}$, satisfies

$$ \|\bar{\theta}_T - \tilde{\theta}\|_{L^2(\Omega)} \le c_{\text{opt}}(T) M^{\frac{2}{3}} \delta^{\frac{1}{3}}. \quad (44) $$

**Remark 3.18.** To conclude these sections linking observability and the robustness of the reconstruction with respect to noise, we would like to emphasize that we could have relaxed the GCC condition. Indeed, there are general log-stability results without conditions on the subdomain $\omega$ in the literature, see in particular the very complete recent paper [37]. Such log-stability conditions could be used in conjunction with the least squares criterion to obtain a reconstruction estimate at logarithmic scales, along the lines of what is done for the heat equation in [56]. Note, however, that the resulting estimates will be logarithmic, hence may be too weak to asymptotically compensate in time for the noise introduced by the observation, which in our framework scales with $T$. This means that a given regularization parameter has less and less influence as time increases, making it more difficult to satisfy all criteria for the definition of an observer, as we will see in the next section.

**4. Observer design.** Our goal in this section is to propose a sequential method for computing $\bar{\theta}_T$ and the associated wave equation trajectory $\bar{z}_T$. By sequential, we mean that we want to update the estimator when measurements are available, without recomputing the estimator each time a new measurement is available. This leads to the definition of an observer in the sense of [36], which is formally defined in a general framework of an observed system of a dynamics of the form

$$ \begin{cases} \dot{z} = F(z, t) \\ z(0) = z_0 \end{cases} \quad (45) $$

where we observed a target trajectory $\dot{z}(t) \in \mathcal{Z}$ a Hilbert space with unknown initial state $\bar{z}_0$ by measuring $(y^\delta(t))_{t \ge 0}$. We here recall such definition adapted to our problem.

**Definition 4.1.** An asymptotic observer of (45) is a function

$$ \mathbb{R}_+ \ni t \mapsto \dot{z}(t; (y^\delta(s))_{s \le t}), $$
such that, for all $0 \le \tau_1 \le \tau_2 \le t$ and, for all observation $y^\delta$ with error support supp($\tilde{y} - y^\delta$) $\subset (\tau_1, \tau_2)$, we have

1. **Robustness to Noise:** for all $\varepsilon > 0$, there exists $\delta, \alpha > 0$, such that

$$ \left. \begin{array}{l} \|\hat{z}(\tau_1) - \tilde{z}(\tau_1)\|_{\mathcal{Z}} \le \alpha \\ \|\tilde{y} - y^\delta\|_{L^2((\tau_1, \tau_2); \mathcal{Y})} \le \delta \end{array} \right\} \Rightarrow \|\tilde{z}(t) - \hat{z}(t)\|_{\mathcal{Z}} \le \varepsilon; $$

2. **Convergence:** for all $\varepsilon, \alpha, \delta > 0$, there exists $T \ge \tau_2$ such that

$$ \left. \begin{array}{l} \|\hat{z}(\tau_1) - \tilde{z}(\tau_1)\|_{\mathcal{Z}} \le \alpha \\ \|\tilde{y} - y^\delta\|_{L^2((\tau_1, \tau_2); \mathcal{Y})} \le \delta \end{array} \right\} \Rightarrow \|\tilde{z}(t) - \hat{z}(t)\|_{\mathcal{Z}} \le \varepsilon, \quad t \ge T. $$

Note that the two above properties correspond respectively to a property of robustness to noise and an asymptotic convergence property in the absence of noise. As the observer can be computed at time $t$ from measurements available at time $s \le t$, we say that it is a causal function of the measurements. Note also that, as a consequence of Definition 4.1-(i), an asymptotic observer satisfies for all $s \ge 0$

$$ \hat{z}(s) = \tilde{z}(s), \quad (y^\delta(t) = \tilde{y}(t), t \ge s) \Rightarrow (\hat{z}(t) = \tilde{z}(t), t \ge s). \quad (46) $$

In our case, we aim to define such an observer for the joint dynamics of $z = (z, \theta)$, where the solution $z$ of the wave equation is extended by the zero dynamics of the parameter $\theta$. Our method is based on a dynamics programming strategy that relates functional minimization to the Kalman observer, which we will apply to the dynamics of the state and parameters. But let us first recall some results about Kalman filtering [12, 5], adapted here to our infinite-dimensional system.

### 4.1. Reduced-order Kalman filters for quasi-evolution operators.
To present our strategy, we first need to review some results in the analysis of evolution problems with non-homogeneous generators.

#### 4.1.1. Quasi-evolution equation.
The aim of this section is to briefly recall some well-known results on quasi-evolution operators, see [47, Chapter 5], for details on this topic. Let us consider the following abstract dynamics defined in a Hilbert space $\mathcal{Z}$

$$ \left\{ \begin{array}{l} \dot{z}(t) = A(t)z(t) + \beta(t), \quad t \in (0, T), \\ z(0) = z_0, \end{array} \right. \qquad (47) $$

from given $z_0 \in \mathcal{Z}$, $\beta \in L^2((0, T); \mathcal{Z})$. Here, the operator

$$ A(t) = A + G(t), $$

where $(A, D(A))$ is maximal dissipative and for all $t > 0$, $G(t)$ is a strongly continuous bounded operator, namely for all $z \in \mathcal{Z}$, $G(\cdot)z \in C^0([0, T], \mathcal{Z})$. Note that by uniform boundedness theorem, there exists $c_{\text{st}} > 0$ such that $\|G(t)\|_{L(\mathcal{Z})} \le c_{\text{st}}$. Classical results for evolution equation give the existence of a solution of (47) knowing $z_0$ and $\beta(t)$ and denoted $z|_{z_0, \beta}$. Indeed, from [13, II-1 Proposition 3.4], if $z_0 \in \mathcal{Z}$ and $\beta \in L^2((0, T); \mathcal{Z})$ then $z|_{z_0, \beta} \in C^0([0, T]; \mathcal{Z})$ is a mild solution

$$ z|_{z_0, \beta}(t) = \Phi(t, 0)z_0 + \int_0^t \Phi(t, s)\beta(s) \, ds, \quad (48) $$

where $\Phi: \Delta_T \to \mathcal{L}(\mathcal{Z})$ with $\Delta_T = \{(t, s) : s \in [0, T], t \in [s, T]\}$ is a quasi-evolution operator in the sense of [22, Definition 1.2].
**Remark 4.2.** Note that it is possible to give an implicit definition of $\Phi$. Indeed, consider the system,

$$ egin{cases} \dot{z}(t) = Az(t) + G(t)z(t), & t \in (s, T), \\ z(s) = z_s \in \mathcal{Z}. \end{cases} $$

By definition of the quasi-evolution operator $\Phi$, we have, using Duhamel's formula

$$ z(t) = \Phi(t, s)z_s = e^{(t-s)A}z_s + \int_s^t e^{(t-\tau)A}G(\tau)z(\tau) \, d\tau, \quad (49) $$

which, using (49), finally becomes

$$ \Phi(t, s)z_s = e^{(t-s)A}z_s + \int_s^t e^{(t-\tau)A}G(\tau)\Phi(\tau, s)z_s \, d\tau. $$

The mild solution (48) is also a weak solution in the sense that

$$ z_{|z_0, \beta} \in L^2((0, T); \mathcal{Z}) \cap H^1((0, T); D(A^*)'), $$

namely for all $q \in D(A^*)$, $(q, z(\cdot))_\mathcal{Z}$ belongs to $H^1(0, T)$ and for almost all $t \in (0, T)$,

$$ \forall q \in D(A^*), \quad \frac{d}{dt}(q, z(t))_\mathcal{Z} + (A(t)^*q, z(t))_\mathcal{Z} = (q, \beta(t))_\mathcal{Z}. \quad (50) $$

Moreover, from [13, II-1 Proposition 3.5], if we further assume that for all $z \in D(A)$, $G(\cdot)z \in C^0([0, T], D(A))$, $z_0 \in D(A)$ and $\beta \in L^2((0, T); D(A))$ then $z_{|z_0, \beta} \in C^0([0, T]; D(A)) \cap H^1((0, T); \mathcal{Z})$. Finally, when $z \in D(A)$, $G(\cdot)z \in C^1([0, T]; \mathcal{Z})$, $z_0 \in D(A)$ and $\beta \in H^1((0, T); D(A))$ then $z_{|z_0, \beta} \in C^0([0, T], D(A)) \cap C^1([0, T]; \mathcal{Z})$.

### 4.1.2. Least-square estimation.
We introduce a Hilbert space $V \subset \mathcal{Z}$ with continuous injection and a continuous, symmetric and coercive bilinear form $m: V \times V \to \mathbb{R}$. We want to minimize a criterion $\mathcal{J}_T: V \to \mathbb{R}$ defined by

$$ \mathcal{J}_T(\zeta) = \frac{1}{2}m(\zeta, \zeta) + \frac{1}{2} \int_0^T \|y^\delta(t) - C(t)z_{|\zeta}(t)\|_{\mathcal{Y}}^2 \, dt, \quad (51) $$

where $z_{|\zeta}$ is the solution of (47) for an initial condition $z_0 = \hat{z}_0 + \zeta$, where $\hat{z}_0$ is a known prior. Note that the problem formulated in (51) is an initial state estimation problem. As will be shown later, our inverse source recovery problem – posed as the minimization of (30) or (40) – can be equivalently reformulated as such initial state recovery problem. The functional $\mathcal{J}_T$ is continuous, differentiable, and strictly convex, so it admits one and only one minimizer denoted $\bar{\zeta}_{|T}$ where we underline the dependency with respect to $T$. Computing the derivative of $\mathcal{J}_T$, we have for all $\xi \in V$,

$$ \langle \mathcal{D}\mathcal{J}_T(\zeta), \xi \rangle_{V', V} = m(\zeta, \xi) + \int_0^T (\xi, \Phi(t, 0)^* C(t)^* (C(t)z_{|\zeta} - y^\delta))_\mathcal{Y} \, dt. $$

Therefore, introducing the adjoint dynamics for any trajectory $z_{|\zeta} \in L^2((0, T); \mathcal{Z})$ and $y^\delta \in L^2((0, T); \mathcal{Y})$,

$$ \left\{ \begin{array}{ll} \dot{q}(t) + A(t)^*q(t) = -C(t)^*(y^\delta(t) - C(t)z_{|\zeta}(t)), & t \in (0, T), \\ q(T) = 0, & \end{array} \right. \quad (52) $$

whose mild solution in $C^0([0, T], \mathcal{Z})$ is given by the Duhamel formulae – adapted to this backward formulation –

$$ q(t) = \int_t^T \Phi(t, s)^* C(t)^* (y^\delta(s) - C(t)z_{|\zeta}(s)) \, ds, \quad t \in [0, T), \quad (53) $$
we find

$$ \forall \xi \in \mathcal{V}, \quad \langle \mathcal{D} \mathcal{J}_T(\zeta), \xi \rangle_{\mathcal{V}', \mathcal{V}} = m(\zeta, \xi) - (\mathbf{q}(0), \xi)_{\mathcal{Z}}. \quad (54) $$

If we now introduce the operator $\Pi_0 \in \mathcal{L}(\mathcal{Z}, \mathcal{V})$ such that for all $\mathbf{q} \in \mathcal{Z}$, $\Pi_0 \mathbf{q}$ is the solution of $m(\Pi_0 \mathbf{q}, \xi) = (\mathbf{q}, \xi)_{\mathcal{Z}}$ for all $\xi \in \mathcal{V}$, we deduce from (54) the optimality condition

$$ \bar{\zeta}_{|T} = \Pi_0 \bar{q}_T(0), $$

where $\bar{q}_T$ is associated with the optimal trajectory $\bar{z}_T = z_{|\bar{\zeta}_{|T}}$ through the two-ends optimality system – as called in [40, 12]

$$ \left\{ \begin{array}{ll} \dot{\bar{z}}_T(t) = \mathbf{A}(t)\bar{z}_T + \beta(t), & t \in (0, T), \\ \dot{\bar{q}}_T(t) + \mathbf{A}(t)^*\bar{q}_T = -\mathbf{C}(t)^*(y^\delta(t) - \mathbf{C}(t)\bar{z}_T(t)), & t \in (0, T), \\ \bar{z}_T(0) = \hat{z}_0 + \Pi_0 \bar{q}_T(0), \\ \bar{q}_T(T) = 0. \end{array} \right. \quad (55) $$

Note finally that if $\mathcal{V} \subset \mathcal{D}(\mathbf{A})$, then $\bar{z}$ and $\bar{q}$ belong to $C^1([0, T], \mathcal{Z}) \cap C^0([0, T], \mathcal{D}(\mathbf{A}))$.

4.2. **Kalman estimator definition.** The objective of this section is to present briefly the Kalman estimator strategy to solve sequentially the two ends problem (55). First, we introduce the Riccati dynamics

$$ \left\{ \begin{array}{l} \dot{\Pi}(t) = \mathbf{A}(t)\Pi(t) + \Pi(t)\mathbf{A}(t)^* - \Pi(t)\mathbf{C}(t)^*\mathbf{C}(t)\Pi(t), \quad t \in (0, T), \\ \Pi(0) = \Pi_0, \end{array} \right. \quad (56) $$

where $\mathbf{C} \in C^0([0, T], \mathcal{L}(\mathcal{Z}, \mathcal{Y}))$. The initial condition $\Pi_0 \in S^+(\mathcal{Z})$ the space of bounded symmetric operators. The following theorem gives the existence of solution of (56) in $S^+(\mathcal{Z})$. The resulting operator $\Pi \in S^+(\mathcal{Z})$ is called Riccati operator or covariance operator for its interpretation in the context of stochastic filtering, see for instance [12, 23].

**Theorem 4.3 (Existence of Riccati solution).**

The Riccati dynamics (56) admits one and only one mild solution $\Pi \in C^0([0, T], S^+(\mathcal{Z}))$ given by

$$ \Pi(t) = \Phi(t, 0)\Pi_0\Phi(t, 0)^* - \int_0^t \Phi(t, s)\Pi(s)C(t)^*C(t)\Pi(s)\Phi(t, s)^* ds. \quad (57) $$

Moreover, for all $(z_1, z_2) \in \mathcal{D}(\mathbf{A}^*)$, $(\Pi(\cdot)z_1, z_2)_{\mathcal{Z}} \in C^1([0, T])$ and satisfies the weak formulation

$$ \frac{d}{dt} (\Pi(t) z_1, z_2)_{\mathcal{Z}} = (\Pi(t) z_1, \mathbf{A}(t)^* z_2)_{\mathcal{Z}} + (\Pi(t) \mathbf{A}(t)^* z_1, z_2)_{\mathcal{Z}} \\ - (\mathbf{C}(t) \Pi(t) z_1, \mathbf{C}(t) \Pi(t) z_2)_{\mathcal{Y}} = 0, \quad t \in [0, T]. \quad (58) $$

*Proof.* We refer to [22, Corollary 2.2], for a proof of existence of a mild solution of a Riccati equation where $\mathbf{A}(t) = \mathbf{A} + \mathbf{G}(t)$ with $(\mathbf{A}, \mathcal{D}(\mathbf{A}))$ unbounded and $\mathbf{G}(t)$ a bounded perturbation. The existence of a weak solution is a direct consequence, see for instance [13, IV-1 Proposition 2.1]. □

From the Riccati operator definition, we can introduce the so-called Kalman observer – also called Kalman estimator in the context of stochastic filtering.
**Theorem 4.4.** There exists one and only one mild solution in $C^0([0, T]; \mathcal{Z})$ of the dynamics

$$ \begin{cases} \dot{\hat{z}} = \mathbf{A}(t)\hat{z} + \mathbf{\Pi}(t)\mathbf{C}(t)^*(y^\delta(t) - \mathbf{C}(t)\hat{z}(t)), & t \in (0, T), \\ \hat{z}(0) = \hat{z}_0, \end{cases} \qquad (59) $$

in the sense that

$$ \hat{z}(t) = \mathbf{\Phi}(t, 0)\hat{z}_0 + \int_0^t \mathbf{\Phi}(t, s)\mathbf{\Pi}(s)\mathbf{C}(s)^*[y^\delta(s) - \mathbf{C}(s)\hat{z}(s)] \, ds. \qquad (60) $$

Moreover, this solution is the unique weak solution in the sense that, (1.) $z \in L^2([0, T]; \mathcal{Z})$, (2.) for all $q \in D(\mathbf{A}^*)$, $(\hat{z}(\cdot), q)_{\mathcal{Z}}$ belongs to $H^1(0, T)$ and (3.) for almost all $t \in (0, T)$,

$$ \forall q \in D(\mathbf{A}^*), \quad \frac{d}{dt}(\hat{z}(t), q)_{\mathcal{Z}} + (\hat{z}, \mathbf{A}(t)^*q)_{\mathcal{Z}} = (y^\delta(t) - \mathbf{C}(t)\hat{z}, \mathbf{C}(t)\mathbf{\Pi}(t)v)_{\mathcal{Y}}. \quad (61) $$

*Proof.* We know that $\mathbf{\Pi} \in C^0([0, T]; S_+(\mathcal{Z}))$. Therefore, for all $t \in [0, T]$, $\mathbf{A}(t) - \mathbf{\Pi}(t)\mathbf{C}(t)^*\mathbf{C}(t)$ is a uniform bounded perturbation of $\mathbf{A}$, the existence enters exactly the framework of quasi-evolution operator. $\square$

### 4.2.1. The optimal filtering property.
We can now move to the understanding on how the Riccati operator and the associated Kalman observer decouples the two ends problems (55). The following proof extends the one proposed in [5] to quasi evolution operator.

**Theorem 4.5.** The Kalman estimator $\hat{z}$ defined by the dynamics (59) satisfies the fundamental identity

$$ \forall t \in [0, T], \quad \bar{z}_T(t) = \hat{z}(t) + \mathbf{\Pi}(t)\bar{q}_T(t), \qquad (62) $$

where $\bar{q}_T$ is solution to (55).

*Proof.* Let us introduce $\eta = \hat{z} - \bar{z}_T + \mathbf{\Pi}\bar{q}_T$ and $v \in D(\mathbf{A}^*)$, and compute

$$ \frac{d}{dt}(\eta(t), v)_{\mathcal{Z}} = (\hat{z}(t), \mathbf{A}(t)^*v)_{\mathcal{Z}} + (y^\delta(t) - \mathbf{C}(t)\hat{z}(t), \mathbf{C}(t)\mathbf{\Pi}(t)v)_{\mathcal{Y}} \\ - (\bar{z}_T(t), \mathbf{A}(t)^*v)_{\mathcal{Z}} + \frac{d}{dt}(\mathbf{\Pi}(t)\bar{q}_T(t), v)_{\mathcal{Z}}, \quad (63) $$

where we need to properly define the last-term time derivative. In fact, as $\bar{q}_T$ can be defined from a Duhamel formula of the form (53), and $\mathbf{\Pi}$ is mild solution of the Riccati equation (56), hence is absolutely continuous, we have from [22, Lemme 2.5] that

$$ \frac{d}{dt}(\mathbf{\Pi}(t)\bar{q}_T(t), v)_{\mathcal{Z}} = \frac{d}{dt}(\mathbf{\Pi}(t)\bar{q}_T(s), v)_{\mathcal{Z}}\Big|_{s=t} + \frac{d}{dt}(\bar{q}_T(t), \mathbf{\Pi}(s)v)_{\mathcal{Z}}\Big|_{s=t} $$

Moreover, as $\mathbf{\Pi}$ is a weak solution of (56), we have from (58),

$$ \frac{d}{dt}(\mathbf{\Pi}(t)\bar{q}_T(s), v)\Big|_{s=t} = (\mathbf{\Pi}(t)\bar{q}_T(t), \mathbf{A}(t)^*v)_{\mathcal{Z}} + (\bar{q}_T(t), \mathbf{A}(t)^*\mathbf{\Pi}(t)v)_{\mathcal{Z}} \\ - (\mathbf{C}(t)\mathbf{\Pi}(t)\bar{q}_T(t), \mathbf{C}(t)\mathbf{\Pi}(t)v)_{\mathcal{Y}}, \quad (64) $$

and, as $\bar{q}_T$ is a weak solution of (55),

$$ \frac{d}{dt}(\bar{q}_T(t), \mathbf{\Pi}(s)v)\Big|_{s=t} = -(\bar{q}_T(t), \mathbf{A}(t)^*\mathbf{\Pi}(t)v)_{\mathcal{Z}} \\ - (y^\delta(t) - \mathbf{C}(t)\bar{z}_T(t), \mathbf{C}(t)\mathbf{\Pi}(t)v)_{\mathcal{Y}}. \quad (65) $$
Gathering (63), (64) and (65), we finally obtain

$$ egin{cases} \frac{d}{dt}(\eta(t), v)_{\mathcal{Z}} = (\eta(t), \mathbf{A}(t)^* - \mathbf{C}(t)^*\mathbf{C}(t)\mathbf{\Pi}(t)v)_{\mathcal{Z}}, & t \in [0, T], \\ \eta(0) = 0, \end{cases} $$

whence, by Theorem 4.4, $\eta = 0$ in $[0, T]$, which concludes the proof. $\square$

<figure><img src="image_2.png" /></figure>

In particular, we see that by taking $t = T$ that for all $t > 0$, $\hat{z}(t) = \bar{z}_t(t)$. This is why the Kalman observer is also called optimal filtering estimator as it consists of sequentially computing the trajectory associated with the least squares estimator at the final time.

### 4.2.2. The reduced order case.
As we saw in the previous section, the inverse problem can be solved using the Kalman filter, but this requires to calculate a Riccati operator in the augmented space $\mathcal{Z}$. In order to reduce the computational cost, we now show that we can instead compute another Riccati operator on a reduced space $\mathcal{Z}_r$ that takes into account prior information on the inverse problem: in our case, described specifically later, only the source term is not known. The reduced space $\mathcal{Z}_r$ is assumed to be a Hilbert space, and we assume given a bounded operator $P_r \in \mathcal{L}(\mathcal{Z}, \mathcal{Z}_r)$. Moreover, we also assume given a closed subspace $\mathcal{V}_r$ of $\mathcal{Z}_r$ (with continuous injection) that is used to define a regularized version of the functional to minimize. The minimization problem reads

$$ \min_{\zeta_r \in \mathcal{V}_r} \left\{ \mathcal{J}_T(\zeta_r) = \frac{1}{2} m_r(\zeta_r, \zeta_r) + \frac{1}{2} \int_0^T \|y^\delta - \mathcal{C}z_{|P_r^*\zeta_r}\|_{\mathcal{Y}}^2 dt \right\}, $$

where $z_{|P_r^*\zeta_r}$ is solution of (47) for an initial condition $z_0 = \hat{z}_0 + P_r^*\zeta_r$ and $m_r$ is a continuous and coercive bilinear form on $\mathcal{V}_r$. The associated two ends problem reads

$$ egin{cases} \dot{\bar{z}}_T = \mathbf{A}(t)\bar{z}_T + \beta(t), & t \in (0, T), \\ \dot{\bar{q}}_T + \mathbf{A}(t)^*\bar{q}_T = -\mathbf{C}(t)^*(y^\delta(t) - \mathbf{C}(t)\bar{z}_T), & t \in (0, T), \\ \bar{z}_T(0) = \hat{z}_0 + P_r^*\Lambda_0 P_r \bar{q}_T(0), \\ \bar{q}_T(T) = 0, \end{cases} \quad (66) $$

with $\Lambda_0 \in \mathcal{L}(\mathcal{V}_r)$ defined such that

$$ \forall (\zeta_r, \xi_r) \in \mathcal{V}_r \times \mathcal{V}_r, \quad m_r(\Lambda_0 \zeta_r, \xi_r) = (\zeta_r, \xi_r)_{\mathcal{Z}_r}. $$

The resulting Kalman observer can still be defined from the Riccati dynamics

$$ egin{cases} \dot{\mathbf{\Pi}}(t) = \mathbf{A}(t)\mathbf{\Pi}(t) + \mathbf{\Pi}(t)\mathbf{A}(t)^* - \mathbf{\Pi}(t)\mathbf{C}^*\mathbf{C}\mathbf{\Pi}(t), & t > 0, \\ \mathbf{\Pi}(0) = P_r^*\Lambda_0 P_r, \end{cases} \quad (67) $$

so that we continue to have $\bar{z}_T(t) = \hat{z}(t) + \mathbf{\Pi}(t)\bar{q}_T(t)$ for all $t \in [0, T]$. But moreover, the Riccati solution of (67) can be deduced from a reduced covariance operator $\Lambda$, solution of a Riccati dynamics in the reduced space $\mathcal{Z}_r$, namely

$$ egin{cases} \dot{\Lambda}(t) = -\Lambda(t)P_r\Phi(t, 0)^*\mathbf{C}^*\mathbf{C}\Phi(t, 0)P_r^*\Lambda(t), & t > 0, \\ \Lambda(0) = \Lambda_0. \end{cases} \quad (68) $$

This dynamics satisfies the condition of existence of Theorem 4.3 since for all $t \in [0, T]$, $\mathbf{C}\Phi(t, 0)P_r^*$ can be considered as a time-dependent bounded observation operator. Moreover, we have the following decomposition.
**Theorem 4.6.** Let $\Pi(t)$ the mild solution of the Riccati dynamics (67) and $\Lambda(t)$ the mild solution of the Riccati dynamics (68), then we have

$$ \Pi(t) = \Phi(t, 0) P_r^* \Lambda(t) P_r \Phi(t, 0)^*. \quad (69) $$

*Proof.* The mild solution of the Riccati dynamics (68) reads

$$ \Lambda(t) = \Lambda(0) - \int_{0}^{t} \Lambda(s) P_{r} \Phi(s, 0)^{*} C^{*} C \Phi(s, 0) P_{r}^{*} \Lambda(s) \mathrm{d}s. $$

Using this mild solution, we compute the following quantity $\Phi(t, 0) P_r^* \Lambda(t) P_r \Phi(t, 0)^*$.

$$ \Phi(t, 0) P_r^* \Lambda(t) P_r \Phi(t, 0)^* = \Phi(t, 0) P_r^* \Lambda(0) P_r \Phi(t, 0)^* \\ - \int_0^t \Phi(t, 0) P_r^* \Lambda(s) P_r \Phi(s, 0)^* C^* C \Phi(s, 0) P_r^* \Lambda(s) P_r \Phi(t, 0)^* \mathrm{d}s. $$

Note that $\Phi(t, 0) = \Phi(t, s)\Phi(s, 0)$, therefore we obtain

$$ \Phi(t, 0) P_r^* \Lambda(t) P_r \Phi(t, 0)^* = \Phi(t, 0) P_r^* \Lambda(0) P_r \Phi(t, 0)^* \\ - \int_0^t \Phi(t, s) \Phi(s, 0) P_r^* \Lambda(s) P_r \Phi(s, 0)^* C^* C \Phi(s, 0) P_r^* \Lambda(s) P_r \Phi(s, 0)^* \Phi(t, s)^* \mathrm{d}s. $$

Setting $\tilde{\Pi}(t) = \Phi(t, 0) P_r^* \Lambda(t) P_r \Phi(t, 0)^*$, we get

$$ \tilde{\Pi}(t) = \tilde{\Pi}(0) - \int_{0}^{t} \Phi(t, s) \tilde{\Pi}(s) C^{*} C \tilde{\Pi}(s) \Phi(t, s)^{*} \mathrm{d}s. $$

Hence, $\tilde{\Pi}$ is a mild solution of (67) and by uniqueness of the solution of the Riccati dynamics, we have

$$ \tilde{\Pi}(t) = \Pi(t) = \Phi(t, 0) P_r^* \Lambda(t) P_r \Phi(t, 0)^*, $$

which concludes the proof. □

Therefore, a computational strategy can be based only on computations restricted to the reduced-space $\mathcal{Z}_r$ using the expression (69) into (59).

4.2.3. *Application to source estimation.* We will now define an observer, which is equivalent to the calculation of the optimum $\theta_T$ with increasing time. To that purpose, we rely on the Kalman estimation theory presented above and more precisely the Reduced-Order Kalman Filtering approach of Section 4.2.2, where the uncertainty space is, here, the parameter space, a subspace of the augmented space $\mathcal{Z} = \mathcal{Z} \times \mathcal{P}$. In particular, we set $\mathcal{Z}_r = L^2(\Omega)$ and $\mathcal{V}_r = H_0^1(\Omega)$. The dynamics in the augmented space reads

$$ \underbrace{\frac{d}{dt} \begin{pmatrix} z \\ \theta \end{pmatrix}}_{\dot{z}} = \underbrace{\begin{pmatrix} A & B(t) \\ 0 & 0 \end{pmatrix} \begin{pmatrix} z \\ \theta \end{pmatrix}}_{\mathbf{A}(t)\mathbf{z}} \quad \text{with} \quad \underbrace{\begin{pmatrix} z(0) \\ \theta(0) \end{pmatrix}}_{\mathbf{z}(0)} = \underbrace{\begin{pmatrix} 0 \\ \zeta \end{pmatrix}}_{\zeta}, \quad (70) $$

and the observation operator is $C = (C \ 0)$. The least squares criteria (30) or (40) rewritten with respect to the augmented variable read

$$ \begin{aligned} \mathcal{J}_T(\theta) &= \frac{\epsilon}{2} \|\theta\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \int_0^T \|y^\delta(t) - Cz_{|_\theta}\|_{\mathcal{Y}}^2 \mathrm{d}t, \\ &= \frac{\epsilon}{2} \|\theta\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \int_0^T \|y^\delta(t) - Cz_{|_\theta}\|_{\mathcal{Y}}^2 \mathrm{d}t, \end{aligned} \quad (71) $$
and we denote by $\bar{\theta}_T \in \mathcal{P} = \text{H}_0^1(\Omega)$ the minimizer of $\mathcal{J}_T(\theta)$. Observe that the problem formulated in (71) is an equivalent reformulation of the original least squares source estimation problem, where the minimization of the functional in (30) or (40) is recast as an initial state estimation problem for an augmented system. Introducing the operator

$$P_{\theta} = (0 \quad \text{Id}_{\mathcal{P}}) \in \mathcal{L}(\boldsymbol{Z}, \boldsymbol{Z}_r), \quad P_{\theta}^{*} = \begin{pmatrix} 0 \\ \text{Id}_{\mathcal{P}} \end{pmatrix} \in \mathcal{L}(\boldsymbol{Z}_r, \boldsymbol{Z}),$$

we have, using Theorem 4.6,

$$\left\{ \begin{array}{l} \boldsymbol{\Pi}(t) = \boldsymbol{\Phi}(t, 0) P_{\theta}^{*} \Lambda(t) P_{\theta} \boldsymbol{\Phi}(t, 0)^{*}, \\ \boldsymbol{\Pi}(0) = P_{\theta}^{*} \Lambda(0) P_{\theta}, \end{array} \right. \qquad (72)$$

where $\Lambda(t)$ is given by (68) with $P_r = P_{\theta}$. Then, we can specify the evolution operator $\boldsymbol{\Phi}$ associated with the following system

$$\mathbf{A}(t) = \begin{pmatrix} A & B(t) \\ 0 & 0 \end{pmatrix}, \quad \left\{ \begin{array}{l} \dot{\mathbf{z}}(t) = \mathbf{A}(t)\mathbf{z}(t), \\ \mathbf{z}(0) = \mathbf{z}_0, \end{array} \right. \quad \Rightarrow \quad \left\{ \begin{array}{l} \dot{\mathbf{z}}(t) = A\mathbf{z}(t) + B(t)\theta, \\ \dot{\theta}(t) = 0, \\ (z, \theta)^t(0) = (z_0, \theta_0)^t. \end{array} \right.$$

Therefore, for initial data of the form $\mathbf{z}_0 = P_{\theta}^* \theta_0 = (0, \theta_0)^t$ the solution reads

$$\begin{pmatrix} z(t) \\ \theta(t) \end{pmatrix} = \boldsymbol{\Phi}(t, 0) \begin{pmatrix} 0 \\ \theta_0 \end{pmatrix},$$

and, using the definition of $L$ given by (5), $z(t) = L(t)\theta_0$ is the mild solution of

$$\left\{ \begin{array}{l} \dot{z} = Az + B\theta_0, \quad t > 0, \\ z(0) = 0. \end{array} \right.$$

Hence, we obtain that

$$\begin{pmatrix} z(t) \\ \theta(t) \end{pmatrix} = \begin{pmatrix} L(t) \\ \text{Id}_{\mathcal{P}} \end{pmatrix} \theta_0.$$

Eventually, we have shown that for all $\theta_0$, $\boldsymbol{\Phi}(t, 0) P_{\theta}^* \theta_0^t$ is given by

$$\begin{pmatrix} z(t) \\ \theta(t) \end{pmatrix} = \boldsymbol{\Phi}(t, 0) P_{\theta}^{*} \theta_{0} = \boldsymbol{\Phi}(t, 0) \begin{pmatrix} 0 \\ \theta_{0} \end{pmatrix} = \begin{pmatrix} L(t) \\ \text{Id}_{\mathcal{P}} \end{pmatrix} \theta_{0}.$$

Therefore, we can simplify the Riccati dynamics (68) into

$$\left\{ \begin{array}{l} \dot{\Lambda} = -\Lambda L^* C^* CL\Lambda, \quad t > 0, \\ \Lambda(0) = \Lambda_0, \end{array} \right. \qquad (73)$$

with $C^* \in \mathcal{L}(\mathcal{Y}, \mathcal{Z})$ the adjoint of $C \in \mathcal{L}(\mathcal{Z}, \mathcal{Y})$ and (72) into

$$\boldsymbol{\Pi}(t) = \begin{pmatrix} L(t) \\ \text{Id}_{\mathcal{P}} \end{pmatrix} \Lambda(t) \begin{pmatrix} L^*(t) & \text{Id}_{\mathcal{P}} \end{pmatrix}. \qquad (74)$$

Note that, in our case $m_r(\cdot, \cdot) = \frac{\epsilon}{2}(\cdot, \cdot)\text{H}_0^1(\Omega)$, therefore we can obtain an explicit expression of $\Lambda_0$ given by

$$\Lambda_0 = -\epsilon^{-1} \Delta_0^{-1}.$$

Finally, we obtain the equivalent sequential estimator with the following result that is a straightforward application of Theorem 4.5.
**Corollary 4.7.** Let $\bar{\theta}_t$ be the minimizer of $\mathcal{J}_t$ introduced in (71). The sequential estimator $\hat{\theta}: \mathbb{R}^+ \ni t \mapsto \bar{\theta}_t(t) \in \mathcal{P}$ is solution to the following dynamics

$$ 
\left\{
\begin{array}{ll}
\dot{\hat{\theta}} = \Lambda L^* C^*(y^\delta - C\hat{z}), & t > 0, \\
\hat{\theta}(0) = \hat{\theta}_0, & \\
\dot{\hat{z}} = A\hat{z} + B\hat{\theta} + L\hat{\dot{\theta}}, & t > 0, \\
z(0) = \hat{z}_0, &
\end{array}
\right.
\qquad (75)
$$

where $L$ is defined by (5) and $\Lambda$ by (73).

Using the results obtained in Sections 3.1.2 and 3.2.2, we can investigate how $(\hat{z}, \hat{\theta})$ can be considered as an observer in the sense of Definition 4.1. Indeed, it satisfies the first property (i) (robustness to noise) and the second property (ii) (convergence) can be fulfilled under additional assumptions.

**Proposition 4.8.** Assuming that we have at our disposal the perfect measurements $y^\delta = \hat{y}$, then, under Assumption 3.6 (resp. Assumption 3.13) if we consider the observation operator (8) (resp. operator (9)) the estimators $(\hat{z}, \hat{\theta})$ asymptotically converge to the target parameter $(\hat{z}, \hat{\theta})$, namely

$$ \lim_{t \to +\infty} \|\hat{\theta}(t) - \check{\theta}(t)\|_{L^2(\Omega)} = 0 \quad \text{and} \quad \lim_{t \to +\infty} \|\hat{z}(t) - \check{z}(t)\|_{Z} = 0. $$

*Proof.* We use here the link between the observer and the functional minimization. Under Assumptions 3.6 and 3.13, we know that (Theorem 3.7 and 3.15), for both types of observation operator, there exists $a > 0$ and $C^{\text{st}}$ such that

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)} \le C^{\text{st}} T^{-a}, $$

which, combined with the fact that, for all $t \ge 0$, $\hat{\theta}(t) = \bar{\theta}_t(t)$, gives the convergence of the parameter estimator

$$ \lim_{t \to +\infty} \|\hat{\theta}(t) - \check{\theta}\|_{L^2(\Omega)} = 0. $$

Moreover, we have that

$$ \forall t \ge 0, \quad \hat{z}(t) = \bar{z}_t(t) := z_{|\bar{\theta}_t}(t). $$

Therefore,

$$ \lim_{t \to +\infty} \|\hat{z}(t) - \check{z}(t)\|_{Z} = \lim_{t \to +\infty} \|z_{|\bar{\theta}_t}(t) - z_{|\check{\theta}}(t)\|_{Z} = 0, $$

which concludes the proof.

□

**Proposition 4.9.** Assuming that we have at our disposal the noisy measurements satisfying (29) or (38), the estimator $(\hat{z}, \hat{\theta})$ is stable with respect to noise, namely there exists a positive scalar $m$ depending only on $t$ such that

$$ \forall t \ge 0, \quad \|\hat{\theta}(t) - \check{\theta}\|_{L^2(\Omega)} \le m(t) \delta \quad \text{and} \quad \|\hat{z}(t) - \check{z}(t)\|_{Z} \le m(t) \delta. $$

*Proof.* We take advantage of the link between the observer and the functional minimization. From Theorem 3.8 and 3.16, for both types of observation operator, there exist $C^{\text{st}}$ such that

$$ \|\bar{\theta}_t - \check{\theta}\|_{L^2(\Omega)} \le C^{\text{st}} \delta, $$

which, combined with $\hat{\theta}(t) = \bar{\theta}_t$ gives the stability of the parameter estimator

$$ \forall t \ge 0, \quad \|\hat{\theta}(t) - \check{\theta}\|_{L^2(\Omega)} \le C^{\text{st}} \delta. $$
Furthermore, we have that

$$ \forall t \geq 0, \quad \hat{z}(t) = \bar{z}_t(t) = z_{|\bar{\theta}_t}(t). $$

Therefore, we obtain

$$ \forall t \geq 0, \quad \|\hat{z}(t) - \bar{z}(t)\|_{\mathcal{Z}} = \|z_{|\bar{\theta}_t}(t) - z_{|\bar{\theta}}(t)\|_{\mathcal{Z}}. $$

Classically using the Grönwall lemma, one can show that there exists a positive scalar $C^{\text{st}}$ depending only on $t$ and $\sigma$ such that

$$ \forall t \geq 0, \quad \|z_{|\bar{\theta}_t}(t) - z_{|\bar{\theta}}(t)\|_{\mathcal{Z}} \leq C^{\text{st}} \|\bar{\theta}_t(t) - \bar{\theta}\|_{L^2(\Omega)}, $$

which concludes the proof. □

**Remark 4.10.** Proposition 4.9 allows us to demonstrate the robustness to noise property of Definition 4.1. On the other hand, Proposition 4.8 is a necessary condition for the convergence property of Definition 4.1. Indeed, the second property of the definition is obtained only in the case $\tau = 0$. We would have to take into account the possibility of uncertainties in the initial condition for dynamics (75) to fall within the asymptotic observer framework of definition 4.1.

### 4.3. Observer discretization.
We can now study the observer discretization, where we face two possible options. First, we could directly propose a time and space discretization of (75), or we could first discretize the wave equation and then reapply the Tikhonov regularization strategy and observer design to the discretized system. Following [61], we are going to follow this second strategy.

#### 4.3.1. Reduced-order Kalman discretization.
As in the time-continuous case, let us first recall the Kalman discretization for a linear discrete-time dynamics. Let $\mathcal{Z}^h \subset \mathcal{V} \subset \mathcal{Z}$ a finite dimensional space equipped with the scalar product of $\mathcal{Z}$. Assuming a dynamics of the following form, with, for all $n \geq 1$ a given source term $\beta_n^h \in \mathcal{Z}^h$

$$ \begin{cases} z_{n+1|\zeta}^h = \mathbf{\Phi}^{h,\tau} z_{n|\zeta}^h + \beta_{n+1}^h, & n \geq 0, \\ z_{0|\zeta}^h = \hat{z}_0^h + \zeta, \end{cases} $$

associated with measurements $y_n^{\delta,h}$ and an observation operator $C^h \in \mathcal{L}(\mathcal{Z}^h, \mathcal{Y}^h)$ where $\mathcal{Y}^h$ is the finite dimensional space in which the observations belong. Above, $\tau$ is a positive parameter that represents a time step. We first consider the minimization of a discrete version of the criterion (51) (see [5])

$$ \min_{\zeta \in \mathcal{Z}^h} \left\{ \mathcal{J}_N(\zeta^h) = \frac{1}{2} m(\zeta^h, \zeta^h) + \sum_{n=1}^{N} \frac{\tau}{2} \| y_n^{\delta,h} - C^h z_{n|\zeta}^h \|_{\mathcal{Y}^h}^2 \right\}, \quad (76) $$

where the time integral is replaced by a simple quadrature formula and $N\tau$ represents a final time of observations. We easily find the corresponding two ends problem associated with the minimization of this criterion,

$$ \left\{ \begin{aligned} \bar{z}_{n+1|N}^h &= \mathbf{\Phi}^{h,\tau} \bar{z}_{n|N}^h + \beta_{n+1}^h, & 0 \le n \le N-1, \\ \bar{z}_{0|N}^h &= \hat{z}_0^h + \Pi_0^h \bar{q}_{0|N}^h, \\ \bar{q}_{n|N}^h &= \mathbf{\Phi}^{h,\tau*} \bar{q}_{n+1|N}^h + \tau C^{h*} (y_n^{\delta,h} - C^h \bar{z}_{n|N}^h), & 0 \le n \le N-1, \\ \bar{q}_{N|N}^h &= 0, \end{aligned} \right. \quad (77) $$
where the notation $|N|$ stresses the fact that the criterion is using the $N$ first observations and $\Pi_0^h \in \mathcal{L}(\mathcal{Z}^h)$ is the operator defined by

$$ \forall (\boldsymbol{q}_h, \xi_h) \in \mathcal{Z}^h \times \mathcal{Z}^h, \quad m(\Pi_0^h \boldsymbol{q}_h, \xi_h) = (\boldsymbol{q}_h, \xi_h)\boldsymbol{z}. $$

Then, we can uncouple these systems by solving the following Riccati equation for all $n \ge 0$,

$$ \left\{ \begin{array}{l} \Pi_{n+1-}^h = \boldsymbol{\Phi}^{h,\tau} \Pi_n^h \boldsymbol{\Phi}^{h,\tau*}, \\ \Pi_{n+1}^h = \Pi_{n+1-}^h - \tau \Pi_{n+1-}^h C^{h*} [\mathrm{Id}_{\mathcal{Y}^h} + \tau C^h \Pi_{n+1-}^h - C^{h*}]^{-1} C^h \Pi_{n+1-}^h, \end{array} \right. \qquad (78) $$

which defines a sequence of operators $\Pi_{n-}^h$ and $\Pi_n^h$. The sequence of $\Pi_{n-}^h$ and $\Pi_n^h$ can be interpreted as a splitting time scheme for the Riccati dynamics (56) [5]. Then, we introduce the estimator

$$ \left\{ \begin{array}{ll} \hat{z}_{n+1-}^h = \boldsymbol{\Phi}^{h,\tau} \hat{z}_n^h + \beta_{n+1}^h, & n \ge 0, \\ \hat{z}_{n+1}^h = \hat{z}_{n+1-}^h + \tau \Pi_{n+1}^h C^{h*} (y_{n+1}^{\delta,h} - C^h \hat{z}_{n+1-}^h), & n \ge 0. \end{array} \right. \qquad (79) $$

Identically, (79) can be interpreted as a splitting time scheme for the estimator dynamics (59), see [5]. Then, we have the following equivalence results proved in [5].

**Theorem 4.11.** We have the following decoupling principle

$$ \bar{z}_{n|N}^h = \hat{z}_{n-}^h + \Pi_{n-}^h \bar{q}_{n|N}^h, \quad \forall 0 \le n \le N+1, \qquad (80) $$

and in particular for all $n \in \mathbb{N}$

$$ \hat{z}_{n-}^h = \bar{z}_{n|N}^h. \qquad (81) $$

Let us now apply such equivalence to a reduced space configuration. For this purpose, we define $\mathcal{Z}_r^h$ a finite dimensional space such that $\mathcal{Z}_r^h \subset \mathcal{V}_r \subset \mathcal{Z}_r$. To simplify the notation, we assume that $P_r(\mathcal{Z}^h) \subset \mathcal{Z}_r^h$ – which is satisfied for our source identification problem – hence $P_r$ can be seen as an operator in $\mathcal{L}(\mathcal{Z}^h, \mathcal{Z}_r^h)$. The discrete criterion here reads

$$ \min_{\zeta_r^h \in \mathcal{Z}_r^h} \left\{ \mathcal{J}_N(\zeta_r^h) = \frac{1}{2} m_r(\zeta_r^h, \zeta_r^h) + \sum_{k=1}^{N} \frac{\tau}{2} \| y_n^{\delta,h} - C \mathcal{Z}_{n|P_r^* \zeta_r^h}^h \|_{\mathcal{Y}^h}^2 \right\}, \qquad (82) $$

and in our identification case, $m_r$ will be the bilinear form associated with the $H_0^1(\Omega)$ norm. We denote $\Lambda_0^h \in \mathcal{L}(\mathcal{V}_r^h)$ the operator given by

$$ \forall (\zeta_r^h, \xi_r^h) \in \mathcal{V}_r^h \times \mathcal{V}_r^h, \quad m_r(\Lambda_0^h \zeta_r^h, \xi_r^h) = (\zeta_r^h, \xi_r^h)\boldsymbol{z}_r. $$

The two ends problem remains as in (77), except for the initial condition that is changed into

$$ \bar{z}_{0|N}^h = \hat{z}_{0-}^h + P_r^* \Lambda_0^h P_r \bar{q}_{0|N}^h. \qquad (83) $$

Moreover, we have the discrete counterpart of Theorem 4.6.

**Theorem 4.12.** The solution of (78) with initial condition defined by

$$ \Pi_0^h = P_r^* \Lambda_0^h P_r $$

satisfies

$$ \Pi_n^h = (\boldsymbol{\Phi}^{h,\tau})^n P_r^* \Lambda_n^h P_r (\boldsymbol{\Phi}^{h,\tau*})^n, \qquad (84) $$

where $\Lambda_n^h$ satisfies $\Lambda_n^h = (U_n^h)^{-1}$ with

$$ U_{n+1}^h = U_n^h + \tau P_r (\boldsymbol{\Phi}^{h,\tau*})^{n+1} C^{h*} C^h (\boldsymbol{\Phi}^{h,\tau})^{n+1} P_r^*, \quad n \ge 0. \qquad (85) $$
*Proof.* We proceed by induction. We have at iteration 0

$$ \Pi_0^h = P_r^* \Lambda_0^h P_r. $$

Let us assume (84) is satisfied at iteration $n$. First, we directly have

$$ \Pi_{n+1-}^h = \Phi^{h,\tau} \Pi_n^h \Phi^{h,\tau*} = (\Phi^{h,\tau})^{n+1} P_r^* \Lambda_n^h P_r (\Phi^{h,\tau*})^{n+1}, $$

Then, using (78) and the expression above of $\Pi_{n+1-}^h$, we have

$$ \Pi_{n+1}^h = (\Phi^{h,\tau})^{n+1} P_r^* \left[ \Lambda_n^h - \tau \Lambda_n^h P_r (\Phi^{h,\tau*})^{n+1} C^{h*} [\mathrm{Id}_{\mathcal{Y}^h} + \tau C^h (\Phi^{h,\tau})^{n+1} \right. \\ \left. \times P_r^* \Lambda_n^h P_r (\Phi^{h,\tau*})^{n+1} C^{h*}]^{-1} C^h (\Phi^{h,\tau})^{n+1} P_r^* \Lambda_n^h \right] P_r (\Phi^{h,\tau})^{n+1}. $$

We therefore have

$$ \Pi_{n+1}^h = (\Phi^{h,\tau})^{n+1} P_r^* \Lambda_{n+1}^h P_r (\Phi^{h,\tau})^{n+1}, $$

as soon as

$$ \Lambda_{n+1}^h = \Lambda_n^h - \Lambda_n^h D_n^{h,*} (\mathrm{Id}_{\mathcal{Y}^h} + D_n^h \Lambda_n^h D_n^{h,*})^{-1} D_n^h \Lambda_n^h, $$

with $D_n^h = \sqrt{\tau} C^h (\Phi^{h,\tau})^{n+1} P_r^*$. By using Sherman-Morrison-Woodbury formula for operators – see for instance [24, Theorem 1.1] – we obtain

$$ (\Lambda_{n+1}^h)^{-1} = (\Lambda_n^h)^{-1} + D_n^{h,*} D_n^h, $$

which ends the proof since it is precisely (85). □

<figure><img src="image_3.png" /></figure>

### 4.3.2. Application to the source identification problem.
In this work, we discretize the wave equation using Lagrange finite element. Let $\theta^h \in \mathcal{P}^h$ the finite dimensional space corresponding to $\mathcal{P}$. We use a conservative mid-point scheme, that ultimately reads

$$ \left\{ \begin{array}{l} \displaystyle \frac{u_{n+1}^h - u_n^h}{\tau} = \frac{v_{n+1}^h + v_n^h}{2}, \\[1em] \displaystyle \frac{v_{n+1}^h - v_n^h}{\tau} - \Delta_0^h \frac{u_{n+1}^h + u_n^h}{2} = \sigma \left( \left( n + \frac{1}{2} \right) \tau \right) \theta^h, \end{array} \right. \qquad (86) $$

where $\Delta_0^h$ is the bounded linear operator in $\mathcal{V}^h \subset H^1(\Omega)$ associated with the conform finite element discretization of the unbounded operator $\Delta_0$ of $L^2(\Omega)$. Note that this system can be rewritten in a first order form by considering $z_n^h \in \mathcal{Z}^h = \mathcal{V}^h \times \mathcal{V}^h$ solution of

$$ z_{n+1}^h = \Phi^{h,\tau} z_n^h + B_{n+1}^{h,\tau} \theta^h, $$

where

$$ \Phi^{h,\tau} = \begin{pmatrix} \mathrm{Id}^h & -\frac{\tau}{2} \mathrm{Id}^h \\ -\frac{\tau}{2} \Delta_0^h & \mathrm{Id}^h \end{pmatrix}^{-1} \begin{pmatrix} \mathrm{Id}^h & \frac{\tau}{2} \mathrm{Id}^h \\ \frac{\tau}{2} \Delta_0^h & \mathrm{Id}^h \end{pmatrix}, $$

and

$$ B_{n+1}^{h,\tau} = \tau \begin{pmatrix} \mathrm{Id}^h & -\frac{\tau}{2} \mathrm{Id}^h \\ -\frac{\tau}{2} \Delta_0^h & \mathrm{Id}^h \end{pmatrix}^{-1} \begin{pmatrix} 0 \\ \sigma \left( \left( n + \frac{1}{2} \right) \tau \right) \end{pmatrix}. $$

Note that $\mathcal{Z}^h$ is equipped with the scalar product on $\mathcal{Z}$ that is the natural scalar product on $H_0^1(\Omega) \times L^2(\Omega)$.
We now apply this result to our source identification problem. The joint state-parameter discrete dynamics in $Z^h = Z^h \times P^h$ reads

$$ \underbrace{\begin{pmatrix} z_{n+1}^h \\ \theta_{n+1}^h \end{pmatrix}}_{z_{n+1}^h} = \underbrace{\begin{pmatrix} \Phi^{h,\tau} & B_{n+1}^{h,\tau} \\ 0 & \mathrm{Id}^h \end{pmatrix}}_{\Phi^{h,\tau} z_n^h} \begin{pmatrix} z_n^h \\ \theta_n^h \end{pmatrix}, \quad \text{with} \quad \underbrace{\begin{pmatrix} z_0^h \\ \theta_0^h \end{pmatrix}}_{z_0^h} = \underbrace{\begin{pmatrix} 0 \\ \theta^h \end{pmatrix}}_{\zeta}. \qquad (87) $$

From Theorem 4.12, we know that

$$ \left\{ \begin{array}{l} \boldsymbol{\Pi}_n^h = (\boldsymbol{\Phi}^{h,\tau})^n P_r^* \Lambda_n^h P_r (\boldsymbol{\Phi}^{h,\tau*})^n \quad \text{and} \quad \boldsymbol{\Pi}_{n+1-}^h = (\boldsymbol{\Phi}^{h,\tau})^{n+1} P_r^* \Lambda_n^h P_r (\boldsymbol{\Phi}^{h,\tau*})^{n+1}, \\ \boldsymbol{\Pi}_0 = P_\theta^*(U_0^h)^{-1}P_\theta, \end{array} \right. \qquad (88) $$

where $U_n^h = (\Lambda_n^h)^{-1}$ is solution of

$$ U_{n+1}^h = U_n^h + \tau P_\theta (\boldsymbol{\Phi}^{h,\tau*})^{n+1} C^{h*} C^h (\boldsymbol{\Phi}^{h,\tau})^{n+1} P_\theta^*, \quad n \ge 0. \qquad (89) $$

Moreover, we have

$$ \begin{pmatrix} z_n^h \\ \theta_n^h \end{pmatrix} = (\boldsymbol{\Phi}^{h,\tau})^n P_\theta^* \theta_0^h = (\boldsymbol{\Phi}^{h,\tau})^n \begin{pmatrix} 0 \\ \theta_0^h \end{pmatrix} = \begin{pmatrix} L_n^h \\ \mathrm{Id}^h \end{pmatrix} \theta_0^h, $$

where the operator $(L_n^h)_{n \in \mathbb{N}}$ is defined by

$$ L_n^h \theta = z_{n|\theta}^h \quad \text{with} \quad \begin{cases} z_{n+1|\theta}^h = \Phi^{h,\tau} z_{n|\theta}^{h,\tau} + B_{n+1}^{h,\tau} \theta, & n \in \mathbb{N}, \\ z_{0|\theta}^h = 0. & \end{cases} \qquad (90) $$

Therefore, we can simplify (89) into

$$ U_{n+1}^h = U_n^h + \tau L_{n+1}^{h*} C^{h*} C^h L_{n+1}^h, \quad n \ge 0, \qquad (91) $$

and (88) into

$$ \boldsymbol{\Pi}_n^h = \begin{pmatrix} L_n^h \\ \mathrm{Id} \end{pmatrix} \Lambda_n^h (L_n^{h*} \quad \mathrm{Id}), \quad \text{and} \quad \boldsymbol{\Pi}_{n+1-}^h = \begin{pmatrix} L_{n+1}^h \\ \mathrm{Id}^h \end{pmatrix} \Lambda_n^h (L_{n+1}^{h*} \quad \mathrm{Id}^h). \qquad (92) $$

Finally, we obtain the equivalent sequential estimator.

**Theorem 4.13.** For all $n \in \mathbb{N}$, let $\bar{\theta}_{n|n}^h$ be the minimizer of (82). The sequential estimator $(\hat{\theta}_n^h)_{n \in \mathbb{N}} = (\bar{\theta}_{n|n}^h)_{n \in \mathbb{N}}$ is solution from $(\hat{z}_0^h, \hat{\theta}_0^h)$ of the recursive dynamics

$$ \left\{ \begin{array}{l} \hat{z}_{n+1-}^h = \Phi^{h,\tau} z_n^h + B_{n+1}^{h,\tau} \hat{\theta}_n^h, \\ \hat{\theta}_{n+1}^h = \hat{\theta}_n^h + \tau (U_{n+1}^h)^{-1} L_{n+1}^{h*} C^{h*} (y_{n+1}^{\delta,h} - C^h \hat{z}_{n+1-}^h), \\ \hat{z}_{n+1}^h = \hat{z}_{n+1-}^h + L_{n+1}^h (\hat{\theta}_{n+1}^h - \hat{\theta}_n^h). \end{array} \right. \qquad (93) $$

We already know that (93) is consistent with (75). A direct convergence study of this splitting scheme could be intricate. Here, by relying on the fundamental relation $\hat{\theta}_n^h = \bar{\theta}_{n|n}^h$, we can justify the stability from the minimization formulation. This could lead – this is a perspective of this work – to the proof of convergence of solutions to the time scheme (93) towards solutions to (75).
5. **Numerical illustrations.** We consider a domain $\Omega$, illustrated in Figure 1, of width 3.3 and of height 2). We assume having access to observations in the observation subdomain $\omega$, hence $T_{gcc} \le 3$. We consider the following wave equation

$$ egin{cases} 
\partial_{tt}\check{u}(x,t) - \Delta\check{u}(x,t) = \sigma(t)\check{\theta}(x), & (x,t) \in \Omega \times (0,T), \\ 
\check{u}(x,t) = 0, & (x,t) \in \partial\Omega \times (0,T), \\ 
\check{u}(x,0) = \check{u}_0(x), & x \in \Omega, \\ 
\partial_t\check{u}(x,0) = 0, & x \in \Omega, 
\end{cases} 
\quad (94) $$

where

$$ \sigma(t) = \frac{1}{2} + \frac{1}{2}e^{-0.5t}, $$

$\check{u}_0$ is solution of

$$ \left\{ \begin{array}{ll} -\Delta \check{u}_0(x) = 10, & x \in \Omega, \\ \check{u}_0(x) = 0, & x \in \partial \Omega, \end{array} \right. \quad (95) $$

and, finally, $\check{\theta}$ is solution of

$$ \left\{ \begin{array}{ll} -\Delta \check{\theta}(x) = 0 & x \in \Omega \setminus \Omega_{\text{hole}}, \\ -\Delta \check{\theta}(x) = 10, & x \in \Omega_{\text{hole}}, \\ \check{\theta}(x) = 0, & x \in \partial \Omega, \end{array} \right. \quad (96) $$

with $\Omega_{\text{hole}}$ pictured in Figure 1.

<figure><img src="image_4.png" /><figcaption>FIGURE 1. Representation of the domain $\Omega$ and the subdomain $\omega$</figcaption></figure>

The initial conditions and the target source term defined respectively by (95) and (96) are illustrated in Figure 2. Note that the choice of $\sigma$ is the same as that of Remark 3.5 with $\beta = 1$ and $\alpha = 0.5$.

<figure><img src="image_5.png" /><figcaption>FIGURE 2. Representation of $\check{u}_0$ (left) and $\check{\theta}$ (right)</figcaption></figure>
We decompose the target source term into

$$ \tilde{\theta}^h := \sum_{i=0}^{N_{\text{mod}}} \tilde{\theta}_i^h \phi_i^h, \qquad (97) $$

where $(\phi_i^h)_{i \in [0, N_{\text{mod}}]}$ are the eigenvectors of $\Delta_0^h$. In the following numerical illustrations, we choose $N_{\text{mod}} = 27$ so that the relative error between $\tilde{\theta}$ and $\tilde{\theta}^h$ is less than 10%, see Figure 3.

<figure><img src="image_6.png" /><figcaption>FIGURE 3. Source term $\tilde{\theta}$ (left) and its decomposition $\tilde{\theta}^h$ on the Laplacian basis with 27 modes (right)</figcaption></figure>

In the following, we compare the two methods presented in this work. We refer to the method where the noisy measurements belong to $H^1(\omega)$ as the $H^1$-method. Similarly, the method presented in Section 3.2 is called the $L^2$-method, since the noisy measurements belong to $L^2(\omega)$.

The refined mesh shown in Figure 4 (left) was used for the simulations. However, as this mesh is very fine and difficult to recognize, we also show a coarser mesh in Figure 4 (right) for better visualization.

<figure><img src="image_7.png" /><figcaption>FIGURE 4. Representation of the mesh of $\Omega$ with 56193 degrees of freedom (left) and with 14169 degrees of freedom</figcaption></figure>

5.1. **Numerical illustrations for noise-free data.** First, to generate the measurements, we implement the direct problem on a time window from 0 to 1 with a time step of $\tau = 0.01$ and with $\mathbb{P}_1$ finite elements and where the space function $\tilde{\theta}^h$ to reconstruct in the source term is defined by (97). We give an illustration of a measurement in Figure 5.

We study the inverse problem for the $H^1$-method in a noise-free case. We implement the observers scheme (93) (where $C$ is defined by (8)) in FreeFem++ [29] over a time window $[0, 1]$ with a time step of 0.01 and with $\mathbb{P}_1$ finite elements. Moreover,
<figure><img src="image_8.png" /><figcaption>FIGURE 5. Solution $\tilde{u}_n^h$ of the direct problem (left) and its corresponding measurements $(\tilde{u}_n^h)|_\omega$ (right) at time $t^n = 0.1$</figcaption></figure>

we start from a null initial guess $\hat{\theta}_0^h = 0$ but with the correct initial conditions $\tilde{u}_0$. Furthermore, we remove the Tikhonov regularization term since we suppose that we have the perfect measurements at our disposal. The same simulation for the $L^2$-method is performed. In other words, the time scheme (93) is implemented, but this time the observation operator $C$ is defined by (9).

Figure 6 shows the relative error between the target source term decomposed on the Laplacian modes and the observer defined by

$$ \|\tilde{\theta}^h - \hat{\theta}_n^h\|_{L^2(\Omega)} / \|\tilde{\theta}^h\|_{L^2(\Omega)} $$

over the time.

<figure><img src="image_9.png" /><figcaption>FIGURE 6. Evolution of the relative error without Tikhonov term for $T = 0.1$ (left) and $T = 1$ (right)</figcaption></figure>

For the $H^1$-method, we can see that the relative error is approximately $10^{-7}$ from $t^n = 0.1$. At the final time $t^n = 1$, the relative error is about $10^{-10}$. For the $L^2$-method, the results are very similar. We can see that both methods converge over time with the same final error level. However, in the short time range (Figure 6 left), the $H^1$-method is more efficient than the $L^2$-method. Note that in both cases the numerical convergence occurs in a time window smaller than the one given by $T_{gcc}$. This may be due to the regularity of the target parameter.
5.2. Numerical illustrations with noisy measurements. We now consider noisy measurements belonging to L²(ω) and solve the inverse problem with Y = L²(ω). We corrupt the synthetic data (y<sup>δ,h</sup>)<sub>0≤n≤N</sub> of sampling rate T/N with a random artificial noise χ<sup>h</sup><sub>n</sub> ∈ L²(ω). To do this, we perturb the measurements at each interpolation point with a random value ξ that follows the Gaussian law N(0, 1) and scale with the noise level δ, i.e., at each interpolation point we have

$$ (y_n^{\delta,h})_i = (\tilde{y}_n^h)_i + \delta\xi_i / |\omega|, \qquad (98) $$

where the scaling by |ω| ensures that the noise is normalized. We represent in Figure 7 an example of a noisy measurement and the associated measurement noise.

<figure><img src="image_10.png" /><figcaption>FIGURE 7. Noisy measurements y<sup>δ,h</sup> (left) and the corresponding noise χ<sup>h</sup> = y<sup>δ,h</sup> - ȳ<sup>h</sup> (right) at time t<sup>n</sup> = 0.1</figcaption></figure>

We solve the inverse problem with the time scheme (93) with a time step of τ = 0.01 and δ = 1. Figure 8 shows the relative error, namely

$$ \frac{\|\tilde{\theta}^h - \hat{\theta}_n^h\|_{L^2(\Omega)}}{\|\tilde{\theta}^h\|_{L^2(\Omega)}} $$

between the target source term decomposed on the basis and the observer at the final time t<sup>n</sup> = 1 as a function of the noise level δ in (98) for different choice of ϵ.

<figure><img src="image_11.png" /><figcaption>FIGURE 8. Representation of the relative error according the values of δ in loglog scale for different values of ϵ</figcaption></figure>
First, we note that the larger the value of $\epsilon$, the lower the weight of the regularization. In particular, we can see that the regularization with $\epsilon = 10^{-8}$ (in red in Figure 8) offers better results. In particular, for the different noise levels considered – from $\delta = 0.01\%$ up to $\delta = 10\%$ in (98) – we reach a reconstruction of the order of the noise level.

**6. Conclusion.** We have proposed a strategy to estimate a source term in a wave equation using observers from internal field measurements. We divided the study into two parts depending on whether we have access to regular or less regular measurements. The regularity of the measurements leads to substantial changes in the definition of the inverse problem through the definition of the observation operator. The proposed approach follows three steps. First, we prove identifiability results based on observability inequalities. These identifiability results give us information about the degree of the ill-posedness of the inverse problem. Then, we propose the minimization problem associated with the reconstruction of the source term with a well-adjusted Tikhonov regularization. This inverse problem is solved with a sequential strategy, which turns out to be equivalent to the minimization of the cost functional. We have assumed that we not only have internal observations about the system, but also perfect knowledge about the initial conditions, which can be a strong assumption in practice. One possible perspective would be to assume that there are uncertainties in the initial conditions. In this case, it would be necessary to further analyze the joint state-source estimation originally proposed in [48, 44].

# Appendix A. Appendix.

## A.1. Proof of Proposition 3.1.

*Proof.* The proof extended the one of [19]. In this proof, we write $a \lesssim b$ if there exists $c > 0$ independent of $T$ such that $a \le cb$.

Since the geometric control condition of [7] holds, we have the classical observability inequality: there exists $\tilde{c} > 0$, such that for all $T \ge T_{gcc}$,

$$ \int_0^T \int_{\omega_{gcc}} |\partial_t u(x, t)|^2 dx dt \ge \tilde{c} \left( \|u_0\|_{H_0^1(\Omega)}^2 + \|v_0\|_{L^2(\Omega)}^2 \right). \quad (99) $$

By definition of $\omega_{gcc}$ and $\omega$ there exists a cutoff function $\psi \in C^2(\bar{\Omega})$ satisfying

$$ \psi(x) = \begin{cases} 0, & \text{if } x \in \Omega \setminus \omega, \\ 1, & \text{if } x \in \omega_{gcc}, \end{cases} $$

and $0 \le \psi(x) \le 1$ for every $x \in \bar{\Omega}$. Such function can be obtained by first, smoothing of the indicator function of the domain

$$ \omega^* = \bigcup_{x \in \omega_{gcc}} B(x; \eta^*), $$

for $\eta^*$ sufficiently small, second restricting the obtained function to $\omega$. For any $\tilde{T} > 0$, we introduce $\varphi(t) = t^2(\tilde{T} - t)^2$. Since $\partial_{tt}u - \Delta u = 0$, we have, by integrating by parts,

$$ \begin{aligned} 0 &= \int_0^{\tilde{T}} \int_{\omega} \varphi \psi (\partial_{tt} u - \Delta u) u \, dx \, dt \\ &= \int_0^{\tilde{T}} \int_{\omega} \ddot{\varphi} \psi \frac{|u|^2}{2} \, dx \, dt - \int_0^{\tilde{T}} \int_{\omega} \varphi \psi |\partial_t u|^2 \, dx \, dt \end{aligned} $$
$$ + \int_{0}^{\bar{T}} \int_{\partial_{\omega}} \varphi \frac{\partial \psi}{\partial n} \frac{|u|^2}{2} \, d\sigma \, dt - \int_{0}^{\bar{T}} \int_{\omega} \varphi \Delta \psi \frac{|u|^2}{2} \, dx \, dt \\ + \int_{0}^{\bar{T}} \int_{\omega} \varphi \psi |\nabla u|^2 \, dx \, dt. $$

Reordering we get,

$$ \int_{0}^{\bar{T}} \int_{\omega} \varphi \psi |\partial_t u|^2 \, dx \, dt = \int_{0}^{\bar{T}} \int_{\omega} \varphi \psi |\nabla u|^2 \, dx \, dt \\ + \int_{0}^{\bar{T}} \int_{\omega} (\ddot{\varphi}\psi - \varphi \Delta \psi) \frac{|u|^2}{2} \, dx \, dt + \int_{0}^{\bar{T}} \int_{\partial \omega} \varphi \frac{\partial \psi}{\partial n} \frac{|u|^2}{2} \, d\sigma \, dt. \quad (100) $$

We recall that for the system (1) with $\theta = 0$, we have classically the conservation energy property namely

$$ \mathcal{E}(t) := \frac{1}{2} \|u_0\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \|v_0\|_{L^2(\Omega)}^2, \quad \mathcal{E}(t) = \mathcal{E}(0). \quad (101) $$

Now, for some $\varepsilon > 0$ we use (99) at the shifted initial time $\varepsilon$ and for the observation time $T = T_{gcc} + \varepsilon$, we find, using (101),

$$ \mathcal{E}(0) = \mathcal{E}(\varepsilon) \lesssim \int_{\varepsilon}^{T_{gcc}+\varepsilon} \int_{\omega_{gcc}} |\partial_t u|^2 \, dx \, dt. $$

Setting $\tilde{T} = T_0 := T_{gcc} + 2\varepsilon$ we use the definition of $\psi$ and $\varphi$ to obtain

$$ \mathcal{E}(0) \lesssim \int_{\varepsilon}^{T_{gcc}+\varepsilon} \int_{\omega_{gcc}} [\varphi\psi |\partial_t u|^2 + |\nabla u|^2] \, dx \, dt \\ \lesssim \int_{0}^{T_0} \int_{\omega} [\varphi\psi |\partial_t u|^2 + |\nabla u|^2] \, dx \, dt. \quad (102) $$

Using (100) and the trace inequality, we get

$$ \int_{0}^{T_0} \int_{\omega} \varphi \psi |\partial_t u|^2 \, dx \, dt \lesssim \int_{0}^{T_0} \int_{\omega} |\nabla u|^2 \, dx \, dt + \int_{0}^{T_0} \int_{\omega} |u|^2 \, dx \, dt. $$

Finally, using the estimate above in (102), we find

$$ \mathcal{E}(0) \lesssim \int_{0}^{T_0} \int_{\omega} [|\nabla u|^2 + |u|^2] \, dx \, dt, \quad (103) $$

which is (13), say, for $T \in [T_0, 2T_0)$. Now, let $T > 2T_0$ and $N \in \mathbb{N}$ such that $T \in [NT_0; (N+1)T_0)$. From (103) and the energy preserving property, we deduce that, for all $t \ge 0$,

$$ \int_{t}^{T+t} \|u\|_{H^1(\omega)}^2 \, dt \ge \tilde{c} \mathcal{E}(t), \quad (104) $$

for some $\tilde{c} > 0$ independent of $T$. We have

$$ \int_{0}^{T} \|u\|_{H^1(\omega)}^{2} \, dt \ge \sum_{n=0}^{N-1} \int_{nT_0}^{(n+1)T_0} \|u\|_{H^1(\omega)}^{2} \, dt. $$

Using (104) with $t = nT_0$, we get, by energy conservation (101),

$$ \int_{0}^{T} \|u\|_{H^1(\omega)}^{2} \, dt \ge \tilde{c} \sum_{n=0}^{N-1} \mathcal{E}(nT_0) = \tilde{c} N \mathcal{E}(0) \ge \tilde{c} \frac{T-T_0}{T_0} \mathcal{E}(0), \quad (105) $$
which gives (13).

### A.2. Proof of Lemma 3.14.

*Proof.* Let $(\mu_i, e_i)$ be the orthonormal basis for the $L^2$ scalar product such that the Laplacian eigenvalues $\mu_i > 0$ satisfies $\mu_{i+1} > \mu_i$, and $\mu_i \to +\infty$. We express the $L^2$-norm of $\theta$ in the orthonormal basis.

$$ \|\theta\|_{L^2(\Omega)}^2 = \sum_{i=0}^{+\infty} |(e_i, \theta)|^2 = \sum_{i=0}^{+\infty} |(e_i, \theta)|^2 \mu_i^{-\frac{2}{3}} \mu_i^{\frac{2}{3}}. $$

Using Hölder inequality we find

$$ \begin{aligned} \|\theta\|_{L^2(\Omega)}^2 &\le \left( \sum_{i=0}^{+\infty} |(e_i, \theta)|^2 \mu_i^{-2} \right)^{\frac{1}{3}} \left( \sum_{i=0}^{+\infty} |(e_i, \theta)|^2 \mu_i \right)^{\frac{2}{3}} \\ &= \left( \|\Delta_0^{-1} \theta\|_{L^2(\Omega)}^2 \right)^{\frac{1}{3}} \left( \|\theta\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}}. \end{aligned} $$

### A.3. Proof of Theorem 3.11.

*Proof.* One can show, as for Theorem 3.4, that we have the inequality

$$ \|u\|_{L^2(0,T;L^2(\omega))} \ge c(T) \|C\bar{L}\theta\|_{L^2(0,T;L^2(\omega))}. \quad (106) $$

As in the proof of Theorem 3.2, we introduce $g \in H_0^1(\Omega) \cap D(-\Delta_0)$ the Riesz representation of $\theta$ (seen as an element of $H^{-1}(\Omega)$) given by $g = -\Delta_0^{-1}\theta$, and $u_g$ solution of the dynamics (15). The difference $\tilde{u}_g = u_g - g$ satisfies the system (17) in the same way. We apply the observability condition of Proposition 3.10 on this system, and we obtain

$$ \|\tilde{u}_g\|_{L^2(0,T;L^2(\omega))}^2 = \|u_g - g\|_{L^2(0,T;L^2(\omega))}^2 \ge c_{\text{obs}}(T) \|g\|_{L^2(\Omega)}^2. \quad (107) $$

Now we want to discard the variable $g$ in the left-hand of (107). That is why, we develop the following scalar product

$$ \int_0^T \|u_g - g\|_{L^2(\omega)}^2 dt = \int_0^T \|u_g\|_{L^2(\omega)}^2 dt + T \|g\|_{L^2(\omega)}^2 - 2 \int_0^T (u_g, g)_{L^2(\omega)} dt. \quad (108) $$

Applying $\Delta_0^{-1}$ to the first equation of (15), then integrate in time and multiplying the resulting equation by the scalar product with $g$ gives

$$ \int_0^T (u_g, g)_{L^2(\omega)} dt = (\Delta_0^{-1} \partial_t u_g(T), g)_{L^2(\omega)} + T \|g\|_{L^2(\omega)}^2. $$

Injecting this result in (108) and using Cauchy-Schwarz and Poincaré inequalities, we obtain

$$ \begin{aligned} \int_0^T \|u_g - g\|_{L^2(\omega)}^2 dt \le \int_0^T \|u_g\|_{L^2(\omega)}^2 dt - T \|g\|_{L^2(\omega)}^2 \\ + 2c_p \|\nabla \Delta_0^{-1} \partial_t u_g(T)\|_{L^2(\Omega)} \|g\|_{L^2(\omega)}^2. \end{aligned} \quad (109) $$

We expect to bound $\|\nabla \Delta_0^{-1} \partial_t u_g(T)\|_{L^2(\Omega)}$ by $\|g\|_{L^2(\Omega)}$. Therefore, we apply $\Delta_0^{-1}$ to the first equation of (15). Integrating in time, the scalar product with $-\partial_t u_g$ leads to

$$ - \int_{\Omega} (\Delta_0^{-1} \partial_t u_g) \partial_t u_g dx + \int_{\Omega} u_g \partial_t u_g dx = \int_{\Omega} g \partial_t u_g dx. $$
Inserting $\Delta_0\Delta_0^{-1} = \text{Id}$ in the first term we get

$$ -\int_{\Omega} (\Delta_0^{-1} \partial_{tt} u_g) \Delta_0 \Delta_0^{-1} \partial_t u_g \, dx + \frac{1}{2} \frac{d}{dt} \int_{\Omega} u_g^2 \, dx = \int_{\Omega} g \partial_t u_g \, dx. $$

Integrating by parts, we obtain

$$ \frac{1}{2} \frac{d}{dt} \int_{\Omega} |\nabla \Delta_0^{-1} \partial_t u_g|^2 \, dx + \frac{1}{2} \frac{d}{dt} \int_{\Omega} u_g^2 \, dx = \int_{\Omega} g \partial_t u_g \, dx. $$

Integrating in time and using Cauchy-Schwarz and Young inequalities, we get

$$ \frac{1}{2} \|\nabla \Delta_0^{-1} \partial_t u_g(T)\|_{L^2(\Omega)}^2 + \frac{1}{2} \|u_g(T)\|_{L^2(\Omega)}^2 = \int_{\Omega} g u_g(T) \, dx \\ \leq \frac{1}{2} \|g\|_{L^2(\Omega)}^2 + \frac{1}{2} \|u_g(T)\|_{L^2(\Omega)}^2, $$

thus,

$$ \|\nabla \Delta_0^{-1} \partial_t u_g(T)\|_{L^2(\Omega)}^2 \leq \|g\|_{L^2(\Omega)}^2. $$

Injecting this result in (109), we have

$$ \int_{0}^{T} \|u_g - g\|_{L^2(\omega)}^2 \, dt \leq \int_{0}^{T} \|u_g\|_{L^2(\omega)}^2 \, dt - T \|g\|_{L^2(\omega)}^2 + 2c_p \|g\|_{L^2(\Omega)} \|g\|_{L^2(\omega)}^2. $$

Using Young inequality, we finally have

$$ \int_{0}^{T} \|u_g - g\|_{L^2(\omega)}^2 \, dt \leq \int_{0}^{T} \|u_g\|_{L^2(\omega)}^2 \, dt - T \|g\|_{L^2(\omega)}^2 \\ + \epsilon c_p \|g\|_{L^2(\Omega)}^2 + \frac{c_p}{\epsilon} \|g\|_{L^2(\omega)}^2. \quad (110) $$

Gathering (107) and (110), we get

$$ \int_{0}^{T} \|u_g\|_{L^2(\omega)}^2 \, dt \geq \left(T - \frac{c_p}{\epsilon}\right) \|g\|_{L^2(\omega)}^2 + (c_{\text{obs}}(T) - \epsilon c_p) \|g\|_{L^2(\Omega)}^2. $$

We choose $\epsilon = \frac{c_{\text{obs}}(T)}{2c_p}$, then we have

$$ \int_{0}^{T} \|u_g\|_{L^2(\omega)}^2 \, dt \geq \left( T - \frac{2c_p^2}{c_{\text{obs}}(T)} \right) \|g\|_{L^2(\omega)}^2 + \frac{c_{\text{obs}}(T)}{2} \|g\|_{L^2(\Omega)}^2. $$

For $T > \frac{2c_p^2}{c_{\text{obs}}(T)}$, and setting $-\Delta_0 g = \theta$, we obtain

$$ \|C\bar{L}\theta\|_{L^2((0,T);L^2(\omega))} = \int_{0}^{T} \|u_g\|_{L^2(\omega)}^2 \, dt \geq \frac{c_{\text{obs}}(T)}{2} \|\Delta_0^{-1}\theta\|_{L^2(\Omega)}^2. \quad (111) $$

Gathering (106) and (111), we conclude the proof. □

### A.4. Proof of Theorem 3.15.

*Proof.* In this proof, we write $a \lesssim b$ if there exists $c > 0$ independent of $T, \epsilon$ and $M$ such that $a \leq cb$.

We use the interpolation inequality of Lemma 3.14 to obtain

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \leq \left( \|\bar{\theta}_T - \check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \left( \|\Delta_0^{-1}(\bar{\theta}_T - \check{\theta})\|_{L^2(\Omega)}^2 \right)^{\frac{1}{3}}. $$
Using the identifiability result given by Theorem 3.11, we have

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \le (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( \|\bar{\theta}_T\|_{H_0^1(\Omega)}^2 + \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \left( \int_0^T \|u_{|\bar{\theta}_T} - \check{u}\|_{L^2(\omega)}^2 dt \right)^{\frac{1}{3}}. $$

Since we consider noise-free observations, we have $y^\delta = \check{u}_{|\omega}$. Therefore, we obtain

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( \|\bar{\theta}_T\|_{H_0^1(\Omega)}^2 + \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \left( \int_0^T \|u_{|\bar{\theta}_T} - y^\delta\|_{L^2(\omega)}^2 dt \right)^{\frac{1}{3}}. $$

Using the definition of the cost functional (40), we obtain

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( 2\epsilon^{-1} \mathcal{J}_T(\bar{\theta}_T) + \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \left( 2\mathcal{J}_T(\bar{\theta}_T) \right)^{\frac{1}{3}}. $$

We can bound the cost functional as follows

$$ \mathcal{J}_T(\bar{\theta}_T) \le \mathcal{J}_T(\check{\theta}) \le \frac{\epsilon}{2} \|\check{\theta}\|_{H_0^1(\Omega)}^2. $$

Finally, we have

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \left( \epsilon \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{1}{3}}. $$

Furthermore, under Assumption 3.13, we have that there exists $c > 0$ independent of $T$ such that

$$ c_{\text{idf}}(T) \ge cT. \quad \square $$

### A.5. Proof of Theorem 3.16.

*Proof.* In this proof, we write $a \lesssim b$ if there exists $c > 0$ independent of $T, \epsilon, \delta$ and $M$ such that $a \le cb$.

On one hand, using the estimate on the amplitude noise (38), we intend to bound the cost functional.

$$ \mathcal{J}_T(\check{\theta}) \le \frac{\epsilon}{2} \|\check{\theta}\|_{H_0^1(\Omega)}^2 + \frac{1}{2} \delta^2 T. \quad (112) $$

On the other hand, as in the proof of Theorem 3.15, one can show that (41) holds. Inserting $y^\delta - y^\delta = 0$ into (41) to reintroduce the cost functional, we get

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( \epsilon^{-1} \mathcal{J}_T(\check{\theta}) + \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{2}{3}} \times \left( \int_0^T \|u_{|\bar{\theta}_T} - y^\delta\|_{L^2(\omega)}^2 dt + \int_0^T \|\check{u} - y^\delta\|_{L^2(\omega)}^2 dt \right)^{\frac{1}{3}}. $$

Again, using again (29) and the cost functional bound (112), we have

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( \|\check{\theta}\|_{H_0^1(\Omega)}^2 + \epsilon^{-1} \delta^2 T \right)^{\frac{2}{3}} \left( \delta^2 T + \epsilon \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right)^{\frac{1}{3}}, \quad (113) $$

which corresponds to

$$ \|\bar{\theta}_T - \check{\theta}\|_{L^2(\Omega)}^2 \lesssim \epsilon^{-\frac{2}{3}} (c_{\text{idf}}(T))^{-\frac{1}{3}} \left( \delta^2 T + \epsilon \|\check{\theta}\|_{H_0^1(\Omega)}^2 \right), $$

hence we get the desired result. Furthermore, under Assumption 3.13, we have that there exists $c > 0$ independent of $T$ such that

$$ c_{\text{idf}}(T) \ge cT. $$
<figure><img src="image_12.png" /></figure>

REFERENCES

[1] M. A. Aguilo, W. Aquino, J.C. Brigham, and M. Fatemi. An inverse problem approach for elasticity imaging through vibroacoustics. *IEEE Transactions on Medical Imaging*, 29(4):1012-1021, 2010.

[2] R. A. Albanese, H. T. Banks, and J. K. Raye. Nondestructive evaluation of materials using pulsed microwave interrogating signals and acoustic wave induced reflections. *Inverse Problems*, 18(6):1935, nov 2002.

[3] C. Alves, A. L. Silvestre, T. Takahashi, and M. Tucnak. Solving inverse source problems using observability. applications to the euler-bernoulli plate equation. *SIAM Journal on Control and Optimization*, 48(3):1632-1659, 2009.

[4] D. Auroux and J. Blum. A nudging-based data assimilation method: the back and forth nudging (bfn) algorithm. *Nonlinear Processes in Geophysics*, 15(2):305-319, 2008.

[5] M. Aussal and P. Moireau. Kernel representation of kalman observer and associated h-matrix based discretization. *ESAIM: Control, Optimisation and Calculus of Variations*, 28:78, 2022.

[6] G. Bal and S. Imperiale. Displacement reconstructions in ultrasound elastography. *SIAM Journal on Imaging Sciences*, 8(2):1070-1089, 2015.

[7] C. Bardos, G. Lebeau, and J. Rauch. Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problemes hyperboliques. *Rend. Sem. Mat. Univ. Politec. Torino*, (Special Issue), 1988.

[8] L. Baudouin, M. De Buhan, and S. Ervedoza. Global carleman estimates for waves and applications. *Communications in Partial Differential Equations*, 38(5):823-859, 2013.

[9] L. Baudouin, M. De Buhan, and S. Ervedoza. Convergent algorithm based on carleman estimates for the recovery of a potential in the wave equation. *SIAM Journal on Numerical Analysis*, 55(4):1578-1613, 2017.

[10] L. Baudouin, M. De Buhan, S. Ervedoza, and A. Osses. Carleman-based reconstruction algorithm for waves. *SIAM Journal on Numerical Analysis*, 59(2):998-1039, 2021.

[11] R. Bellman. Dynamic programming. *Science*, 153(3731):34-37, 1966.

[12] A. Bensoussan. *Filtrage optimal des systèmes linéaires*, volume 3. Dunod, 1971.

[13] A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter. *Representation and control of infinite dimensional systems*. Springer Science & Business Media, 2007.

[14] Maîtine Bergounioux, Xavier Bonnefond, Thomas Haberkorn, and Yannick Privat. An optimal control problem in photoacoustic tomography. *Mathematical Models and Methods in Applied Sciences*, 24(12):2525-2548, 2014.

[15] J. Blum, F.-X. Le Dimet, and I. M. Navon. Data assimilation for geophysical fluids. In *Handbook of numerical analysis*, volume 14, pages 385-441. Elsevier, 2009.

[16] M. Bonnet and W. Aquino. Three-dimensional transient elastodynamic inversion using an error in constitutive relation functional. *Inverse Problems*, 31:035010, 2015.

[17] E. Burman, A. Feizmohammadi, A. Münch, and L. Oksanen. Space time stabilized finite element methods for a unique continuation problem subject to the wave equation. *ESAIM: Mathematical Modelling and Numerical Analysis*, 55:S969-S991, 2021.

[18] N. Burq and P. Gérard. Stabilization of wave equations on the torus with rough dampings. *Pure and Applied Analysis*, 2(3):627-658, 2020.

[19] D. Chapelle, N. Cîndea, M. De Buhan, and P. Moireau. Exponential convergence of an observer based on partial field measurements for the wave equation. *Mathematical Problems in Engineering*, 2012, 2012.

[20] N. Cîndea and A. Münch. Inverse problems for linear hyperbolic equations using mixed formulations. *Inverse Problems*, 31(7):075001, 2015.

[21] J.-M. Coron. *Control and Nonlinearity*. Number 136. American Mathematical Soc., 2007.

[22] R. Curtain and A. J. Pritchard. The infinite-dimensional Riccati equation for systems defined by evolution operators. *SIAM Journal on Control and Optimization*, 14(5):951-983, 1976.

[23] R. F. Curtain. A survey of infinite-dimensional filtering. *SIAM Review. A Publication of the Society for Industrial and Applied Mathematics*, 17:395-411, 1975.

[24] C. Y. Deng. A generalization of the Sherman-Morrison-Woodbury formula. *Applied Mathematics Letters*, 24(9):1561-1564, 2011.

[25] H. W. Engl, M. Hanke, and A. Neubauer. *Regularization of inverse problems*, volume 375 of *Mathematics and its Applications*. Kluwer Academic Publishers Group, Dordrecht, 1996.
[26] B. Gebauer and O. Scherzer. Impedance-acoustic tomography. *SIAM Journal on Applied Mathematics*, 69(2):565-576, 2008.

[27] G. Gripenberg, S. O. Londen, and O. Staffans. *Volterra Integral and Functional Equations*. Cambridge University Press, 1990.

[28] G. Haine. An observer-based approach for thermoacoustic tomography. *The 21st International Symposium on Mathematical Theory of Networks and Systems*, 2014.

[29] F. Hecht. New development in freefem++. *Journal of numerical mathematics*, 20(3-4):251-266, 2012.

[30] E. Humbert, Y. Privat, and E. Trélat. Observability properties of the homogeneous wave equation on a closed manifold. *Communications in Partial Differential Equations*, (9):749-772, 2019.

[31] A. Imperiale, D. Chapelle, and P. Moireau. Sequential data assimilation for mechanical systems with complex image data: application to tagged-MRI in cardiac mechanics. *Advanced Modeling and Simulation in Engineering Sciences*, 8, January 2021.

[32] S. Imperiale, P. Moireau, and A. Tonnoir. Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains. *ESAIM: Control, Optimisation and Calculus of Variations*, 26:45, 2020.

[33] D. Jellali. An inverse problem for the acoustic wave equation with finite sets of boundary data. *Journal of Inverse & Ill-Posed Problems*, 14(7), 2006.

[34] V. Komornik and M. Yamamoto. Upper and lower estimates in determining point sources in a wave equation. *Inverse Problems*, 18(2):319, 2002.

[35] V. Komornik and M. Yamamoto. Estimation of point sources and applications to inverse problems. *Inverse Problems*, 21(6):2051, 2005.

[36] A. J. Krener. A lyapunov theory of nonlinear observers, 1999.

[37] C. Laurent and M. Léautaud. Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves. *Journal of the European Mathematical Society*, 21(4):957-1069, 2018.

[38] F.-X. Le Dimet and O. Talagrand. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. *Tellus A: Dynamic Meteorology and Oceanography*, 38(2):97-110, 1986.

[39] J. M. Lewis and J. C. Derber. The use of adjoint equations to solve a variational adjustment problem with advective constraints. *Tellus A*, 37(4):309-322, 1985.

[40] 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. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.

[41] K Liu, M Yamamoto, and X Zhang. Observability inequalities by internal observation and their applications. *Journal of optimization theory and applications*, 116:621-645, 2003.

[42] D. Luenberger. An introduction to observers. *IEEE Transactions on automatic control*, 16(6):596-602, 1971.

[43] D. G. Luenberger. *Determining the state of a linear system with observers of low dynamic order*. Stanford University, 1963.

[44] P. Moireau, D. Chapelle, and P. Le Tallec. Joint state and parameter estimation for distributed mechanical systems. *Computer methods in applied mechanics and engineering*, 197(6-8):659-677, 2008.

[45] B. Mukanova and V. G. Romanov. Inverse source problem for wave equation and gpr data interpretation problem. *arXiv preprint arXiv:1609.03555*, 2016.

[46] F. Natterer. Error bounds for tikhonov regularization in hilbert scales. *Applicable Analysis*, 18(1-2):29-37, 1984.

[47] A. Pazy. *Semigroups of Linear Operators and Applications to Partial Differential Equations*. Springer New York, NY, 1992.

[48] D. T. Pham, J. Verron, and L. Gourdeau. Filtres de kaiman singuliers évolutifs pour l'assimilation de données en océanographie. *Comptes Rendus de l'Académie des Sciences-Series IIA-Earth and Planetary Science*, 326(4):255-260, 1998.

[49] H. Rabier, F. and Järvinen, E. Klinker, J.-F. Mahfouf, and A. Simmons. The ecmwf operational implementation of four-dimensional variational assimilation. i: Experimental results with simplified physics. *Quarterly Journal of the Royal Meteorological Society*, 126(564):1143-1170, 2000.
[50] K. Ramdani, M. Tucsnak, and G. Weiss. Recovering the initial state of an infinite-dimensional system using observers. *Automatica*, 46(10):1616–1625, 2010.

[51] C. Roman, F. Ferrante, and C. Prieur. Parameter identification of a linear wave equation from experimental boundary data. *IEEE Transactions on Control Systems Technology*, 29(5):2166–2179, 2020.

[52] Y. Sasaki. A fundamental study of the numerical prediction based on the variational principle. *Journal of the Meteorological Society of Japan. Ser. II*, 33(6):262–275, 1955.

[53] Y. Sasaki. Some basic formalisms in numerical variational analysis. *Monthly Weather Review*, 98(12):875–883, 1970.

[54] O. Talagrand and P. Courtier. Variational assimilation of meteorological observations with the adjoint vorticity equation. i: Theory. *Quarterly Journal of the Royal Meteorological Society*, 113(478):1311–1328, 1987.

[55] M. Tucsnak and G. Weiss. *Observation and control for operator semigroups*. Springer Science & Business Media, 2009.

[56] Thi Minh Nhat Vo. *Construction of a control and reconstruction of a source for linear and nonlinear heat equations*. PhD thesis, Université d'Orléans, October 2018.

[57] M. Yamamoto. Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method. *Inverse problems*, 11(2):481, 1995.

[58] M. Yamamoto. Uniqueness and stability in multidimensional hyperbolic inverse problems. *Journal de mathématiques pures et appliquées*, 78(1):65–98, 1999.

[59] Q. Zhang. Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems. *IEEE transactions on automatic control*, 47(3):525–529, 2002.

[60] X. Zou, I.M. Navon, and F.X. LeDimet. An optimal nudging data assimilation scheme using parameter estimation. *Quarterly Journal of the Royal Meteorological Society*, 118(508):1163–1186, 1992.

[61] E. Zuazua. Propagation, observation, control and numerical approximation of waves. *SIAM Review*, 47(2):197–243, 2005.

Received xxxx 20xx; revised xxxx 20xx; early access xxxx 20xx.