### Appendix A: The illuminating example of efficient state feedback in elastodynamics

In the following very simplified example, we aim at mathematically proving why a simple state feedback can have a comparable efficiency with respect to a Kalman-based feedback in the context of elastodynamics problems. Let us then consider a simplified configuration where the model and the observation operator are linear. More precisely, we assume a linear elastic system in which the Cauchy stress tensor is given by

$$\begin{aligned} \varvec{\sigma }= \varvec{A}: \varvec{\varepsilon }(\varvec{u}), \end{aligned}$$

where the elasticity tensor \(\varvec{A}\) is assumed to be constant and isotropic. Defining \(\Delta _e(\cdot ) = \mathop {\mathbf {div}}( \varvec{A}: \varvec{\varepsilon }(\cdot ) )\), our model simply reads

$$\begin{aligned}&\left\{ \begin{array}{ll} {\dot{\varvec{u}}} = \varvec{v}, &{} \quad \text{ in } \Omega _0\times (0,T) \\ \rho {\dot{\varvec{v}}} - \Delta _e \varvec{u}= \varvec{f},&{} \quad \text{ in } \Omega _0\times (0,T) \\ \varvec{u}= 0, &{} \quad {{\text {on }}} \partial \Omega _0\times (0,T). \end{array} \right. \end{aligned}$$

(29)

The external load is a time-dependent regular function \(\varvec{f} \in C^1([0,T],{\mathcal {L}}^2(\Omega _0))\). We introduce \({\mathcal {Y}}^{v}= {\mathcal {L}}^2(\Omega _0)^3\), the displacement space \({\mathcal {Y}}^{u}= {\mathcal {H}}^1_0(\Omega _0)^3\), and \( {\mathcal {Y}}= {\mathcal {Y}}^{u}\times {\mathcal {Y}}^{v}\). Using the Korn and Poincaré inequalities, \({\mathcal {Y}}^{u}\) is an Hilbert space with the following scalar product

$$\begin{aligned} \forall (\varvec{u}_1,\varvec{u}_2) \in {\mathcal {Y}}^{u}, \quad (\varvec{u}_1, \varvec{u}_2 )_{{\mathcal {E}}_0} = \int _{\Omega _0} \varvec{\varepsilon }(\varvec{u}_1) : \varvec{A}: \varvec{\varepsilon }(\varvec{u}_2) \, {{\mathrm {d}}}\Omega . \end{aligned}$$

In a semi-group theory context we introduce the semi-group generator \({{\mathrm {A}}}\in {\mathcal {L}}({\mathcal {D}}({{\mathrm {A}}}),{\mathcal {Y}})\) with

$$\begin{aligned} {{\mathrm {A}}}= \begin{pmatrix} 0 &{} \mathbb {1} \\ \tfrac{1}{\rho } \Delta _e &{} 0 \end{pmatrix}, \end{aligned}$$

and we can prove that (29) admits a classical solution \(C^0([0,T],{\mathcal {Y}}^{u}) \cap C^1([0,T],{\mathcal {Y}}^{v})\) for every initial condition in the domain [72]

$$\begin{aligned} {\mathcal {D}}({{\mathrm {A}}}) = \left\{ (\varvec{u},\varvec{v}) \in {\mathcal {Y}}^{u}\times {\mathcal {Y}}^{v}, \quad \mathop {\mathbf {div}}( \varvec{A}: \varvec{\varepsilon }(\varvec{u}) ) \in {\mathcal {Y}}^{v}\right\} . \end{aligned}$$

Moreover, the operator \({{\mathrm {A}}}\) is skew-adjoint, implying that

$$\begin{aligned} \forall t > 0, \quad \left\| {{\mathrm {y}}}(t)\right\| _{{\mathcal {Y}}} = \left\| {{\mathrm {y}}}(0)\right\| _{{\mathcal {Y}}}, \end{aligned}$$

corresponding to the energy balance on the system (29) with \({\mathcal {E}} = \frac{1}{2} \left\| {{\mathrm {y}}}\right\| _{{\mathcal {Y}}}^2\). Using the semi-group theory, we rewrite the dynamics of the model in the abstract state-space form

$$\begin{aligned} {\dot{{{\mathrm {y}}}}} = {{\mathrm {A}}}{{\mathrm {y}}}+ {{\mathrm {R}}}, \quad {{\mathrm {y}}}(0)= {{\mathrm {y}}}_0+ \zeta _{{{\mathrm {y}}}}. \end{aligned}$$

Concerning this model, we assume that we have at our disposal some measurements of the displacements. We introduce the observation operator \({{\mathrm {H}}}= ({{\mathrm {H}}}_{|{\mathcal {Y}}^{u}} \quad 0)\) such that

$$\begin{aligned} {{\mathrm {H}}}_{|{\mathcal {Y}}^{u}} : \left| \begin{aligned} {\mathcal {Y}}^{u}&\rightarrow {\mathcal {Z}}\\ \varvec{u}&\mapsto \mathbb {1}_{\omega _0} \varvec{u}, \end{aligned} \right. \end{aligned}$$

where \(\omega _0\) is an internal open subdomain of \(\Omega _0\) and \({\mathcal {Z}}= {\mathcal {H}}^1(\omega _0)^3\). Here \({{\mathrm {H}}}\) and \({{\mathrm {H}}}_{|{\mathcal {Y}}^{u}}\) as \({{\mathrm {H}}}= ({{\mathrm {H}}}_{|{\mathcal {Y}}^{u}} \quad 0)\) correspond to the same operators with different input spaces.

Using the extension \(\varvec{\psi } = \mathbf {Ext}_{\omega _0}(\varvec{\varphi })\) defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta _e(\varvec{\psi }) = 0, &{} \quad {{\text {in}}} ~\Omega _0\\ \varvec{\psi } = \varvec{\varphi }, &{}\quad {{\text {in}}} ~\omega _0\\ \varvec{\psi } = 0, &{}\quad {{\text {on}}} ~\partial \Omega _0, \end{array}\right. } \end{aligned}$$

(30)

we can prove the following property.

### Proposition A.1

For any \((\varvec{\varphi }_1,\varvec{\varphi }_2) \in {\mathcal {Z}}^2\), the bilinear form \((\mathbf {Ext}_{\omega _0}(\varvec{\varphi }_1),\mathbf {Ext}_{\omega _0}(\varvec{\varphi }_2))_{{\mathcal {E}}_0}\) defines a scalar product on \({\mathcal {Z}}= {\mathcal {H}}^1(\omega _0)^3\).

### Proof

The proof is a simple extension of the property proven by [73] for scalar equations. Let \(\varvec{\varphi }\) be an element of \({\mathcal {Z}}\). The only difficulty lies in proving the norm equivalence with \(\left\| \varvec{\varphi }\right\| _{{\mathcal {H}}^1(\omega _0)^3}^2\). First, we have

$$\begin{aligned} \left\| \varvec{\varphi }\right\| _{{\mathcal {H}}^1(\omega _0)^3}^2&= \left\| \varvec{\nabla }\,\varvec{\varphi }\right\| _{{\mathcal {L}}^2(\omega _0)^3}^2 + \left\| \varvec{\varphi }\right\| _{{\mathcal {L}}^2(\omega _0)^3}^2 \\&\le \left\| \varvec{\nabla }\, \mathbf {Ext}_{\omega _0}(\varvec{\varphi })\right\| _{{\mathcal {L}}^2(\Omega _0)^3}^2 +\left\| \mathbf {Ext}_{\Omega _0}(\varvec{\varphi })\right\| _{{\mathcal {L}}^2(\Omega _0)^3}^2\\&\le (1+C_p) \left\| \varvec{\nabla }\, \mathbf {Ext}_{\omega _0}(\varvec{\varphi })\right\| _{{\mathcal {L}}^2(\Omega _0)^3}^2\\&\le C_k(1+C_p) \left\| \mathbf {Ext}_{\omega _0}(\varvec{\varphi })\right\| _{{\mathcal {E}}_0}^2, \end{aligned}$$

with \(C_p\) given by the Poincaré inequality and \(C_k\) given by Korn inequality and a bound \(C_a\) on the elasticity tensor. Conversely, by continuity of the extension on \(\Omega _0\backslash \omega _0\) with respect to the data, there exists a constant \(C_d >0\) such that for any \(\varvec{\psi } = \mathbf {Ext}_{\omega _0}(\varvec{\varphi }) \) we have

$$\begin{aligned} \int _{\Omega _0\backslash \omega _0} \varvec{\varepsilon }(\varvec{\psi }) : \varvec{A}: \varvec{\varepsilon }(\varvec{\psi }) \, {{\mathrm {d}}}\Omega \le C_d \Vert \varvec{\varphi }_{|\partial \omega _0}\Vert _{{\mathcal {H}}^{\frac{1}{2}}(\partial \omega _0)^3}^2. \end{aligned}$$

Hence, denoting by \(C_t\) the constant arising from the continuity of the trace operator, we have

$$\begin{aligned} \left\| \mathbf {Ext}_{\omega _0}(\varvec{\varphi })\right\| _{{\mathcal {E}}_0}^2&\le \int _{\omega _0} \varvec{\varepsilon }(\varvec{\psi }) : \varvec{A}: \varvec{\varepsilon }(\varvec{\psi }) \, {{\mathrm {d}}}\Omega + C_d \left\| \varvec{\varphi }\right\| _{{\mathcal {H}}^{\frac{1}{2}}(\partial \omega _0)^3}^2 \\&\le C_a \left\| \varvec{\nabla }\, \varvec{\varphi }\right\| _{{\mathcal {L}}^{2}(\omega _0)^3}^2 + C_d \Vert \varvec{\varphi }_{|\partial \omega _0}\Vert ^2_{{\mathcal {H}}^{\frac{1}{2}}(\partial \omega _0)^3} \\&\le (C_a + C_d C_t) \Vert \varvec{\varphi } \Vert ^2_{{\mathcal {H}}^{1}( \omega _0)^3}, \end{aligned}$$

which completes the proof. \(\square \)

It is now possible to define the adjoint of the observation operator.

### Proposition A.2

The operator \({{\mathrm {H}}}\) is bounded from \({\mathcal {Y}}\) to \({\mathcal {Z}}\) and \({{\mathrm {H}}}^*\) is given by

$$\begin{aligned} {{\mathrm {H}}}^* = \begin{pmatrix} {{\mathrm {H}}}_{|{\mathcal {Y}}^{u}}^* \\ 0 \end{pmatrix} {{\text { with }}} {{\mathrm {H}}}_{|{\mathcal {Y}}^{u}}^*\, :\, \left| \begin{aligned} {\mathcal {Z}}&\rightarrow {\mathcal {Y}}^{u}\\ \varvec{\varphi }&\mapsto \mathbf {Ext}_{\omega _0}(\varvec{\varphi }) \end{aligned} \right. \end{aligned}$$

### Proof

Let us first prove that \({{\mathrm {H}}}\) is bounded. We consider \(\varvec{\psi }\in {\mathcal {Y}}^{u}\) and \(\varvec{\varphi }\) such that \(\varvec{\varphi } = {{\mathrm {H}}}_{|{\mathcal {Y}}^{u}}\,\varvec{\psi }\). We have directly, from norm equivalences,

$$\begin{aligned} \left\| \varvec{\varphi }\right\| _{\mathcal {Z}}^2 = \left\| \mathbf {Ext}_{\omega _0}(\varvec{\varphi })\right\| _{{\mathcal {E}}_0}^2 \le C_1 \left\| \varvec{\varphi }\right\| _{{\mathcal {H}}^1(\omega _0)^3}^2 \le C_1 \left\| \varvec{\psi }\right\| _{{\mathcal {H}}^1_0(\Omega _0)^3}^2. \end{aligned}$$

Then, we have that for all \(\varvec{\varphi } \in {\mathcal {Z}}\) and \(\varvec{v}^{\sharp }\in {\mathcal {Y}}^{u}\)

$$\begin{aligned} (\varvec{\varphi }, {{\mathrm {H}}}_{|{\mathcal {Y}}^{u}}\, \varvec{v}^{\sharp })_{{\mathcal {Z}}} = \int _{\Omega _0} \varvec{\varepsilon }(\mathbf {Ext}_{\omega _0}(\varvec{\varphi })) : \varvec{A}: \varvec{\varepsilon }(\mathbf {Ext}_{\omega _0}(\varvec{v}^{\sharp }_{|\omega _0})) \, {{\mathrm {d}}}\Omega . \end{aligned}$$

By the variational characterization of the extension (5) we have

$$\begin{aligned} \int _{\Omega _0} \varvec{\varepsilon }(\mathbf {Ext}_{\omega _0}(\varvec{\varphi })) : \varvec{A}: \varvec{\varepsilon }(\mathbf {Ext}_{\omega _0}(\varvec{v}^{\sharp }_{|\omega _0})) \, {{\mathrm {d}}}\Omega = \int _{\Omega _0} \varvec{\varepsilon }(\mathbf {Ext}_{\omega _0}(\varvec{\varphi })) : \varvec{A}: \varvec{\varepsilon }(\varvec{v}^{\sharp }) \, {{\mathrm {d}}}\Omega , \end{aligned}$$

since \(\varvec{v}^{\sharp }_{|\omega _0} - \mathbf {Ext}_{\omega _0}(\varvec{v}^{\sharp }_{|\omega _0}) = 0\) on \(\omega _0\). Therefore \((\varvec{\varphi }, {{\mathrm {H}}}_{|{\mathcal {Y}}^{u}}\, \varvec{v}^{\sharp })_{{\mathcal {Z}}} = ({{\mathrm {H}}}_{|{\mathcal {Y}}^{u}}^* \varvec{\varphi }, \varvec{v}^{\sharp })_{{\mathcal {E}}_0}, \) and \({{\mathrm {H}}}^*\) is given by

$$\begin{aligned} {{\mathrm {H}}}^* : \left| \begin{aligned} {\mathcal {Z}}&\rightarrow {\mathcal {Y}}^{u}\\ \varvec{\varphi }&\mapsto \begin{pmatrix}{{\text {Ext}}}_{\omega _0}(\varvec{\varphi })\\ 0\end{pmatrix}. \end{aligned} \right. \end{aligned}$$

\(\square \)

We can now define the observer by the dynamics

which in strong form reads

which converges to the solution of (29) under the observability condition given by the next theorem, see e.g. [74] for a proof.

### Theorem A.3

Let \({{\mathrm {A}}}\) be a time-independent skew-adjoint operator generating a group and \({{\mathrm {H}}}\in {\mathcal {L}}({\mathcal {Y}},{\mathcal {Z}})\). The error system \({\widetilde{{{\mathrm {y}}}}}\) of dynamics

$$\begin{aligned} \dot{{\widetilde{{{\mathrm {y}}}}}} = ({{\mathrm {A}}}- \gamma {{\mathrm {H}}}^* {{\mathrm {H}}}) {\widetilde{{{\mathrm {y}}}}} \end{aligned}$$

is exponentially stable if the following observability condition is satisfied: there exists two constants \((C_{{{\text {st}}}},T)\) such that for any solution \({{\mathrm {y}}}\) of \({\dot{{{\mathrm {y}}}}} = {{\mathrm {A}}}{{\mathrm {y}}}\), we have

$$\begin{aligned} \int _0^T \left\| {{\mathrm {H}}}{{\mathrm {y}}}(s)\right\| ^2_{\mathcal {Z}}\, {{\mathrm {d}}}s \ge C_{{{\text {st}}}} \left\| {{\mathrm {y}}}(0)\right\| ^2_{{\mathcal {Y}}}. \end{aligned}$$

(32)

We can now make explicit the specific observability condition in our configuration that will allow us to invoke Theorem A.3 of Appendix A.

### Theorem A.4

If there exists a constant \(C_{{\text {st}}}\) and a time *T* such that every solution of

$$\begin{aligned}&\left\{ \begin{array}{ll} {\dot{\varvec{u}}} = \varvec{v}, &{} \quad \text{ in } \Omega _0\\ \rho {\dot{\varvec{v}}} - \Delta _e \varvec{u}= 0,&{} \quad \text{ in } \Omega _0\\ \varvec{u}= 0, &{} \quad {{\text {on }}} \partial \Omega _0\end{array} \right. \end{aligned}$$

satisfies the observability condition

$$\begin{aligned} \int _{0}^T&\left\| \mathbf {Ext}_{\omega _0}(\mathbb {1}_{\omega _0} \varvec{u})\right\| _{{\mathcal {E}}_0}^2 \, {{\mathrm {d}}}t \ge c^{{\text {st}}} \Bigl ( \left\| \varvec{u}(0)\right\| _{{\mathcal {E}}_0}^2 + \left\| \varvec{v}(0)\right\| _{{\mathcal {L}}^2}^2 \Bigr ), \end{aligned}$$

(33)

then, in the absence of observation error, the observer given by the dynamics (31) converges to the solution \({{\mathrm {y}}}^{{{\text {ref}}}}\) of (29) such that

$$\begin{aligned} {{\mathrm {z}}}= {{\mathrm {H}}}{{\mathrm {y}}}^{{{\text {ref}}}}. \end{aligned}$$

### Proof

We have defined the reference trajectory as the solution of

$$\begin{aligned} {\dot{{{\mathrm {y}}}}}^{{{\text {ref}}}} = {{\mathrm {A}}}{{\mathrm {y}}}^{{{\text {ref}}}} + {{\mathrm {R}}}, \end{aligned}$$

and the observer as the solution of

The error \({\widetilde{{{\mathrm {y}}}}} = {{\mathrm {y}}}^{{{\text {ref}}}} - {\widehat{{{\mathrm {y}}}}}\) is then solution of

which, from Theorem A.3 of Appendix A, converges exponentially to 0 for every initial condition when the observability condition (33) is verified. \(\square \)

Following [75], we define the elastic geometric control condition:

### Definition A.1

(*Elastic Geometric Control Condition*) The elastic geometric control condition is satisfied if every combination of pressure (P) and shear (S) waves ray encounters—in the sense of [75]—the subdomain of observation.

Readers may refer to [75] for a complete description of such rays. This condition generalizes to the vectorial case the so-called geometric control condition (GCC) introduced by [76], allowing to control any solution of the acoustic wave equation from the observations of the time derivative of the wave in a subdomain.

### Theorem A.5

The observability condition of Theorem A.4 holds on a subdomain \(\omega _0\) as soon as the elastic geometric control condition is satisfied with \({\check{\omega }}_0\) and \({{\text {dist}}}(\Omega _0\backslash \omega _0,{\check{\omega }}_0)>0\).

### Proof

For technical reasons, we assume that the elastic geometric control condition is satisfied for an observation domain \({\check{\omega }}_0\) slightly smaller than \(\omega _0\), namely, with \({\check{\omega }}_0\subset \omega _0\) and \({{\text {dist}}}(\Omega _0\backslash \omega _0,{\check{\omega }}_0)>0\). We first recall the classical observability result when the velocity is observed. In fact there exists a constant \(C_{{{\text {st}}}}\) and a time *T* such that every solution of (29) satisfies the observability condition

$$\begin{aligned} \int _{0}^{\breve{T}} \left\| \varvec{v}\right\| _{{\mathcal {L}}^2({\check{\omega }}_0)^3}^2 \, {{\mathrm {d}}}t\ge C_{{{\text {st}}}}\Bigl ( \left\| \varvec{u}(0)\right\| _{{\mathcal {E}}_0}^2 + \left\| \varvec{v}(0)\right\| _{{\mathcal {L}}^2(\Omega _0)^3}^2 \Bigr ), \end{aligned}$$

(34)

with \(\breve{T}=T-\delta \) for \(\delta >0\) sufficiently small, as soon as the elasticity geometric control condition is verified in the time interval [0, *T*[ [75]. Following what was already done for acoustic waves by [73] we will use a property of equirepartition (over time) of the total energy localized within the observation subdomain between the kinetic and elastic contributions to infer (33) from (34).

Let \(\psi \in C_c^\infty ({\overline{\Omega }}_0)\) be a cutoff function satisfying

$$\begin{aligned} \psi (\varvec{\xi }) = \left\{ \begin{array}{l} 0, \quad \text {if } \varvec{\xi }\in \Omega _0\!\backslash \! \omega _0\\ 1, \quad \text {if } \varvec{\xi }\in \breve{\omega }_0 \end{array} \right. \end{aligned}$$

and \(0 \le \psi (\varvec{\xi }) \le 1\) for every \(\varvec{\xi }\in {\overline{\Omega }}_0\). Denote also \(\phi (t) = t^2 ({\breve{T}} - t)^2\). Then, by repeated integrations by parts we obtain

$$\begin{aligned} 0&= \int _0^{\breve{T}} \int _{\omega _0} \phi \psi (\ddot{\varvec{u}} - \Delta _e \varvec{u}) \cdot \varvec{u}\, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}t \\&= \int _0^{\breve{T}} \int _{\omega _0} \ddot{\phi } \psi \frac{|\varvec{u}|^2}{2} \, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}t - \int _0^{\breve{T}} \int _{\omega _0} \phi \psi |{\dot{\varvec{u}}}|^2 \, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}t \\&\quad - \int _0^{\breve{T}} \int _{\omega _0} \phi \, \varvec{\varepsilon }(\varvec{u}) : \varvec{A}: (\varvec{\nabla } \psi \otimes \varvec{u})\, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}t \\&\quad + \int _0^{\breve{T}} \int _{\omega _0} \phi \psi \, \varvec{\varepsilon }(\varvec{u}) : \varvec{A}: \varvec{\varepsilon }(\varvec{u}) \, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}t. \end{aligned}$$

Moreover,

$$\begin{aligned}&\int _{\omega _0} \varvec{\varepsilon }(\varvec{u}) : \varvec{A}: (\varvec{\nabla } \psi \otimes \varvec{u}) \, {{\mathrm {d}}}\Omega \\&\quad \le C_{{{\text {st}}}} \left\| \psi \right\| _{{\mathcal {W}}^{1,\infty }} \left\| \varvec{\varepsilon }(\varvec{u})\right\| _{{\mathcal {L}}^2(\omega _0)^3} \left\| \varvec{u}\right\| _{{\mathcal {L}}^2(\omega _0)^3} \\&\quad \le C_{{{\text {st}}}} \left\| \psi \right\| _{{\mathcal {W}}^{1,\infty }} \left\| \varvec{u}\right\| _{{\mathcal {H}}^1(\omega _0)^3}^2. \end{aligned}$$

where \(C_{{{\text {st}}}}\) represents a different constant in each line. This identity combined with the properties of the cutoff functions \(\phi \) and \(\psi \) provides, for any strictly positive \(\varepsilon \), the existence of a constant \(C_{{{\text {st}}}} > 0\) such that

$$\begin{aligned} \int _\varepsilon ^{\breve{T}-\varepsilon } \int _{{\check{\omega }}_0} |{\dot{\varvec{u}}}|^2 \, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}t \le C_{{{\text {st}}}} \int _0^{\breve{T}} \int _{\omega _0} \Vert \varvec{u}(\cdot ,t)\Vert _{{\mathcal {H}}^1(\omega _0)^3}^2 \, {{\mathrm {d}}}t . \end{aligned}$$

Substituting \({\breve{T}}+2\varepsilon \) for \({\breve{T}}\) in all the above computations gives

$$\begin{aligned} \int _\varepsilon ^{{\breve{T}} + \varepsilon } \int _{\breve{\omega }_0} |{\dot{\varvec{u}}}|^2\, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}t \le C_{{{\text {st}}}}\int _0^{{\breve{T}}+2\varepsilon } \Vert \varvec{u}(\cdot ,t)\Vert _{{\mathcal {H}}^1(\omega _0)^3}^2 \, {{\mathrm {d}}}t. \end{aligned}$$

We proceed by making the change of variable \(\tau = t - \varepsilon \) in the left-hand side integral, yielding

$$\begin{aligned} \int _0^{\breve{T}} \int _{\breve{\omega }_0} |{\dot{\varvec{u}}}(\varvec{\xi }, \tau + \varepsilon )|^2 \, {{\mathrm {d}}}\Omega \, {{\mathrm {d}}}\tau \le C_{{{\text {st}}}} \int _0^{{\breve{T}}+2\varepsilon } \Vert \varvec{u}\Vert _{{\mathcal {H}}^1(\omega _0)^3}^2 \, {{\mathrm {d}}}t. \end{aligned}$$

(35)

Noting that \(\varvec{u}(\varvec{\xi }, t + \varepsilon )\) satisfies the elastodynamics system with initial data \((\varvec{u}(\varvec{\xi },\varepsilon ),{\dot{\varvec{u}}}(\varvec{\xi },\varepsilon ))\) and applying (34) with this shifted solution, we obtain that there exists also \(C_{{{\text {st}}}}\) such that

$$\begin{aligned} \int _0^{\breve{T}} \int _{\breve{\omega }_0} |{\dot{\varvec{u}}}(\varvec{\xi }, \tau + \varepsilon )|^2 \, {{\mathrm {d}}}\Omega \, \, {{\mathrm {d}}}\tau \ge C_{{{\text {st}}}} \left( \Vert \varvec{u}(\varepsilon )\Vert ^2_{{\mathcal {E}}_0} + \Vert {\dot{\varvec{u}}}(\varepsilon )\Vert ^2_{{\mathcal {L}}^2(\Omega _0)^3} \right) . \end{aligned}$$

(36)

Combining (35), (36) and the fact that the energy of the solution of the elastodynamics equation is exactly conserved over time, we have our observability inequality (33) upon choosing \(\varepsilon =\frac{\delta }{2}\). \(\square \)

### Appendix B: Analysis of the prediction–correction scheme

This appendix is dedicated to the analysis of the prediction–correction scheme proposed in “Time discretization of the state observer” section. More specifically, we are interested in showing that the linearization procedure applied in the context of nonlinear observation operators induces a dissipative behavior for the linearized time-discrete estimation error. This property is a crucial aspect when building relevant state observers.

### Linear model and linear observation operator

To start with, we assume that the dynamical system satisfied by the target trajectory is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{{{\mathrm {y}}}}}(t) = ({{\mathrm {A}}}+ \upeta {{\mathrm {V}}}){{\mathrm {y}}}(t)\\ {{\mathrm {y}}}(0) = {{\mathrm {y}}}_0 + \zeta _{{\mathrm {y}}}, \end{array}\right. } \end{aligned}$$

(37)

where \({{\mathrm {A}}}\) is a skew-adjoint operator, \({{\mathrm {V}}}\) is a self-adjoint and semi-negative operator and \(\upeta \ge 0\) is a viscosity coefficient. Note that (37) can be interpreted as a linearization of (3) around the stress-free configuration. Additionally, we consider a linear observation operator and we neglect, for simplicity, the observation noise. Denoting by \(\Delta {{\mathrm {t}}}\) the (constant) time step of the numerical procedure, the time-discrete observations read \({{\mathrm {z}}}^n = {{\mathrm {H}}}{{\mathrm {y}}}(n\Delta {{\mathrm {t}}})\). The prediction–correction scheme for the observer reads

Relation (38a) corresponds to the prediction step, with the operators driving the target system, while (38b) is the correction step. Defining the discrete estimation error from the correction step

$$\begin{aligned} {\widetilde{{{\mathrm {y}}}}}^{n}_+ = {{\mathrm {y}}}(n\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n}_+, \end{aligned}$$

(39)

and associating a prediction error with

$$\begin{aligned} {\widetilde{{{\mathrm {y}}}}}^{n+1}_- = {{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n+1}_-, \end{aligned}$$

(40)

we can determine the time-discrete dynamics satisfied by the estimation error.

### Proposition B.1

Assuming that \({{\mathrm {y}}}\in C^3([0,T],{\mathcal {Y}})\), then the estimation error satisfies the following discrete dynamical system

with \(\varepsilon ^n=O\bigl (\Delta {{\mathrm {t}}}^2\left\| \dddot{{{\mathrm {y}}}}\right\| _{C^3([0,T],{\mathcal {Y}})}\bigr )\).

### Proof

(41b) is directly inferred from the definition of the prediction estimation error and using (38b), namely,

$$\begin{aligned} \begin{aligned} {\widetilde{{{\mathrm {y}}}}}^{n+1}_-&= {{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n+1}_- \\&= {{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n+1}_+ + \gamma \Delta {{\mathrm {t}}}{{{\mathrm {H}}}}^{*} {{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+\\&= (\mathbb {1} + \gamma \Delta {{\mathrm {t}}}{{{\mathrm {H}}}}^{*} {{\mathrm {H}}}){\widetilde{{{\mathrm {y}}}}}^{n+1}_+. \end{aligned} \end{aligned}$$

We now have to work our way to (41a). First, we remark that

$$\begin{aligned} \dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- - {\widetilde{{{\mathrm {y}}}}}^{n}_+}{\Delta {{\mathrm {t}}}} = \dfrac{{{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {{\mathrm {y}}}(n\Delta {{\mathrm {t}}})}{\Delta {{\mathrm {t}}}} - \dfrac{{\widehat{{{\mathrm {y}}}}}^{n+1}_- - {\widehat{{{\mathrm {y}}}}}^n_+}{\Delta {{\mathrm {t}}}}. \end{aligned}$$

(42)

Using centered Taylor expansions we can easily see that

$$\begin{aligned} \dfrac{{{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {{\mathrm {y}}}(n\Delta {{\mathrm {t}}})}{\Delta {{\mathrm {t}}}} = ({{\mathrm {A}}}+ \upeta {{\mathrm {V}}})\dfrac{{{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) + {{\mathrm {y}}}(n\Delta {{\mathrm {t}}})}{2} + \varepsilon ^n, \end{aligned}$$

(43)

with \(\varepsilon ^n=O\bigl (\Delta {{\mathrm {t}}}^2\left\| \dddot{{{\mathrm {y}}}}\right\| _{C^3([n\Delta {{\mathrm {t}}},(n+1)\Delta {{\mathrm {t}}}],{\mathcal {Y}})}\bigr )\). Therefore, feeding equation (42) with (43) and (38a), we obtain

$$\begin{aligned} \dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- - {\widetilde{{{\mathrm {y}}}}}^{n}_+}{\Delta {{\mathrm {t}}}} = ({{\mathrm {A}}}+ \upeta {{\mathrm {V}}}) \dfrac{{{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n+1}_-}{2} + ({{\mathrm {A}}}+ \upeta {{\mathrm {V}}})\dfrac{{{\mathrm {y}}}(n\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n}_+}{2}+ \varepsilon ^n, \end{aligned}$$

(44)

hence, (41a) holds. \(\square \)

We can now establish the energy estimate associated with (41a)–(41b)

### Proposition B.2

The norm of the estimation error, namely

$$\begin{aligned} {\widetilde{{\mathcal {E}}}}^{n}_+ = \dfrac{1}{2} \left\| {\widetilde{{{\mathrm {y}}}}}^{n}_+\right\| ^2_{{\mathcal {Y}}}, \end{aligned}$$

satisfies the following estimate

$$\begin{aligned} \dfrac{{\widetilde{{\mathcal {E}}}}^{n+1}_+ - {\widetilde{{\mathcal {E}}}}^n_+}{\Delta {{\mathrm {t}}}}= & {} - \upeta \left\| \big (\sqrt{-{{\mathrm {V}}}}\big ) \dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- + {\widetilde{{{\mathrm {y}}}}}^n_+}{2}\right\| ^2_{{\mathcal {Y}}} - \gamma \left\| {{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+\right\| ^2_{{\mathcal {Z}}}\nonumber \\&- \gamma ^2\dfrac{\Delta {{\mathrm {t}}}}{2}\left\| {{{\mathrm {H}}}}^{*} {{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+\right\| ^2_{{\mathcal {Z}}} + (\varepsilon ^n,\dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- + {\widetilde{{{\mathrm {y}}}}}^n_+}{2})_{{\mathcal {Y}}}. \end{aligned}$$

(45)

### Proof

Denoting \({\widetilde{{\mathcal {E}}}}^{n+1}_- = \dfrac{1}{2} \left\| {\widetilde{{{\mathrm {y}}}}}^{n+1}_-\right\| ^2_{{\mathcal {Y}}}\), we have, from system (41a)-(41b),

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{{\widetilde{{\mathcal {E}}}}^{n+1}_- - {\widetilde{{\mathcal {E}}}}^{n}_+}{\Delta {{\mathrm {t}}}} = - \upeta \left\| \big (\sqrt{-{{\mathrm {V}}}}\big ) \dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- + {\widetilde{{{\mathrm {y}}}}}^n_+}{2}\right\| ^2_{{\mathcal {Y}}} + \Bigl (\varepsilon ^n,\dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- + {\widetilde{{{\mathrm {y}}}}}^n_+}{2}\Bigr )_{{\mathcal {Y}}}\\ \dfrac{{\widetilde{{\mathcal {E}}}}^{n+1}_+ - {\widetilde{{\mathcal {E}}}}^{n+1}_-}{\Delta {{\mathrm {t}}}} = - \gamma \Bigl ({{{\mathrm {H}}}}^{*} {{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+,\dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_+ + {\widetilde{{{\mathrm {y}}}}}^{n+1}_-}{2}\Bigr )_{{\mathcal {Y}}}\!\! \end{array}\right. } \end{aligned}$$

(46)

Equation (41b) leads to

$$\begin{aligned} \dfrac{{\widetilde{{\mathcal {E}}}}^{n+1}_+ - {\widetilde{{\mathcal {E}}}}^{n+1}_-}{\Delta {{\mathrm {t}}}}&= - \gamma ( {{{\mathrm {H}}}}^{*} {{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+,{\widetilde{{{\mathrm {y}}}}}^{n+1}_+ + \frac{\gamma }{2} \Delta {{\mathrm {t}}}{{{\mathrm {H}}}}^{*} {{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+)_{{\mathcal {Y}}} \\&= - \gamma \left\| {{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+\right\| ^2_{{\mathcal {Z}}} - \gamma ^2\dfrac{\Delta {{\mathrm {t}}}}{2} \left\| {{{\mathrm {H}}}}^{*}{{\mathrm {H}}}{\widetilde{{{\mathrm {y}}}}}^{n+1}_+\right\| ^2_{{\mathcal {Y}}}, \end{aligned}$$

which, by regrouping both equations in (46), entails the desired estimate. \(\square \)

In (45) we see the effect of the correction step (38b) leading to some dissipation terms brought by the observation operator. The expression is the abstract and discrete version of expression (7), perturbed with natural consistency terms.

### Linear model and nonlinear observation operator

As a second step, we consider the case of a nonlinear observation operator, so that the observations are obtained through (18), while the model operator remains linear. In this case, the related prediction–correction time-scheme is

As for the linear case, one can derive the time-discrete dynamics satisfied by the estimation error.

### Proposition B.3

Assuming that \({{\mathrm {y}}}\in C^3([0,T],{\mathcal {Y}})\), then the estimation error associated with (47a)–(47b) satisfies the discrete dynamical system

with the source term in (48a) is identical to the one in (41a), and the source term in (48b) is given by

$$\begin{aligned} \lambda ^n = O(\left\| {{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^e_+\right\| ^2_{{\mathcal {Y}}}). \end{aligned}$$

### Proof

Similarly to the linear case, the prediction estimation error is

$$\begin{aligned} {\widetilde{{{\mathrm {y}}}}}^{n+1}_- = {{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n+1}_-. \end{aligned}$$

This entails (48b) since from equation (47b) and the linearization of \({{\mathrm {H}}}({{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}))\) around the extrapolated trajectory \({\widehat{{{\mathrm {y}}}}}^e_+\) we have

$$\begin{aligned} \begin{aligned} {\widetilde{{{\mathrm {y}}}}}^{n+1}_-&= {{\mathrm {y}}}((n+1)\Delta {{\mathrm {t}}}) - {\widehat{{{\mathrm {y}}}}}^{n+1}_+ + \gamma \Delta {{\mathrm {t}}}{D{{\mathrm {H}}}^{e}_+}^{*}\big ({{\mathrm {z}}}^{n+1} - {{\mathrm {H}}}({\widehat{{{\mathrm {y}}}}}^{e}_+) - D{{\mathrm {H}}}^{e}_+({\widehat{{{\mathrm {y}}}}}^{n+1}_+ - {\widehat{{{\mathrm {y}}}}}^{e}_+)\big )\\&= (\mathbb {1} + \gamma \Delta {{\mathrm {t}}}{D{{\mathrm {H}}}^{e}_+}^{*} D{{\mathrm {H}}}^{e}_+ ){\widetilde{{{\mathrm {y}}}}}^{n+1}_+ + \Delta {{\mathrm {t}}}\lambda ^n. \end{aligned} \end{aligned}$$

All the other computations previously presented to prove Proposition B.1 still hold, so that we obtain the dynamical system (48a)–(48b) satisfied by the estimation error. \(\square \)

We can now establish the energy estimate associated with (48a)–(48b).

### Proposition B.4

The norm of the estimation error in the case of a nonlinear observation operator satisfies the following estimate

$$\begin{aligned} \dfrac{{\widetilde{{\mathcal {E}}}}^{n+1}_+ - {\widetilde{{\mathcal {E}}}}^n_+}{\Delta {{\mathrm {t}}}}= & {} - \upeta \left\| \big (\sqrt{-{{\mathrm {V}}}}\big ) \dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- + {\widetilde{{{\mathrm {y}}}}}^n_+}{2}\right\| ^2_{{\mathcal {Y}}} \nonumber \\&- \gamma \left\| D{{\mathrm {H}}}^{e}_+ {\widetilde{{{\mathrm {y}}}}}^{n+1}_+\right\| ^2_{{\mathcal {Z}}} - \gamma ^2\dfrac{\Delta {{\mathrm {t}}}}{2}\left\| {D{{\mathrm {H}}}^{e}_+}^{*} D{{\mathrm {H}}}^{e}_+ {\widetilde{{{\mathrm {y}}}}}^{n+1}_+\right\| ^2_{{\mathcal {Z}}} \nonumber \\&+ \left( {\varepsilon ^n},{\dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_- + {\widetilde{{{\mathrm {y}}}}}^n_+}{2}}\right) _{{\mathcal {Y}}} + \left( { \lambda ^n},{\dfrac{{\widetilde{{{\mathrm {y}}}}}^{n+1}_+ + {\widetilde{{{\mathrm {y}}}}}^{n+1}_-}{2}}\right) _{{\mathcal {Y}}}. \end{aligned}$$

(49)

### Proof

We remark that the system satisfied by the linearized estimation error (48a)–(48b) can be obtained from system (41a)–(41b) by replacing the observation operator by its tangent around the linearization trajectory, and by adding the linearization term \(\lambda ^n\) in the correction step. Hence, by following the demonstration of Proposition B.2 of Appendix B, one can obtain estimate (49). \(\square \)

Finally, let us remark that the extension of these results to the case of a nonlinear model does not entail specific challenges, within the context of the presented analysis. After linearization, one can expect to produce another linearization source term, this time in the prediction step (48a)—instead of the correction step. Hence, the dissipation property of the time-discrete observer still holds, up to this additional linearization term.