<figure><img src="image_2.png" /><figcaption>FIGURE 2. Estimation of the test function supports in the case $\varepsilon \le 1$.</figcaption></figure>

with $\alpha^1(x, x^0) \in \mathbb{R}$ and $\alpha'(x, x^0) \in \mathbb{R}^{d-1}$. By (4.9), for $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$ we have $\|x - x^0\| \lesssim \delta$. From the uniform continuity of $X$ in any compact set we conclude that

$$ (4.12) \qquad |\alpha^1(x, x^0)| + \|\alpha'(x, x^0)\| = o(1) \quad \text{as } \delta \to 0^+, $$

uniformly<fnref n="3" /> with respect to $x^0 \in K$ and $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$. Using that $X(x^0) = \partial_{x_1}$ and the form of $v$ given above, we write

$$ \begin{aligned} Xv(x) &= (X(x)v)(x) = (\partial_{x_1}v + (X(x) - X(x^0))v)(x) \\ &= -\delta^{-1}(1 + \alpha^1(x, x^0) - 16\varepsilon^{-1}(\varepsilon\delta)^{-1}\alpha'(x, x^0) \cdot (x' - x^{0'})). \end{aligned} $$

Using again (4.9), we thus find for $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$

$$ |\alpha^1(x, x^0) - 16\varepsilon^{-1}(\varepsilon\delta)^{-1}\alpha'(x, x^0) \cdot (x' - x^{0'})| \lesssim |\alpha^1(x, x^0)| + \varepsilon^{-1}\|\alpha'(x, x^0)\|. $$

With $\varepsilon$ fixed above and with (4.12) we find that $Xv(x) \sim -\delta^{-1}$ as $\delta \to 0^+$ uniformly with respect to $x^0 \in K$ and $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$.

Finally, we have $g_{\varepsilon, \delta, x^0}(x^0) = -\delta^{-1}\chi'(1/2)\beta(2\varepsilon^{-1}) > 0$ and thus $g_{\varepsilon, \delta, x^0}$ is positive in a neighborhood of $x^0$. $\square$

We are now in a position to conclude the proof of Proposition 4.2. Note that it suffices to prove the result for $0 < \varepsilon \le 1$. We choose $\delta_0 > 0$ as given by Lemma 4.3. Let then $x^0 \in K \cap \text{supp}(\mu)$. We apply (4.1) to the family $q_{\varepsilon, \delta, x^0}$ of test functions with $0 < \delta \le \delta_0$:

$$ (4.13) \qquad 0 = \langle \mu, X(q_{\varepsilon, \delta, x^0}) \rangle = \langle \mu, g_{\varepsilon, \delta, x^0} \rangle + \langle \mu, h_{\varepsilon, \delta, x^0} \rangle. $$

By Lemma 4.3, $g_{\varepsilon, \delta, x^0} \ge 0$ and $g_{\varepsilon, \delta, x^0}$ is positive in a neighborhood of $x^0$. As $x^0 \in \text{supp}(\mu)$ we find $\langle \mu, g_{\varepsilon, \delta, x^0} \rangle > 0$. Consequently, $\langle \mu, h_{\varepsilon, \delta, x^0} \rangle \ne 0$. By the support estimate for $h_{\varepsilon, \delta, x^0}$ given in (4.11) the conclusion follows: $\text{supp}(\mu) \cap B(x^0 + \delta X(x^0), \varepsilon\delta) \ne \emptyset$. $\square$

<footnote n="3">³Observe that the change of variables made above for $X(x^0) = (1, 0, \dots, 0)$ does not affect uniformity since the dilation is made by a factor in $[c_K, C_K]$.</footnote>
<figure><img src="image_2.png" /><figcaption>FIGURE 2. Estimation of the test function supports in the case $\varepsilon \le 1$.</figcaption></figure>

with $\alpha^1(x, x^0) \in \mathbb{R}$ and $\alpha'(x, x^0) \in \mathbb{R}^{d-1}$. By (4.9), for $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$ we have $\|x - x^0\| \lesssim \delta$. From the uniform continuity of $X$ in any compact set we conclude that

$$ (4.12) \qquad |\alpha^1(x, x^0)| + \|\alpha'(x, x^0)\| = o(1) \quad \text{as } \delta \to 0^+, $$

uniformly<fnref n="3" /> with respect to $x^0 \in K$ and $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$. Using that $X(x^0) = \partial_{x_1}$ and the form of $v$ given above, we write

$$ \begin{aligned} Xv(x) &= (X(x)v)(x) = (\partial_{x_1}v + (X(x) - X(x^0))v)(x) \\ &= -\delta^{-1}\left(1 + \alpha^1(x, x^0) - 16\varepsilon^{-1}(\varepsilon\delta)^{-1}\alpha'(x, x^0) \cdot (x' - x^{0'})\right). \end{aligned} $$

Using again (4.9), we thus find for $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$

$$ |\alpha^1(x, x^0) - 16\varepsilon^{-1}(\varepsilon\delta)^{-1}\alpha'(x, x^0) \cdot (x' - x^{0'})| \lesssim |\alpha^1(x, x^0)| + \varepsilon^{-1}\|\alpha'(x, x^0)\|. $$

With $\varepsilon$ fixed above and with (4.12) we find that $Xv(x) \sim -\delta^{-1}$ as $\delta \to 0^+$ uniformly with respect to $x^0 \in K$ and $x \in \text{supp}(\chi' \circ v)(\beta \circ w)$.

Finally, we have $g_{\varepsilon, \delta, x^0}(x^0) = -\delta^{-1}\chi'(1/2)\beta(2\varepsilon^{-1}) > 0$ and thus $g_{\varepsilon, \delta, x^0}$ is positive in a neighborhood of $x^0$. $\square$

We are now in a position to conclude the proof of Proposition 4.2. Note that it suffices to prove the result for $0 < \varepsilon \le 1$. We choose $\delta_0 > 0$ as given by Lemma 4.3. Let then $x^0 \in K \cap \text{supp}(\mu)$. We apply (4.1) to the family $q_{\varepsilon, \delta, x^0}$ of test functions with $0 < \delta \le \delta_0$:

$$ (4.13) \qquad 0 = \langle \mu, X(q_{\varepsilon, \delta, x^0}) \rangle = \langle \mu, g_{\varepsilon, \delta, x^0} \rangle + \langle \mu, h_{\varepsilon, \delta, x^0} \rangle. $$

By Lemma 4.3, $g_{\varepsilon, \delta, x^0} \ge 0$ and $g_{\varepsilon, \delta, x^0}$ is positive in a neighborhood of $x^0$. As $x^0 \in \text{supp}(\mu)$ we find $\langle \mu, g_{\varepsilon, \delta, x^0} \rangle > 0$. Consequently, $\langle \mu, h_{\varepsilon, \delta, x^0} \rangle \ne 0$. By the support estimate for $h_{\varepsilon, \delta, x^0}$ given in (4.11) the conclusion follows: $\text{supp}(\mu) \cap B(x^0 + \delta X(x^0), \varepsilon\delta) \ne \emptyset$. $\square$

<footnote n="3">³Observe that the change of variables made above for $X(x^0) = (1, 0, \dots, 0)$ does not affect uniformity since the dilation is made by a factor in $[c_K, C_K]$.</footnote>
5. EXACT CONTROLLABILITY: PROOF OF THEOREM 1.12

Let $(\kappa^0, g^0) \in \mathcal{X}^1(\mathcal{M})$ and assume that $(\omega, T)$ fulfills the geometric control condition of Definition 1.8$^{\prime\prime}$.

Let also $(\kappa, g) \in \mathcal{Y}(\mathcal{M})$. With Proposition 1.6, the result of Theorem 1.12 follows if we prove that there exists $\varepsilon > 0$ and $C_{\text{obs}} > 0$ such that

$$ \mathcal{E}_{\kappa,g}(u)(0) \leq C_{\text{obs}} \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L}, \kappa \mu_g dt)}^2, $$

for any weak solution $u$ of the wave equation associated with $(\kappa, g)$ chosen such that

$$ \|(\kappa, g) - (\kappa^0, g^0)\|_{\mathcal{Y}(\mathcal{M})} \leq \varepsilon. $$

The $L^2$-norm on the r.h.s. is associated with $(\kappa, g)$, that is,

$$ \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L}, \kappa \mu_g dt)}^2 = \int_0^T \int_\omega |\partial_t u|^2 \kappa \mu_g dt. $$

Yet, for $\varepsilon > 0$ chosen sufficiently small one has $\|\cdot\|_{L^2(\mathcal{L}, \kappa^0 \mu_{g^0})} \stackrel{\sim}{=} \|\cdot\|_{L^2(\mathcal{L}, \kappa \mu_g)}$, where $A \stackrel{\sim}{=} B$ means $c_1 \le A/B \le c_2$ for some $c_1, c_2 > 0$. In other words, we have equivalence with constants uniform with respect to $(\kappa, g)$. In what follows, $L^2$- and more generally $H^s$-norms on $\mathcal{M}$ are chosen with respect to $\kappa^0 \mu_{g^0}$ unless explicitly written. Our goal is thus to prove the following observability inequality

$$ (5.1) \qquad \mathcal{E}_{\kappa^0, g^0}(u)(0) \leq C_{\text{obs}} \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L})}^2. $$

The Bardos-Lebeau-Rauch uniqueness compactness argument reduces the proof of (5.1) to the proof of the weaker estimate

$$ (5.2) \qquad \mathcal{E}_{\kappa^0, g^0}(u)(0) \leq C \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L})}^2 + C' \|(u(0), \partial_t u(0))\|_{L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})}^2, $$

that exhibits an additional compact term, and expresses observability for high-frequencies. Low frequencies are dealt with by means of a unique continuation argument.

To prove (5.2) we argue by contradiction and we assume that there exists a sequence $(\kappa_k, g_k)_{k \in \mathbb{N}} \subset \mathcal{Y}(\mathcal{M})$ such that

$$ (5.3) \qquad \lim_{k \to +\infty} \|(\kappa_k, g_k) - (\kappa^0, g^0)\|_{\mathcal{Y}(\mathcal{M})} = 0, $$

and yet for each $k \in \mathbb{N}$ the associated observability inequality does not hold. Thus, for each $k \in \mathbb{N}$, there exists a sequence of initial data $(v^{k,p,0}, v^{k,p,1})_{p \in \mathbb{N}} \subset H^1(\mathcal{M}) \times L^2(\mathcal{M})$ with associated solution $(v^{k,p})_{p \in \mathbb{N}}$, that is,

$$ \begin{cases} P_k v^{k,p} = 0 & \text{in } (0, +\infty) \times \mathcal{M}, \\ v^{k,p}_{|t=0} = v^{k,p,0}, \quad \partial_t v^{k,p}_{|t=0} = v^{k,p,1} & \text{in } \mathcal{M}, \end{cases} $$

with $P_k = P_{\kappa_k, g_k}$, that moreover has the properties

$$ \mathcal{E}_{\kappa^0, g^0}(v^{k,p})(0) = 1 \quad \text{and} \quad \| \mathbf{1}_{(0,T) \times \omega} \partial_t v^{k,p} \|_{L^2(\mathcal{L})} + \|(v^{k,p,0}, v^{k,p,1})\|_{L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})} \le \frac{1}{p+1}. $$

We take $p = k$ and we set $(u^{k,0}, u^{k,1}) = (v^{k,k,0}, v^{k,k,1})$ and $u^k = v^{k,k}$, one obtains $P_k u^k = 0$ in $\mathcal{L}$ and

$$ (5.4) \qquad \mathcal{E}_{\kappa^0, g^0}(u^k)(0) = 1 \quad \text{and} \quad \| \mathbf{1}_{(0,T) \times \omega} \partial_t u^k \|_{L^2(\mathcal{L})} + \|(u^{k,0}, u^{k,1})\|_{L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})} \le \frac{1}{k+1}. $$
5. EXACT CONTROLLABILITY: PROOF OF THEOREM 1.12

Let $(\kappa^0, g^0) \in \mathcal{X}^1(\mathcal{M})$ and assume that $(\omega, T)$ fulfills the geometric control condition of Definition 1.8''.

Let also $(\kappa, g) \in \mathcal{Y}(\mathcal{M})$. With Proposition 1.6, the result of Theorem 1.12 follows if we prove that there exists $\varepsilon > 0$ and $C_{\text{obs}} > 0$ such that

$$ \mathcal{E}_{\kappa,g}(u)(0) \leq C_{\text{obs}} \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L}, \kappa \mu_g dt)}^2, $$

for any weak solution $u$ of the wave equation associated with $(\kappa, g)$ chosen such that

$$ \|(\kappa, g) - (\kappa^0, g^0)\|_{\mathcal{Y}(\mathcal{M})} \leq \varepsilon. $$

The $L^2$-norm on the r.h.s. is associated with $(\kappa, g)$, that is,

$$ \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L}, \kappa \mu_g dt)}^2 = \int_0^T \int_\omega |\partial_t u|^2 \kappa \mu_g dt. $$

Yet, for $\varepsilon > 0$ chosen sufficiently small one has $\|\cdot\|_{L^2(\mathcal{L}, \kappa^0 \mu_{g^0})} \stackrel{\sim}{=} \|\cdot\|_{L^2(\mathcal{L}, \kappa \mu_g)}$, where $A \stackrel{\sim}{=} B$ means $c_1 \le A/B \le c_2$ for some $c_1, c_2 > 0$. In other words, we have equivalence with constants uniform with respect to $(\kappa, g)$. In what follows, $L^2$- and more generally $H^s$-norms on $\mathcal{M}$ are chosen with respect to $\kappa^0 \mu_{g^0}$ unless explicitly written. Our goal is thus to prove the observability inequality

$$ (5.1) \qquad \mathcal{E}_{\kappa^0, g^0}(u)(0) \leq C_{\text{obs}} \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L})}^2. $$

The Bardos-Lebeau-Rauch uniqueness compactness argument reduces the proof of (5.1) to the proof of the weaker estimate

$$ (5.2) \qquad \mathcal{E}_{\kappa^0, g^0}(u)(0) \leq C \| \mathbf{1}_{(0,T) \times \omega} \partial_t u \|_{L^2(\mathcal{L})}^2 + C' \|(u(0), \partial_t u(0))\|_{L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})}^2, $$

which exhibits an additional compact term, and expresses observability for high-frequencies. Low frequencies are dealt with by means of a unique continuation argument.

To prove (5.2) we argue by contradiction and we assume that there exists a sequence $(\kappa_k, g_k)_{k \in \mathbb{N}} \subset \mathcal{Y}(\mathcal{M})$ such that

$$ (5.3) \qquad \lim_{k \to +\infty} \|(\kappa_k, g_k) - (\kappa^0, g^0)\|_{\mathcal{Y}(\mathcal{M})} = 0, $$

and yet for each $k \in \mathbb{N}$ the associated observability inequality does not hold. Thus, for each $k \in \mathbb{N}$, there exists a sequence of initial data $(v^{k,p,0}, v^{k,p,1})_{p \in \mathbb{N}} \subset H^1(\mathcal{M}) \times L^2(\mathcal{M})$ with associated solution $(v^{k,p})_{p \in \mathbb{N}}$, that is,

$$ \left\{ \begin{array}{ll} P_k v^{k,p} = 0 & \text{in } (0, +\infty) \times \mathcal{M}, \\ v^{k,p}_{|t=0} = v^{k,p,0}, \quad \partial_t v^{k,p}_{|t=0} = v^{k,p,1} & \text{in } \mathcal{M}, \end{array} \right. $$

with $P_k = P_{\kappa_k, g_k}$, that moreover has the properties

$$ \mathcal{E}_{\kappa^0, g^0}(v^{k,p})(0) = 1 \quad \text{and} \quad \| \mathbf{1}_{(0,T) \times \omega} \partial_t v^{k,p} \|_{L^2(\mathcal{L})} + \|(v^{k,p,0}, v^{k,p,1})\|_{L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})} \le \frac{1}{p+1}. $$

We take $p = k$ and we set $(u^{k,0}, u^{k,1}) = (v^{k,k,0}, v^{k,k,1})$ and $u^k = v^{k,k}$, one obtains $P_k u^k = 0$ in $\mathcal{L}$ and

$$ (5.4) \qquad \mathcal{E}_{\kappa^0, g^0}(u^k)(0) = 1 \quad \text{and} \quad \| \mathbf{1}_{(0,T) \times \omega} \partial_t u^k \|_{L^2(\mathcal{L})} + \|(u^{k,0}, u^{k,1})\|_{L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})} \le \frac{1}{k+1}. $$
From (5.4) one has $u^k \to 0$ weakly in $H_{\text{loc}}^1(\mathcal{L})$. With (3.1)-(3.2), we can associate with (a subsequence of) $(u^k)_k$ a $H^1$-microlocal defect measure $\mu$ on $S^*(\mathcal{L})$. Here, the measure is understood with respect to $L^2(\mathcal{L}, \kappa^0 \mu_{g^0} dt)$. From the second part of (5.4) one has

$$ (5.5) \qquad \mu = 0 \text{ in } S^*((0,T) \times \omega). $$

In fact, for any $\psi \in \mathcal{C}^\infty((0,T) \times \omega)$ one has $\|\psi \partial_t u^k\|_{L^2(\mathcal{L})} \sim 0$ and thus $\langle \mu, \tau^2 \psi^2 \rangle = 0$. Hence, $\text{supp}(\mu) \cap S^*((0,T) \times \omega) \subset \{\tau = 0\}$. Since $\{\tau = 0\} \cap \text{Char}(p^0) \cap S^*(\mathcal{L}) = \emptyset$ with (3.9) one obtains (5.5).

With the first part of (5.4) one has the following lemma.

**Lemma 5.1.** The measure $\mu$ does not vanish on $S^*(\mathcal{L})$.

A proof is given below.

We now use Proposition 3.7 to obtain a precise description of the measure $\mu$. First, one has $\text{supp}(\mu) \cap S^*((0,T) \times \mathcal{M}) \subset \text{Char}(p^0)$. Furthermore, one has ${}^t\text{H}_{p^0}\mu = 0$ in the sense of distributions on $S^*((0,T) \times \mathcal{M})$. Since $\text{H}_{p^0}$ is a $\mathcal{C}^0$-vector field on the manifold $S^*\mathcal{L}$, Theorem 1.10 implies that $\text{supp}(\mu)$ is a union of maximally extended bicharacteristics in $S^*((0,T) \times \mathcal{M})$.

Under the geometric control condition of Definition 1.8'', any maximal bicharacteristic meets $S^*((0,T) \times \omega)$ where $\mu$ vanishes by (5.5). Thus $\text{supp}(\mu) = \emptyset$ yiedling a contradiction with the result of Lemma 5.1. We thus obtain that (5.1) holds. This concludes the proof of Theorem 1.12. $\square$

*Proof of Lemma 5.1.* Let $T_1 < T_2$ and $\phi \in \mathcal{C}_c^\infty(\mathbb{R})$ nonnegative and equal to 1 on a neighborhood of $[T_1, T_2]$. On $\mathcal{L}$, consider the elliptic operator $Q = -\partial_t^2 - A_{\kappa^0, g^0} + 1$ with symbol $q = \tau^2 + \sum_{p,q} g^{0,p,q}(x) \xi_p \xi_q$. Taking (3.2) and Lemma 3.6 into account one can write

$$ (5.6) \qquad \langle \phi^2 Q u^k, \overline{u^k} \rangle_{H_{\text{comp}}^{-1}(\mathcal{L}), H_{\text{loc}}^1(\mathcal{L})} \underset{k \to +\infty}{\sim} \langle \mu, \phi^2 q \rangle_{S^*\mathcal{L}}. $$

Integrating by parts one obtains

$$ \begin{align*} \langle \phi^2 Q u^k, \overline{u^k} \rangle_{H_{\text{comp}}^{-1}(\mathcal{L}), H_{\text{loc}}^1(\mathcal{L})} & = \int_{\mathcal{L}} \phi(t)^2 (|\partial_t u^k|^2 + g^0(\nabla_{g^0} u^k, \nabla_{g^0} \overline{u^k}) + |u^k|^2) \kappa^0 \mu_{g^0} dt + 2(\phi' \phi \partial_t u^k, u^k)_{L^2(\mathcal{L})} \\ & = \int_{\mathbb{R}} \phi(t)^2 \mathcal{E}_{\kappa^0, g^0}(u^k)(t) dt + 2(\phi' \phi \partial_t u^k, u^k)_{L^2(\mathcal{L})}. \end{align*} $$

Since the energy built on $\kappa^p, g^p$ is preserved by the evolution given by $P_p$, we have by (5.4)

$$ (5.7) \qquad \begin{aligned} \mathcal{E}_{\kappa^0, g^0}(u^k)(t) &= \mathcal{E}_{\kappa^k, g^k}(u^k)(t) + o(1) = \mathcal{E}_{\kappa^k, g^k}(u^k)(0) + o(1) = \mathcal{E}_{\kappa^0, g^0}(u^k)(0) + o(1) \\ &= 1 + o(1) \end{aligned} $$

and since $(\phi' \phi \partial_t u^k, u^k)_{L^2(\mathcal{L})} \to 0$ as $u^k \to 0$ strongly in $L^2_{\text{loc}}(\mathcal{L})$, one obtains

$$ \langle \phi^2 Q u^k, \overline{u^k} \rangle_{H_{\text{comp}}^{-1}(\mathcal{L}), H_{\text{loc}}^1(\mathcal{L})} \underset{k \to +\infty}{\sim} \|\phi\|_{L^2(\mathbb{R})}^2. $$

With (5.6) this proves that $\mu \neq 0$. $\square$

<figure><img src="image_3.png" /></figure>
From (5.4) one has $u^k \to 0$ weakly in $H_{\text{loc}}^1(\mathcal{L})$. With (3.1)-(3.2), we can associate with (a subsequence of) $(u^k)_k$ an $H^1$-microlocal defect measure $\mu$ on $S^*(\mathcal{L})$. Here, the measure is understood with respect to $L^2(\mathcal{L}, \kappa^0 \mu_{g^0} dt)$. From the second part of (5.4) one has

$$ (5.5) \qquad \mu = 0 \text{ in } S^*((0,T) \times \omega). $$

In fact, for any $\psi \in \mathcal{C}^\infty((0,T) \times \omega)$ one has $\|\psi \partial_t u^k\|_{L^2(\mathcal{L})} \sim 0$ and thus $\langle \mu, \tau^2 \psi^2 \rangle = 0$. Hence, $\text{supp}(\mu) \cap S^*((0,T) \times \omega) \subset \{\tau = 0\}$. Since $\{\tau = 0\} \cap \text{Char}(p^0) \cap S^*(\mathcal{L}) = \emptyset$ with (3.9) one obtains (5.5).

With the first part of (5.4) one has the following lemma.

**Lemma 5.1.** The measure $\mu$ does not vanish on $S^*(\mathcal{L})$.

A proof is given below.

We now use Proposition 3.7 to obtain a precise description of the measure $\mu$. First, one has $\text{supp}(\mu) \cap S^*((0,T) \times \mathcal{M}) \subset \text{Char}(p^0)$. Furthermore, one has ${}^t\text{H}_{p^0}\mu = 0$ in the sense of distributions on $S^*((0,T) \times \mathcal{M})$. Since $\text{H}_{p^0}$ is a $\mathcal{C}^0$-vector field on the manifold $S^*\mathcal{L}$, Theorem 1.10 implies that $\text{supp}(\mu)$ is a union of maximally extended bicharacteristics in $S^*((0,T) \times \mathcal{M})$.

Under the geometric control condition of Definition 1.8'', any maximal bicharacteristic meets $S^*((0,T) \times \omega)$ where $\mu$ vanishes by (5.5). Thus $\text{supp}(\mu) = \emptyset$ yielding a contradiction with the result of Lemma 5.1. We thus obtain that (5.1) holds. This concludes the proof of Theorem 1.12. $\square$

*Proof of Lemma 5.1.* Let $T_1 < T_2$ and $\phi \in \mathcal{C}_c^\infty(\mathbb{R})$ nonnegative and equal to 1 on a neighborhood of $[T_1, T_2]$. On $\mathcal{L}$, consider the elliptic operator $Q = -\partial_t^2 - A_{\kappa^0, g^0} + 1$ with symbol $q = \tau^2 + \sum_{p,q} g^{0,p,q}(x) \xi_p \xi_q$. Taking (3.2) and Lemma 3.6 into account one can write

$$ (5.6) \qquad \langle \phi^2 Q u^k, \overline{u^k} \rangle_{H_{\text{comp}}^{-1}(\mathcal{L}), H_{\text{loc}}^1(\mathcal{L})} \underset{k \to +\infty}{\sim} \langle \mu, \phi^2 q \rangle_{S^*\mathcal{L}}. $$

Integrating by parts one obtains

$$ \begin{align*} \langle \phi^2 Q u^k, \overline{u^k} \rangle_{H_{\text{comp}}^{-1}(\mathcal{L}), H_{\text{loc}}^1(\mathcal{L})} & = \int_{\mathcal{L}} \phi(t)^2 (|\partial_t u^k|^2 + g^0(\nabla_{g^0} u^k, \nabla_{g^0} \overline{u^k}) + |u^k|^2) \kappa^0 \mu_{g^0} dt + 2(\phi' \phi \partial_t u^k, u^k)_{L^2(\mathcal{L})} \\ & = \int_{\mathbb{R}} \phi(t)^2 \mathcal{E}_{\kappa^0, g^0}(u^k)(t) dt + 2(\phi' \phi \partial_t u^k, u^k)_{L^2(\mathcal{L})}. \end{align*} $$

Since the energy built on $\kappa^p, g^p$ is preserved by the evolution given by $P_p$, we have by (5.4)

$$ (5.7) \qquad \begin{aligned} \mathcal{E}_{\kappa^0, g^0}(u^k)(t) &= \mathcal{E}_{\kappa^k, g^k}(u^k)(t) + o(1) = \mathcal{E}_{\kappa^k, g^k}(u^k)(0) + o(1) = \mathcal{E}_{\kappa^0, g^0}(u^k)(0) + o(1) \\ &= 1 + o(1) \end{aligned} $$

and since $(\phi' \phi \partial_t u^k, u^k)_{L^2(\mathcal{L})} \to 0$ as $u^k \to 0$ strongly in $L^2_{\text{loc}}(\mathcal{L})$, one obtains

$$ \langle \phi^2 Q u^k, \overline{u^k} \rangle_{H_{\text{comp}}^{-1}(\mathcal{L}), H_{\text{loc}}^1(\mathcal{L})} \underset{k \to +\infty}{\sim} \|\phi\|_{L^2(\mathbb{R})}^2. $$

With (5.6) this proves that $\mu \neq 0$. $\square$

<figure><img src="image_3.png" /></figure>
## 6. LACK OF CONTINUITY OF THE CONTROL OPERATOR WITH RESPECT TO COEFFICIENTS

### 6.1. Proof of Theorems 1.14 and 1.14'.
We prove the result of both theorems, that is, in the case $k \ge 1$. In the case $k = 1$ we are simply required to prove additionally that the geometric control condition of Definition 1.8 is fulfilled for geodesics given by the chosen metric $\tilde{g}$; see Remark 1.17.

Let $\varepsilon > 0$. We set $\tilde{g} = (1 + \varepsilon)g$. Given any neighborhood $\mathcal{U}$ of $(\kappa, g)$ in $\mathcal{X}^k(\mathcal{M})$, for $\varepsilon > 0$ chosen sufficiently small one has $(\kappa, \tilde{g}) \in \mathcal{U}$.

Moreover, observe that for $\varepsilon > 0$ chosen sufficiently small geodesics associated with $\tilde{g}$ can be made arbitrary close to those associated with $g$ uniformly in $t \in [0, T]$. Hence, for such $\varepsilon > 0$ the geometric control condition is fulfilled for geodesics associated with $\tilde{g}$.

Observe that one has

$$ (6.1) \qquad \mathrm{Char}(p_{\kappa,g}) \cap \mathrm{Char}(p_{\kappa,\tilde{g}}) \cap S^*\mathcal{L} = \emptyset. $$

We consider a sequence $(y^{k,0}, y^{k,1}) \to (0, 0)$ weakly in $H^1(\mathcal{M}) \oplus L^2(\mathcal{M})$ such that

$$ \frac{1}{2} (\|y^{k,0}\|_{H^1(\mathcal{M})}^2 + \|y^{k,1}\|_{L^2(\mathcal{M})}^2) = 1. $$

$L^2$- and $H^1$-norms are based on the $\kappa\mu_g dt$ measure on $\mathcal{L}$.

Setting $f_{\kappa,g}^k = H_{\kappa,g}(y^{k,0}, y^{k,1}) \in L^2((0,T) \times \mathcal{M})$ with $H_{\kappa,g}$ defined in (1.7), one obtains a sequence of control functions. According to the HUM method [26], $f_{\kappa,g}^k$ is itself a (weak) solution to the following free wave equation

$$ (6.2) \qquad P_{\kappa,g} f_{\kappa,g}^k = 0, $$

in the energy space $L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})$, that is, $(f_{\kappa,g}^k(0), \partial_t f_{\kappa,g}^k(0)) \in L^2(\mathcal{M}) \times H^{-1}(\mathcal{M})$. Moreover, $(f_{\kappa,g}^k(0), \partial_t f_{\kappa,g}^k(0))$ depend continuously on $(y^{k,0}, y^{k,1})$. The function $f_{\kappa,g}^k$ is thus bounded in $\mathcal{C}^0((T_1, T_2), L^2(\mathcal{M}))$ uniformly with respect to $k$ for any $T_1 < T_2$. Since the map $H_{\kappa,g}$ is continuous $f_{\kappa,g}^k \to 0$ weakly in $L^2_{\mathrm{loc}}(\mathcal{L})$. Up to extraction of a subsequence, it is associated with a $L^2$-microlocal defect measure $\mu_f$. With Proposition 3.7' one has

$$ (6.3) \qquad \mathrm{supp}(\mu_f) \subset \mathrm{Char}(p_{\kappa,g}). $$

We consider the sequences of solutions $(y^k)_k$ and $(\tilde{y}^k)_k$ to

$$ \left\{ \begin{array}{ll} P_{\kappa,g} y^k = \mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k & \text{in } \mathcal{L}, \\ (y^k, \partial_t y^k)|_{t=0} = (y^{k,0}, y^{k,1}) & \text{in } \mathcal{M}, \end{array} \right. \qquad \left\{ \begin{array}{ll} P_{\kappa,\tilde{g}} \tilde{y}^k = \mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k & \text{in } \mathcal{L}, \\ (\tilde{y}^k, \partial_t \tilde{y}^k)|_{t=0} = (y^{k,0}, y^{k,1}) & \text{in } \mathcal{M}. \end{array} \right. $$

Both are bounded and weakly converges to 0 in $H_{\mathrm{loc}}^1(\mathcal{L})$. Up to extraction of subsequences, both are associated with $H^1$-microlocal defect density measures $\mu$ and $\tilde{\mu}$ respectively. Since $\mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k \to 0$ weakly in $L_{\mathrm{loc}}^2(\mathcal{L})$ then $\mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k \to 0$ strongly in $H_{\mathrm{loc}}^{-1}(\mathcal{L})$ and, with Proposition 3.7, one finds $\mathrm{supp}(\tilde{\mu}) \subset \mathrm{Char}(p_{\kappa,\tilde{g}})$. Thus one has

$$ (6.4) \qquad \mathrm{supp}(\tilde{\mu}) \cap \mathrm{supp}(\mu_f) = \emptyset. $$

The sequence $(\partial_t \tilde{y}^k)$ converges to 0 weakly in $L_{\mathrm{loc}}^2(\mathcal{L})$ and can be associated with a $L^2$-microlocal defect density measure whose support is given by $\mathrm{supp}(\tilde{\mu})$.

**Lemma 6.1.** One has $(\mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L}, \kappa\mu_{\tilde{g}}dt)} \to 0$ as $k \to +\infty$.
## 6. LACK OF CONTINUITY OF THE CONTROL OPERATOR WITH RESPECT TO COEFFICIENTS

### 6.1. Proof of Theorems 1.14 and 1.14'.
We prove the result of both theorems, that is, in the case $k \ge 1$. In the case $k = 1$ we are simply required to prove additionally that the geometric control condition of Definition 1.8 is fulfilled for geodesics given by the chosen metric $\tilde{g}$; see Remark 1.17.

Let $\varepsilon > 0$. We set $\tilde{g} = (1 + \varepsilon)g$. Given any neighborhood $\mathcal{U}$ of $(\kappa, g)$ in $\mathcal{X}^k(\mathcal{M})$, for $\varepsilon > 0$ chosen sufficiently small one has $(\kappa, \tilde{g}) \in \mathcal{U}$.

Moreover, observe that for $\varepsilon > 0$ chosen sufficiently small geodesics associated with $\tilde{g}$ can be made arbitrarily close to those associated with $g$ uniformly in $t \in [0, T]$. Hence, for such $\varepsilon > 0$ the geometric control condition is fulfilled for geodesics associated with $\tilde{g}$.

Observe that one has

$$ (6.1) \qquad \mathrm{Char}(p_{\kappa,g}) \cap \mathrm{Char}(p_{\kappa,\tilde{g}}) \cap S^*\mathcal{L} = \emptyset. $$

We consider a sequence $(y^{k,0}, y^{k,1}) \to (0,0)$ weakly in $H^1(\mathcal{M}) \oplus L^2(\mathcal{M})$ such that

$$ \frac{1}{2} (\|y^{k,0}\|_{H^1(\mathcal{M})}^2 + \|y^{k,1}\|_{L^2(\mathcal{M})}^2) = 1. $$

$L^2$- and $H^1$-norms are based on the $\kappa\mu_g dt$ measure on $\mathcal{L}$.

Setting $f_{\kappa,g}^k = H_{\kappa,g}(y^{k,0}, y^{k,1}) \in L^2((0,T) \times \mathcal{M})$ with $H_{\kappa,g}$ defined in (1.7), one obtains a sequence of control functions. According to the HUM method [20], $f_{\kappa,g}^k$ is itself a (weak) solution to the following free wave equation

$$ (6.2) \qquad P_{\kappa,g} f_{\kappa,g}^k = 0, $$

in the energy space $L^2(\mathcal{M}) \oplus H^{-1}(\mathcal{M})$, that is, $(f_{\kappa,g}^k(0), \partial_t f_{\kappa,g}^k(0)) \in L^2(\mathcal{M}) \times H^{-1}(\mathcal{M})$. Moreover, $(f_{\kappa,g}^k(0), \partial_t f_{\kappa,g}^k(0))$ depend continuously on $(y^{k,0}, y^{k,1})$. The function $f_{\kappa,g}^k$ is thus bounded in $\mathcal{C}^0((T_1, T_2), L^2(\mathcal{M}))$ uniformly with respect to $k$ for any $T_1 < T_2$. Since the map $H_{\kappa,g}$ is continuous $f_{\kappa,g}^k \to 0$ weakly in $L_{\mathrm{loc}}^2(\mathcal{L})$. Up to extraction of a subsequence, it is associated with an $L^2$-microlocal defect measure $\mu_f$. With Proposition 3.7' one has

$$ (6.3) \qquad \mathrm{supp}(\mu_f) \subset \mathrm{Char}(p_{\kappa,g}). $$

We consider the sequences of solutions $(y^k)_k$ and $(\tilde{y}^k)_k$ to

$$ \left\{ \begin{array}{ll} P_{\kappa,g} y^k = \mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k & \text{in } \mathcal{L}, \\ (y^k, \partial_t y^k)|_{t=0} = (y^{k,0}, y^{k,1}) & \text{in } \mathcal{M}, \end{array} \right. \qquad \left\{ \begin{array}{ll} P_{\kappa,\tilde{g}} \tilde{y}^k = \mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k & \text{in } \mathcal{L}, \\ (\tilde{y}^k, \partial_t \tilde{y}^k)|_{t=0} = (y^{k,0}, y^{k,1}) & \text{in } \mathcal{M}. \end{array} \right. $$

Both are bounded and weakly converge to 0 in $H_{\mathrm{loc}}^1(\mathcal{L})$. Up to extraction of subsequences, both are associated with $H^1$-microlocal defect density measures $\mu$ and $\tilde{\mu}$ respectively. Since $\mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k \to 0$ weakly in $L_{\mathrm{loc}}^2(\mathcal{L})$, we have $\mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k \to 0$ strongly in $H_{\mathrm{loc}}^{-1}(\mathcal{L})$ and, with Proposition 3.7, one finds $\mathrm{supp}(\tilde{\mu}) \subset \mathrm{Char}(p_{\kappa,\tilde{g}})$. Thus one has

$$ (6.4) \qquad \mathrm{supp}(\tilde{\mu}) \cap \mathrm{supp}(\mu_f) = \emptyset. $$

The sequence $(\partial_t \tilde{y}^k)$ converges to 0 weakly in $L_{\mathrm{loc}}^2(\mathcal{L})$ and can be associated with an $L^2$-microlocal defect density measure whose support is given by $\mathrm{supp}(\tilde{\mu})$.

**Lemma 6.1.** One has $(\mathbf{1}_{(0,T) \times \omega} f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L}, \kappa\mu_{\tilde{g}}dt)} \to 0$ as $k \to +\infty$.
A proof is given below.

Using the density of strong solutions of the wave equation, with integration by parts, one finds the following classical energy estimate

$$ \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(T) - \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(0) = (\mathbf{1}_{(0,T)\times\omega} f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\kappa\mu_{\tilde{g}}dt)}. $$

With Lemma 6.1 one obtains

$$ \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(T) \underset{k \to +\infty}{\sim} \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(0). $$

With the form of $\tilde{g}$ chosen above one has

$$ \mathcal{E}_{\kappa, g}(\tilde{y}^k)(t) = (1 + \mathcal{O}(\varepsilon))\mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(t), $$

uniformly with respect to $t \in [0, T]$. Choosing $\varepsilon > 0$ sufficiently small and $k$ sufficiently large, the first part of Theorem 1.14 follows since $\mathcal{E}_{\kappa,g}(\tilde{y}^k)(0) = 1$.

We use the values of $\varepsilon$ and $k$ chosen above. To prove (1.10), we write $\tilde{y}^k$ in the form $\tilde{y}^k = v_1 + v_2$ where $v_1$ and $v_2$ are solution to

$$ (6.5) \quad \begin{cases} P_{\kappa, \tilde{g}} v_1 = \mathbf{1}_{(0,T)\times\omega} f_{\kappa, \tilde{g}}^k & \text{in } \mathcal{L}, \\ (v_1, \partial_t v_1)|_{t=0} = (y^{k,0}, y^{k,1}) & \text{in } \mathcal{M}, \end{cases} \quad \begin{cases} P_{\kappa, \tilde{g}} v_2 = \mathbf{1}_{(0,T)\times\omega} (f_{\kappa,g}^k - f_{\kappa, \tilde{g}}^k) & \text{in } \mathcal{L}, \\ (v_2, \partial_t v_2)|_{t=0} = (0,0) & \text{in } \mathcal{M}. \end{cases} $$

with $f_{\kappa, \tilde{g}}^k = H_{\kappa, \tilde{g}}(y^{k,0}, y^{k,1})$. A hyperbolic energy estimation for $v_2$ solution to the second equation in (6.5) gives

$$ \mathcal{E}_{\kappa, \tilde{g}}(v_2)(T) \leq C_T \|\mathbf{1}_{(0,T)\times\omega}(f_{\kappa,g}^k - f_{\kappa, \tilde{g}}^k)\|_{L^2(\mathcal{L})}^2. $$

Since one has $(v_1(T), \partial_t v_1(T)) = (0,0)$ because of the definition of $f_{\kappa, \tilde{g}}^k$ one finds

$$ \mathcal{E}_{\kappa, \tilde{g}}(v_2)(T) = \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(T) \geq 1/2, $$

which gives the second result of Theorem 1.14. $\square$

*Proof of Lemma 6.1.* The key point in the proof is the following lemma.

**Lemma 6.2** ([18, Proposition 3.1]). Assume that $u_k$ and $v_k$ are two sequences bounded in $L_{loc}^2$ that converge weakly to zero and are associated with defect measures $\mu$ and $\nu$ respectively. Assume that $\mu \perp \nu$, that is, $\mu$ and $\nu$ are supported on disjoint sets. Then, for any $\psi \in \mathcal{C}_c^0$,

$$ \lim_{k \to +\infty} (\psi u_k, v_k)_{L^2} = 0. $$

To apply this result, we just need to exchange the rough cutoff $\mathbf{1}_{(0,T)\times\omega}$ for a smooth cutoff $\psi(t, x)$. First, note that one has

$$ (\mathbf{1}_{(0,T)\times\omega} f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L}, \kappa\mu_{\tilde{g}}dt)} \underset{k \to +\infty}{\sim} (\mathbf{1}_{(0,T)\times\omega} f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L}, \kappa\mu_g dt)}. $$

We may thus simply consider the $L^2$-norm and inner product associated with $\kappa\mu_g dt$.

Second, let $\delta > 0$. Since $(f_{\kappa,g}^k)_k$ and $(\tilde{y}^k)_k$ are both bounded in $\mathcal{C}^0((0,T), L^2(\mathcal{M}))$ uniformly with respect to $k$, there exists $0 < T_1 < T_2 < T$ and $\mathcal{O} \in \omega$ such that

$$ \iint_K |f_{\kappa,g}^k| |\partial_t \tilde{y}_k| \kappa \mu_g dt \leq \delta. $$
A proof is given below.

Using the density of strong solutions of the wave equation, with integration by parts, one finds the classical energy estimate

$$ \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(T) - \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(0) = (\mathbf{1}_{(0,T)\times\omega} f_{\kappa, g}^k, \partial_t \tilde{y}^k)_{L^2(\kappa\mu_{\tilde{g}}dt)}. $$

With Lemma 6.1 one obtains

$$ \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(T) \underset{k \to +\infty}{\sim} \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(0). $$

With the form of $\tilde{g}$ chosen above one has

$$ \mathcal{E}_{\kappa, g}(\tilde{y}^k)(t) = (1 + \mathcal{O}(\varepsilon))\mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(t), $$

uniformly with respect to $t \in [0, T]$. Choosing $\varepsilon > 0$ sufficiently small and $k$ sufficiently large, the first part of Theorem 1.14 follows since $\mathcal{E}_{\kappa, g}(\tilde{y}^k)(0) = 1$.

We use the values of $\varepsilon$ and $k$ chosen above. To prove (1.10), we write $\tilde{y}^k$ in the form $\tilde{y}^k = v_1 + v_2$ where $v_1$ and $v_2$ are solutions to

$$ (6.5) \quad \left\{ \begin{array}{ll} P_{\kappa, \tilde{g}} v_1 = \mathbf{1}_{(0,T)\times\omega} f_{\kappa, \tilde{g}}^k & \text{in } \mathcal{L}, \\ (v_1, \partial_t v_1)|_{t=0} = (y^{k,0}, y^{k,1}) & \text{in } \mathcal{M}, \end{array} \right. \quad \left\{ \begin{array}{ll} P_{\kappa, \tilde{g}} v_2 = \mathbf{1}_{(0,T)\times\omega} (f_{\kappa, g}^k - f_{\kappa, \tilde{g}}^k) & \text{in } \mathcal{L}, \\ (v_2, \partial_t v_2)|_{t=0} = (0, 0) & \text{in } \mathcal{M}. \end{array} \right. $$

with $f_{\kappa, \tilde{g}}^k = H_{\kappa, \tilde{g}}(y^{k,0}, y^{k,1})$. A hyperbolic energy estimation for the solution $v_2$ to the second equation in (6.5) gives

$$ \mathcal{E}_{\kappa, \tilde{g}}(v_2)(T) \leq C_T \| \mathbf{1}_{(0,T)\times\omega} (f_{\kappa, g}^k - f_{\kappa, \tilde{g}}^k) \|_{L^2(\mathcal{L})}^2. $$

Since one has $(v_1(T), \partial_t v_1(T)) = (0, 0)$ because of the definition of $f_{\kappa, \tilde{g}}^k$ one finds

$$ \mathcal{E}_{\kappa, \tilde{g}}(v_2)(T) = \mathcal{E}_{\kappa, \tilde{g}}(\tilde{y}^k)(T) \geq 1/2, $$

which gives the second result of Theorem 1.14. $\square$

*Proof of Lemma 6.1.* The key point in the proof is the following lemma.

**Lemma 6.2** ([17, Proposition 3.1]). Assume that $u_k$ and $v_k$ are two sequences bounded in $L_{loc}^2$ that converge weakly to zero and are associated with defect measures $\mu$ and $\nu$ respectively. Assume that $\mu \perp \nu$, that is, $\mu$ and $\nu$ are supported on disjoint sets. Then, for any $\psi \in \mathcal{C}_c^0$,

$$ \lim_{k \to +\infty} (\psi u_k, v_k)_{L^2} = 0. $$

To apply this result, we just need to exchange the rough cutoff $\mathbf{1}_{(0,T)\times\omega}$ for a smooth cutoff $\psi(t, x)$. First, note that one has

$$ (\mathbf{1}_{(0,T)\times\omega} f_{\kappa, g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L}, \kappa\mu_{\tilde{g}}dt)} \underset{k \to +\infty}{\sim} (\mathbf{1}_{(0,T)\times\omega} f_{\kappa, g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L}, \kappa\mu_g dt)}. $$

We may thus simply consider the $L^2$-norm and inner product associated with $\kappa\mu_g dt$.

Second, let $\delta > 0$. Since $(f_{\kappa, g}^k)_k$ and $(\tilde{y}^k)_k$ are both bounded in $\mathcal{C}^0((0, T), L^2(\mathcal{M}))$ uniformly with respect to $k$, there exists $0 < T_1 < T_2 < T$ and $\mathcal{O} \in \omega$ such that

$$ \iint_K |f_{\kappa, g}^k| |\partial_t \tilde{y}_k| \kappa \mu_g dt \leq \delta. $$
with $K = ((0, T) \times \omega) \setminus ((T_1, T_2) \times \mathcal{O})$. Let $\psi \in \mathcal{C}_c^\infty((0, T) \times \omega)$ such that $0 \le \psi \le 1$ and equal to 1 in a neighborhood of $[T_1, T_2] \times \mathcal{O}$. One thus has

$$ \begin{aligned} |(\mathbf{1}_{(0,T)\times\omega} f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| &\le |(\psi f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| + |((\mathbf{1}_{(0,T)\times\omega} - \psi)f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| \\ &\le |(\psi f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| + \delta. \end{aligned} $$

With (6.4) and Lemma 6.2, one finds

$$ (6.6) \qquad (\psi f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})} \underset{k \to +\infty}{\to} 0, $$

and the conclusion of the lemma follows. □

## 6.2. Proof of Theorem 1.19.
We consider first the case $\alpha = 1$. As proven in [15] one has $f_{\kappa,g}^{y^0,y^1} \in \mathcal{C}^0([0,T], H^1(\mathcal{M}))$ and the estimate

$$ \|f_{\kappa,g}^{y^0,y^1}\|_{L^\infty(0,T;H^1(\mathcal{M}))} \lesssim \|(y^0,y^1)\|_{H^2(\mathcal{M})\oplus H^1(\mathcal{M})}. $$

With this regularity of the source term in the right-hand-side of the wave equations in (1.8), one finds $y, \tilde{y} \in \mathcal{C}^0([0,T], H^2(\mathcal{M}))$. Computing the difference in (1.8) one writes

$$ (6.7) \qquad P_{\kappa,g}(y - \tilde{y}) = (A_{\kappa,g} - A_{\tilde{\kappa},\tilde{g}})\tilde{y}. $$

A hyperbolic energy estimate yields

$$ \begin{aligned} \mathcal{E}_{\kappa,g}(y - \tilde{y})(T)^{1/2} &\lesssim \|(A_{\kappa,g} - A_{\tilde{\kappa},\tilde{g}})\tilde{y}\|_{L^\infty(0,T;L^2(\mathcal{M}))} \lesssim \|(\kappa,g) - (\tilde{\kappa},\tilde{g})\|_{X^1}\|\tilde{y}\|_{L^\infty(0,T;H^2(\mathcal{M}))} \\ &\lesssim \|(\kappa,g) - (\tilde{\kappa},\tilde{g})\|_{X^1}\|f_{\kappa,g}^{y^0,y^1}\|_{L^\infty(0,T;H^1(\mathcal{M}))} \\ &\lesssim \|(\kappa,g) - (\tilde{\kappa},\tilde{g})\|_{X^1}\|(y^0,y^1)\|_{H^2(\mathcal{M})\oplus H^1(\mathcal{M})}. \end{aligned} $$

In the case $\alpha = 0$, one writes

$$ \begin{aligned} \mathcal{E}_{\kappa,g}(y - \tilde{y})(T)^{1/2} &\lesssim \mathcal{E}_{\kappa,g}(y)(T)^{1/2} + \mathcal{E}_{\kappa,g}(\tilde{y})(T)^{1/2} \lesssim \mathcal{E}_{\kappa,g}(y)(T)^{1/2} + \mathcal{E}_{\tilde{\kappa},\tilde{g}}(\tilde{y})(T)^{1/2} \\ &\lesssim \|(y^0,y^1)\|_{H^1(\mathcal{M})\oplus L^2(\mathcal{M})}. \end{aligned} $$

Finally, the result follows from interpolation between the two cases $\alpha = 0$ and $\alpha = 1$. □

## REFERENCES

[1] L. Ambrosio and G. Crippa. Continuity equations and ODE flows with non-smooth velocity *Proc. R. Soc. Edinb., Sect. A, Math.*, 144(6): 1191–1244, 2014.

[2] C. Bardos, G. Lebeau, and J. Rauch. Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. *Rend. Sem. Mat. Univ. Politec. Torino*, (Special Issue):11–31 (1989), 1988. Nonlinear hyperbolic equations in applied sciences.

[3] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. *SIAM J. Control and Optim.*, 30(5):1024–1065, 1992.

[4] N. Burq. Contrôle de l'équation des ondes dans des ouverts peu réguliers. *Asymptotic Analysis*, 14: 157–191, 1997.

[5] N. Burq, Mesures semi-classiques et mesures de défaut *Séminaire Bourbaki, Astérisque* Vol. 1996/97. No. 245, Exp. No. 826, 4, 167–195, 1997.

[6] N. Burq, B. Dehman and J. Le Rousseau. Measure and continuous vector field at a boundary I: propagation equation and wave observability, 2023. Submitted.

[7] N. Burq, B. Dehman and J. Le Rousseau. Measure and continuous vector field at a boundary II: geodesics and support propagation, 2023. Submitted.

[8] N. Burq and P. Gérard. Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. *C. R. Acad. Sci. Paris Sér. I Math.*, 325(7):749–752, 1997.
with $K = ((0, T) \times \omega) \setminus ((T_1, T_2) \times \mathcal{O})$. Let $\psi \in \mathcal{C}_c^\infty((0, T) \times \omega)$ such that $0 \le \psi \le 1$ and equal to 1 in a neighborhood of $[T_1, T_2] \times \mathcal{O}$. One thus has

$$ \begin{aligned} |(\mathbf{1}_{(0,T)\times\omega} f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| &\le |(\psi f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| + |((\mathbf{1}_{(0,T)\times\omega} - \psi)f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| \\ &\le |(\psi f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})}| + \delta. \end{aligned} $$

With (6.4) and Lemma 6.2, one finds

$$ (6.6) \qquad (\psi f_{\kappa,g}^k, \partial_t \tilde{y}^k)_{L^2(\mathcal{L})} \xrightarrow{k \to +\infty} 0, $$

and the conclusion of the lemma follows. □

6.2. **Proof of Theorem 1.19.** We consider first the case $\alpha = 1$. As proven in [15] one has $f_{\kappa,g}^{y^0,y^1} \in \mathcal{C}^0([0,T], H^1(\mathcal{M}))$ and the estimate

$$ \|f_{\kappa,g}^{y^0,y^1}\|_{L^\infty(0,T;H^1(\mathcal{M}))} \lesssim \|(y^0,y^1)\|_{H^2(\mathcal{M})\oplus H^1(\mathcal{M})}. $$

With this regularity of the source term in the right-hand-side of the wave equations in (1.8), one finds $y, \tilde{y} \in \mathcal{C}^0([0,T], H^2(\mathcal{M}))$. Computing the difference in (1.8) one writes

$$ (6.7) \qquad P_{\kappa,g}(y - \tilde{y}) = (A_{\kappa,g} - A_{\tilde{\kappa},\tilde{g}})\tilde{y}. $$

A hyperbolic energy estimate yields

$$ \begin{aligned} \mathcal{E}_{\kappa,g}(y - \tilde{y})(T)^{1/2} &\lesssim \|(A_{\kappa,g} - A_{\tilde{\kappa},\tilde{g}})\tilde{y}\|_{L^\infty(0,T;L^2(\mathcal{M}))} \lesssim \|(\kappa,g) - (\tilde{\kappa},\tilde{g})\|_{X^1}\|\tilde{y}\|_{L^\infty(0,T;H^2(\mathcal{M}))} \\ &\lesssim \|(\kappa,g) - (\tilde{\kappa},\tilde{g})\|_{X^1}\|f_{\kappa,g}^{y^0,y^1}\|_{L^\infty(0,T;H^1(\mathcal{M}))} \\ &\lesssim \|(\kappa,g) - (\tilde{\kappa},\tilde{g})\|_{X^1}\|(y^0,y^1)\|_{H^2(\mathcal{M})\oplus H^1(\mathcal{M})}. \end{aligned} $$

In the case $\alpha = 0$, one writes

$$ \begin{aligned} \mathcal{E}_{\kappa,g}(y - \tilde{y})(T)^{1/2} &\lesssim \mathcal{E}_{\kappa,g}(y)(T)^{1/2} + \mathcal{E}_{\kappa,g}(\tilde{y})(T)^{1/2} \lesssim \mathcal{E}_{\kappa,g}(y)(T)^{1/2} + \mathcal{E}_{\tilde{\kappa},\tilde{g}}(\tilde{y})(T)^{1/2} \\ &\lesssim \|(y^0,y^1)\|_{H^1(\mathcal{M})\oplus L^2(\mathcal{M})}. \end{aligned} $$

Finally, the result follows from interpolation between the two cases $\alpha = 0$ and $\alpha = 1$. □

## REFERENCES

[1] L. Ambrosio and G. Crippa. Continuity equations and ODE flows with non-smooth velocity *Proc. R. Soc. Edinb., Sect. A, Math.*, 144(6): 1191–1244, 2014.

[2] C. Bardos, G. Lebeau, and J. Rauch. Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. *Rend. Sem. Mat. Univ. Politec. Torino*, (Special Issue):11–31 (1989), 1988. Nonlinear hyperbolic equations in applied sciences.

[3] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. *SIAM J. Control and Optim.*, 30(5):1024–1065, 1992.

[4] N. Burq. Contrôlabilité exacte des ondes dans des ouverts peu réguliers. *Asymptotic Analysis*, 14: 157–191, 1997.

[5] N. Burq, Mesures semi-classiques et mesures de défaut *Séminaire Bourbaki, Astérisque* Vol. 1996/97. No. 245, Exp. No. 826, 4, 167–195, 1997.

[6] N. Burq, B. Dehman and J. Le Rousseau. Measure and continuous vector field at a boundary I: propagation equation and wave observability, preprint 2024, arXiv 2407.02255

[7] N. Burq, B. Dehman and J. Le Rousseau. Measure and continuous vector field at a boundary II: geodesics and support propagation, preprint 2024, arXiv 2407.02259

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[13] F. Colombini, D. Del Santo, F. Fanelli, and G. Métivier. Time-dependent loss of derivatives for hyperbolic operators with non regular coefficients. *Comm. Partial Differential Equations*, 38(10):1791–1817, 2013.

[14] B. Dehman and S. Ervedoza. Observability estimates for the wave equation with rough coefficients. *C. R. Acad. Sci. Paris*, Ser. I 355: 499–514, 2017.

[15] B. Dehman and G. Lebeau. Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time. *SIAM J. Control Optim.*, 48(2):521–550, 2009.

[16] F. Fanelli and E. Zuazua. Weak observability estimates for 1-D wave equations with rough coefficients. *Ann. Inst. H. Poincaré Anal. Non Linéaire*, 32(2):245–277, 2015.

[17] P. Gérard, Microlocal defect measures. *Comm. Partial Differential Equations* 16, no. 11, 1761–1794, 1991.

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[22] J.Rauch and M. Taylor Exponential decay of solutions to hyperbolic equations in bounded domains. 24, 79–86,1974. *Indiana Univ. Math. J.*, 24, 1–68, 1988.

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[24] H. Zhu Stabilization of damped waves on spheres and Zoll surfaces of revolution. *ESAIM Control Optim. Calc. Var.* 24 no. 4, 1759–1788, 2018.

NICOLAS BURQ. LABORATOIRE DE MATHÉMATIQUES D'ORSAY, UNIVERSITÉ PARIS-SACLAY, BÂTIMENT 307, 91405 ORSAY CEDEX & CNRS UMR 8628 & INSTITUT UNIVERSITAIRE DE FRANCE

Email address: nicolas.burq@universite-paris-saclay.fr

BELHASSEN DEHMAN. UNIVERSITÉ DE TUNIS EL MANAR, FACULTÉ DES SCIENCES DE TUNIS, 2092 EL MANAR & ECOLE NATIONALE D'INGÉNIEURS DE TUNIS, ENIT-LAMSIN, B.P. 37, 1002 TUNIS, TUNISIA.

Email address: Belhassen.Dehman@fst.utm.tn

JÉRÔME LE ROUSSEAU. UNIVERSITÉ SORBONNE PARIS NORD, LABORATOIRE ANALYSE, GÉOMÉTRIE ET APPLICATIONS, LAGA, CNRS, UMR 7539, F-93430, VILLETANEUSE, FRANCE.

Email address: jlr@math.univ-paris13.fr
NICOLAS BURQ. LABORATOIRE DE MATHÉMATIQUES D'ORSAY, UNIVERSITÉ PARIS-SACLAY, BÂTIMENT 307,
91405 ORSAY CEDEX & CNRS UMR 8628 & INSTITUT UNIVERSITAIRE DE FRANCE

*E-mail address: nicolas.burq@universite-paris-saclay.fr*

BELHASSEN DEHMAN. UNIVERSITÉ DE TUNIS EL MANAR, FACULTÉ DES SCIENCES DE TUNIS, 2092 EL MANAR & ECOLE NATIONALE D'INGÉNIEURS DE TUNIS, ENIT-LAMSIN, B.P. 37, 1002 TUNIS, TUNISIA.

*E-mail address: Belhassen.Dehman@fst.utm.tn*

JÉRÔME LE ROUSSEAU. UNIVERSITÉ SORBONNE PARIS NORD, LABORATOIRE ANALYSE, GÉOMÉTRIE ET APPLICATIONS, LAGA, CNRS, UMR 7539, F-93430, VILLETANEUSE, FRANCE.

*E-mail address: jlr@math.univ-paris13.fr*